The inherent community structure of hyperbolic networks
TThe inherent community structure of hyperbolic networks
Bianka Kov´acs and Gergely Palla , , Dept. of Biological Physics, E¨otv¨os Lor´and University, H-1117 Budapest, P´azm´any P. stny.1/A, Hungary MTA-ELTE Statistical and Biological Physics Research Group, H-1117 Budapest, P´azm´any P.stny. 1/A, Hungary Health Services Management Training Centre, Semmelweis University, H-1125, K´utv¨olgyi ´ut 2,Budapest, HungaryE-mail: [email protected]
Keywords: hyperbolic networks; communities; PSO model; S / H model Abstract.
A remarkable approach for grasping the relevant statistical features of real networkswith the help of random graphs is offered by hyperbolic models, centred around the idea ofplacing nodes in a low-dimensional hyperbolic space, and connecting node pairs with a probabilitydepending on the hyperbolic distance. It is widely appreciated that these models can generaterandom graphs that are small-world, highly clustered and scale-free at the same time; thus,reproducing the most fundamental common features of real networks. In the present work, wefocus on a less well-known property of the popularity-similarity optimisation (PSO) model andthe S / H model from this model family, namely that the networks generated by these approachesalso contain communities for a wide range of the parameters, which was certainly not an intentionat the design of the models. We extracted the communities from the studied networks usingwell-established community finding methods such as Louvain, Infomap and label propagation.The observed high modularity values indicate that the community structure can become verypronounced under certain conditions. In addition, the modules found by the different algorithmsshow good consistency, implying that these are indeed relevant and apparent structural units. Sincethe appearance of communities is rather common in networks representing real systems as well,this feature of hyperbolic models makes them even more suitable for describing real networks thanthought before.
1. Introduction
Complex network theory is a rapidly expanding interdisciplinary field, strongly interwoven withstatistical physics, concentrating on the interesting non-trivial statistical features of the graphsrepresenting the connections/interactions between entities of complex systems [1–5]. Over thelast two decades, the vast number of studies of real networks have shown that some of thesefeatures seem to be almost universal, such as the small-world property [6, 7], the relativelyhigh clustering coefficient [8], the inhomogeneous degree distribution [9, 10], and the presence ofcommunities [11–13]. Grasping these properties in a unified modelling framework is a non-trivialproblem; however, a very notable approach pointing in this direction is given by hyperbolic networkmodels [14–21], centred around the idea of placing nodes on a hyperbolic plane, and drawing linkswith a probability depending on the metric distance. a r X i v : . [ phy s i c s . s o c - ph ] J a n he inherent community structure of hyperbolic networks S / H model [19, 22]. In the S modelnodes are placed on a circle and are given a hidden variable drawn from a power-law distribution.Here the connection probability depends on the angular distance between the nodes and the hiddenvariables. By converting the hidden variables to radial coordinates in the native disk representationof the hyperbolic plane, we arrive to the equivalent H model, where the connection probabilitiesdepend on the hyperbolic distance between the nodes in a similar way as in the PSO model.In parallel with the success of hyperbolic models, there have also been several studies carriedout focusing on possible hidden metric spaces behind real networks, starting with the examinationof the self-similarity of scale-free networks [19], followed by reports on the hyperbolicity of proteininteraction networks [23, 24], the Internet [25–29], brain networks [30, 31], or the world tradenetwork [32]. Furthermore, a connection between the navigability of networks and hyperbolicspaces was shown [25, 33], the geometric nature of weights [34] and clustering [35, 36] wasdemonstrated, methods for measuring the hyperbolicity of networks were introduced [37, 38],and practical fast algorithms for generating hyperbolic networks were proposed [21]. Hyperbolicnetworks are also closely related to network models based on simplicial complexes [39, 40], wherethe emergent geometry of the generated random graphs was shown to be hyperbolic. In addition,significant achievements were obtained related to the problem of hyperbolic embedding as well[15, 22, 26, 41–44], where the task is to find the most suitable node coordinates in a hyperbolicspace given an input network topology.Returning to hyperbolic network models, in the recent years there have also been effortsdevoted to the development of generative methods capable of producing hyperbolic random graphswith an apparent community structure [16–18,20]. Clusters or communities in hyperbolic networksusually correspond to separated angular regions [45–50]. In accordance with this, in Refs. [17, 18]the uniform angular distribution of the nodes was replaced by a multimodal distribution, wherecommunities arise naturally at the peaks. The appearance of communities in Refs. [16, 20] wasachieved by applying a geometric preferential attachment process, also inducing the formation ofdenser angular regions corresponding to communities.Although the above-mentioned ideas do provide very interesting models with ’built-in’community formation, in the present paper we would like to draw the attention to the lesser- he inherent community structure of hyperbolic networks S / H model as well. This was first shownfor the E-PSO model (a generalisation of the PSO model [15]) in Refs. [45, 47] and for the S / H model in Ref. [46], along with the proposition of the ”Community-Sector hypothesis”, supposingthat most members of a community gather in the same angular sector on the hyperbolic plane.In the closely related study of Ref. [48], the dependence of the modularity (a commonly usedquality score for communities introduced in Ref. [51]) on the temperature parameter T ∈ [0 , . T is low, and gradually decreases when T is increased; however, can still stayabove 0 . T approaches 1. In parallel with these studies, in Ref. [53] the analogy betweenthe hyperbolic embedding and the community structure was studied mostly for real networks andpartly for synthetic graphs generated by the PSO model, where again, the PSO networks wasobserved to have a notable community structure, just like the real networks.Even though the above results already provide important signs related to the presence ofcommunities in hyperbolic networks with homogeneous angular node distribution, here we revisitthis phenomenon in a detailed in-depth study, motivated by the following. First of all, in spite thata modularity value above 0 . . S / H model have basically two parameters: one controlling the decay exponent γ of the scale-free degree distribution and the other controlling the clustering coefficient. Byanalysing the effect of these parameters on the communities, we can gain a clear picture aboutwhat sort of modular structure can be expected when the aim is to generate a hyperbolic randomgraph with specified γ and clustering coefficient values.Along this line, here we generate random graphs according to the PSO and the S / H models ina wide range of parameter settings and examine their community structure with the help of threewell-established community finding algorithms given by the Louvain method [52], the Infomapalgorithm [59] and asynchronous label propagation [60]. The Louvain approach is known to be avery efficient modularity maximising method, while the other two algorithms included do not buildon the modularity and extract the modular structure of the studied networks based on differentconcepts. By applying independent community finding methods, the comparison between thefound modules can reveal whether they correspond to strong, significant structures that can belocated consistently in several different ways or not. In order to gain a quantitative comparisonbetween the communities found by the different methods, we rely on the concept of the adjusted he inherent community structure of hyperbolic networks S / H models usedfor network generation, together with a short summary of the applied community finding methodsand the quality measures used for evaluating the detected community structures. This is followedby the results in section 3, whereas we discuss the implications of our findings in section 4.
2. Methods and preliminaries
We begin the description of the used methods with a brief introduction to hyperbolic networkmodels in section 2.1, including both the PSO model in section 2.1.1 and the S / H model insection 2.1.2. The community related measures and algorithms are summarised in section 2.2,starting with the concept of modularity in section 2.2.1, the angular separation index insection 2.2.2 and the adjusted mutual information in section 2.2.3, followed by the descriptionof the used community finding algorithms in sections 2.2.4–2.2.6. When studying the underlying hyperbolic geometry of complex networks, commonly the nativerepresentation of the two-dimensional hyperbolic space is used [62], in which the hyperbolic planeof constant curvature
K < R in the Euclidean plane (forwhich K = 0). In this representation the Euclidean angles between hyperbolic lines are equal totheir hyperbolic values, and the radial coordinate r of a point (defined as its Euclidean distancefrom the disk centre) is equal to its hyperbolic distance from the disk centre. The hyperbolicdistance between two points is measured along their connecting hyperbolic line, which is eitherthe arc of the Euclidean circle going through the given points and intersecting the disk’s boundaryperpendicularly or – if the disk centre falls on the Euclidean line connecting the two points inquestion – the corresponding diameter of the disk. The hyperbolic distance x between two pointsat polar coordinates ( r, θ ) and ( r (cid:48) , θ (cid:48) ) fulfills the hyperbolic law of cosines written ascosh( ζx ) = cosh( ζr ) cosh( ζr (cid:48) ) − sinh( ζr ) sinh( ζr (cid:48) ) cos(∆ θ ) , (1)where ζ = √− K and ∆ θ = π − | π − | θ − θ (cid:48) || is the angle between the examined points. Accordingto Ref. [62], for 2 · √ e − ζr + e − ζr (cid:48) < ∆ θ and sufficiently large ζr and ζr (cid:48) , the hyperbolic distancecan be approximated as x ≈ r + r (cid:48) + 2 ζ · ln (cid:18) ∆ θ (cid:19) . (2) In the popularity-similarity optimisation model,nodes are placed one by one in the above described native disk representation of the hyperbolicplane and connected with probabilities depending on the hyperbolic distance. The parameters ofthe model can be listed as follows: he inherent community structure of hyperbolic networks • The curvature
K < ζ = √− K >
0. Changing thevalue of ζ corresponds to a simple rescaling of the hyperbolic distances; the usual custom isto set the value of ζ to 1 (i.e. set K to − • The final number of nodes N ∈ Z + in the network. • The number of connections m ∈ Z + established by the newly appearing nodes, correspondingto the half of the average degree (cid:104) k (cid:105) . (The first m nodes of the network form a completegraph). • The popularity fading parameter β ∈ (0 , γ of the power-law decaying tail of the degree distribution isrelated to the popularity fading parameter as γ = 1 + 1 /β . • The temperature T ∈ [0 , i = 1 , , ..., N a new node joins the network as follows:(i) The new node i appears at polar coordinates ( r ii , θ i ), where the radial coordinate r ii is set to ζ ln( i ) and the angular coordinate θ i is sampled from [0 , π ) uniformly at random.(ii) The radial coordinate of each previously (at time j < i ) appeared node j is increased accordingto the formula r ji = βr jj + (1 − β ) r ii in order to simulate popularity fading.(iii) The new node i establishes connections with previously appeared nodes. Only single links arepermitted.(a) If the number of previously appeared nodes is not larger than m , node i connects to allof them.(b) Otherwise, the new node i connects to m of the previously appeared nodes, where theconnection probabilities are determined by the hyperbolic distances between the nodepairs, which can be calculated based on equation (1). If T = 0, node i simply connects tothe m hyperbolically closest nodes, whereas at temperatures T >
0, any previous node j = 1 , , ..., i − i with probability p ( x ij ) = 11 + e ζ T ( x ij − R i ) , (3)where the cutoff distance R i is set to R i = r ii − ζ ln (cid:18) T sin( T π ) · − e − ζ − β ) rii m (1 − β ) (cid:19) if β < ,r ii − ζ ln (cid:16) T sin( T π ) · ζr ii m (cid:17) if β = 1 , (4)ensuring that the expected number of nodes connecting to the new node i at its arrivalis equal to m . S / H model for network generation In the S model [19], first the N number ofnodes are placed on a one-dimensional sphere (i.e. a circle) and each is given a hidden variable κ i ∈ [ κ , ∞ ) , i = 1 , , ..., N . Then, each pair of nodes becomes connected with a probabilitytaking into account both the angular distance and the hidden variables. In the below described he inherent community structure of hyperbolic networks κ i corresponds to the expected degree ¯ k i of node i in the thermodynamic limit.Thus, the connection rule can be phrased in a simple, intuitive way, namely the nodes that arecloser in the hidden metric space underlying the network are more likely to be connected, but in themeantime nodes with higher degree obtain farther-reaching connections as well. In the equivalent H model [22], the hidden variable κ i is converted into the radial coordinate r i in the nativerepresentation of the hyperbolic plane, and the connection probability depends on the hyperbolicdistance between the nodes, that expresses the effect of both the similarity and the node degrees(the popularity).The parameters of these models can be listed as follows: • The total number of nodes N . • The average degree (cid:104) k (cid:105) . • The exponent 2 < γ of the tail of the degree distribution following a power law of the form P ( k ) ∼ k − γ . • The parameter 1 < α , controlling the average clustering coefficient (cid:104) c (cid:105) of the generated network(lim α → (cid:104) c (cid:105) = 0).In the S model, a network of N number of nodes – each of them indexed by i ∈ [1 , N ] – isgenerated through the following steps:(i) For each node i an angular coordinate θ i is sampled from the interval [0 , π ) uniformly atrandom.(ii) For each node i a hidden variable κ i is sampled from the interval [ κ , ∞ ) according to thedistribution ρ ( κ ) = ( γ − · κ − γ κ − γ , where κ = γ − γ − · (cid:104) k (cid:105) .(iii) Each pair of nodes i − j is connected with probability p ij = 11 + (cid:16) N · ∆ θ ij π · µ · κ i · κ j (cid:17) α , (5)where ∆ θ ij = π −| π −| θ i − θ j || is the angular distance between the nodes, and µ = α π (cid:104) k (cid:105) · sin (cid:0) πα (cid:1) .To facilitate a straightforward comparison with the PSO model, we converted the hiddenvariable associated to the nodes into a radial coordinate in the native representation of thehyperbolic plane (at K = − r i = ˆ R − (cid:18) κ i κ (cid:19) , (6)where ˆ R = 2 ln (cid:16) Nµπκ (cid:17) . Note that using this hyperbolic representation (i.e. the H model) theconnection probability (5) becomes p ij = (cid:104) e α · ( x ij − ˆ R ) (cid:105) − , depending on the hyperbolic distance x ij in the same way as the connection probability in equation (3). Communities (also referred to as modules, cohesive groups, clusters) are frequently occurringstructural units in complex networks having usually a larger internal and a smaller external link he inherent community structure of hyperbolic networks
Probably the most well-known quality measure for communities is given bythe modularity [51], comparing the observed density of links between the members of the samecommunity with the expected link density based on some random null model, written in generalas Q = 12 L N (cid:88) i =1 N (cid:88) j =1 [ A ij − P ij ] δ c i ,c j , (7)where N is the number of nodes in the network, A ij denotes an element the adjacency matrix( A ij ≡ A ji = 1 if i is connected to j and otherwise A ij ≡ A ji = 0), P ij gives the connectionprobability between nodes i and j in the null model, L stands for the total number of links in thenetwork, c i is the community to which node i belongs and the Kronecker delta δ c i ,c j ensures thatnon-zero contribution can come only from node pairs in the same community. This quality measurecan take values in the Q ∈ [ − / ,
1] interval, where larger values of Q indicate stronger communitiesthat have a significantly larger internal link density compared to the random expectation.In practice, a natural choice for the null model is provided by the configuration model, wherethe connection probability between nodes i and j can be given with the node degrees k i and k j simply as P ij = k i k j L . This form has also been extended to weighted networks [64], where thenumber of links L is replaced by M = · N (cid:80) i =1 N (cid:80) j =1 w ij (with w ij denoting the link weight betweennodes i and j ), and the node degrees are replaced by the node strengths defined e.g. for node i as s i = (cid:80) N(cid:96) =1 w i(cid:96) , resulting in Q = 12 M · N (cid:88) i =1 N (cid:88) j =1 (cid:104) w ij − s i s j M (cid:105) δ c i ,c j . (8)In order to take into account the hyperbolic distances along the links, we adopted the practicesuggested in Ref. [42], and used in our community analysis a link weight defined as w ij ≡ w ji = 11 + x ij (9)for adjacent nodes i and j , where the hyperbolic distance x ij was calculated based on equation (1)using ζ = 1. In our analysis, we used the code available from Ref. [65] for calculating the weightedmodularity of the detected community structures. he inherent community structure of hyperbolic networks In networks embedded into the hyperbolic disk, communitiesusually occupy well-defined angular regions, having little or no overlap with the region of the othercommunities [45–50]. A quantitative score characterising this tendency is given by the angularseparation index (ASI) [50]. Its basic idea is to compare the number of ”mistakes” in the angulararrangement – i.e. the number o i of nodes belonging to other communities falling between theboundaries of the given module i – summed over all the C communities of the network with thehighest total number of mistakes obtained with the same clustering of the nodes when the angularcoordinates are shuffled at random. Formally, the ASI can be expressed asASI = 1 − C (cid:80) i =1 o i max r (cid:18) C (cid:80) i =1 o ( r ) i (cid:19) , (10)where the maximisation in the denominator is over a fixed number of random shuffles (we used1000 shuffles, i.e. r = 1 , , ..., In the field of community detection, together with the rapidincrease in the number of different algorithms proposed, came the need for well-grounded methodsfor comparing the results of the different approaches. Since e.g. the number of found communitiesand the sizes of the modules can show large variations across the different methods, judging theextent of similarity between two community partitions is non-trivial. Given two sets of communities A and B over the same network, hosting C A and C B number of communities each, a well-known information theoretic similarity measure is offered by the normalised mutual information(NMI) [66, 67], that can be defined based on the mutual informationMI( A, B ) = − C A (cid:88) i =1 C B (cid:88) j =1 N ij N ln (cid:18) N ij NN i N j (cid:19) (11)and the entropies H ( A ) = − C A (cid:88) i =1 N i N ln (cid:18) N i N (cid:19) , H ( B ) = − C B (cid:88) j =1 N j N ln (cid:18) N j N (cid:19) , (12)where N ij denotes the number of shared members of communities i and j , N i and N j stand forthe number of nodes in the individual communities, and the total number of nodes in the networkis given by N . There are several different possibilities for normalising the mutual informationMI( A, B ), e.g. we can divide it by the maximum, the arithmetic mean or the geometric mean ofthe entropies H ( A ) and H ( B ) [61]. In the present study we used the maximum of the entropies;thus, throughout the paperNMI( A, B ) ≡ MI(
A, B )max [ H ( A ) , H ( B )] . (13) he inherent community structure of hyperbolic networks A and B are identical, otherwise its value islower than 1.The concept of adjusted mutual information (AMI) supplements this consistent upper boundwith a consistent zero expectation corresponding to the similarity we can expect by randomchance [61, 68]. To achieve this, the average mutual information of random partitions A (cid:48) and B (cid:48) is subtracted from the nominator, and the average maximum entropy of random partitions issubtracted from the denominator yieldingAMI( A, B ) = MI(
A, B ) − (cid:104) MI( A (cid:48) , B (cid:48) ) (cid:105) rand max [ H ( A ) , H ( B )] − (cid:104) max [ H ( A (cid:48) ) , H ( B (cid:48) )] (cid:105) rand . (14)In our analysis, we used the code available from Ref. [69] for calculating the AMI between thefound community partitions. The asynchronous labelpropagation algorithm [60] simulates the diffusion of labels along the links in the examined network,where the nodes are labelled by the identifier of the community to which they belong, and theselabels are regularly updated based on the labels of the neighbouring nodes using a majority rule.The idea behind this method is that as the labels propagate, the densely connected groups of nodeswill reach a consensus on a unique label. This approach is not aimed at optimising any predefinedmeasure or function.Initially, a unique community label is assigned to each node in the network. Afterwards, thefollowing asynchronous update process is repeated until every node in the network has at least asmany neighbours within its own community as it has in any other communities:(i) Nodes are arranged in a random order.(ii) According to this order, we iterate over the nodes and update their label one by one based ontheir neighbours: each node joins the community to which most of its neighbours currentlybelong. Note that the label of the neighbours may have already been updated in the giveniteration. The neighbouring labels are weighted based on the strength of their link connectedto the current node, and ties in the weighted number of neighbours are broken at random.Due to the random propagation of the labels, in this approach it is possible that distinctcommunities may eventually settle to the same label. Therefore, after the termination of theabove algorithm (where we used the code available from Ref. [70] with link weights calculatedfrom equation (9)), we also applied a breadth-first search on the subgraphs of each individualcommunity to separate the disconnected (i.e. connected only via nodes of different communitiesin the original network) groups of nodes having the same label, as suggested in Ref. [60].
Though finding the exact maximum ofmodularity is a computationally hard problem [71], over the years several heuristic modularityoptimisation methods were proposed [11, 12], and one of the most popular among these is theLouvain algorithm [52]. This approach is capable of unfolding a complete hierarchical communitystructure (where modules can be composed of submodules) within a relatively short time even forextremely large networks. The algorithm is repeating two phases iteratively until the modularitystops improving: he inherent community structure of hyperbolic networks • First, a unique community is assigned to each node of the current network. • This is followed by a repeated iteration over the nodes until the modularity does notincrease any further (or, in our case, until the gain in the modularity does not decreasebelow a threshold of ∆ Q min = 10 − ). – We evaluate the changes in the modularity that would take place if the current node i was transferred to the community of each of its neighbours. – If all the calculated modularity changes are negative, node i stays in its currentcommunity. Otherwise, we carry out the transfer of node i where the improvementin the modularity is the largest.(ii) Moving up to the next organisation level of the system represented by the network betweenthe just found communities: • Each community is considered as a single node. • A self-loop is created for each new node, weighted by twice the sum of the link weightswithin the corresponding community. • The new nodes are connected by links weighted by the sum of the link weights betweenthe corresponding community members on the previous organisation level.In our investigations, we weighted the links in the examined hyperbolic networks according toequation (9) and considered only the final partition (i.e. the top-level community structure, havingthe highest modularity among the different organisation levels) found by the implementation ofthe algorithm available from Ref. [72].
The Infomap algorithm, as suggested byits name, provides an information-theoretic approach for finding communities in networks [59]based on a correspondence between the optimal community structure and the most parsimoniousdescription of an infinitely long random walk trajectory on the network. The random walk canbe considered as a proxy for the flow in the network (travelling passengers, spreading ideas,etc.), making its components interdependent to varying extents. It is intuitive to assume thatcommunities correspond to localized regions of the network where random walkers spend a lot oftime. We can take advantage of this property of communities when aiming for the most compactdescription of a random walker trajectory as follows.In a simple approach, the trajectory is corresponding to the sequence of the visited nodes,each labelled with a unique codeword. However, trajectories can be defined more concisely byusing a map-like description following the principle of geographic maps, where e.g. the same streetnames appear in multiple cities. In a similar manner, after naming the communities, the codewords of the nodes can be recycled among the different communities, and only the members of thesame community have to be given unique names. By limiting the number of different code wordsused to denote the nodes, the length of these code words can be reduced, leading to a considerablesaving in the length of the trajectory description. Naturally, the recycling of the code words alsocomes at a cost, namely one has to indicate when the random walker leaves a given community toenter a new one by specifying the code word of the new community. Nevertheless, if communities he inherent community structure of hyperbolic networks
3. Results
We generated random graphs using the PSO and the S / H models in a wide range of parametersettings, and used the obtained networks as inputs for the community finding methods given bythe asynchronous label propagation, the Louvain and the Infomap algorithms. According to theresults, the hyperbolic random graphs seemed to possess a strong community structure for quitea few combinations of the network generation parameters.As an illustration, in figure 1 we show the partition found by the Louvain algorithm innetworks of size N = 1000 both according to the layout in the native disk representation ofthe two-dimensional hyperbolic space and according to a standard layout in the Euclidean plane.In figures 1(a) and 1(c), the sets of nodes grouped together by Louvain occupy well-definedangular regions in the hyperbolic disk with barely any overlap with the region of the neighbouringcommunities. However, according to figures 1(b) and 1(d), the detected communities are clearlyoutlined even in such layouts which do not build on the hyperbolic origin of the networks.We found that the angular separation of the detected modules exemplified by figures 1(a) and1(c) is quite general in the hyperbolic disk. Using the angular separation index (ASI) described insection 2.2.2, we evaluated quantitatively the angular separation of the modules obtained with theasynchronous label propagation, the Louvain and the Infomap algorithms for a large variety of thenetwork generation parameters. In the case of the PSO model, for both the temperature T andthe popularity fading parameter β we took 10 equidistant data points between 0 and 1 (altogether100 parameter combinations in the T − β parameter plane) and generated 100 networks with eachparameter setting. In the case of the S / H model, to allow a straightforward comparison withthe results seen for the PSO model, instead of the original model parameters α and γ we changedto 1 /α (analogous to the temperature T in the PSO model) and 1 / ( γ −
1) (equivalent to thepopularity fading parameter β in the PSO model). Similarly to the studies of the PSO model, weconsidered a 9 × /α − / ( γ −
1) parameter plane (in the S / H model 2 < γ and α is finite; hence, this model is not defined for β = 1 and T = 0), and generated 100 networksfor each parameter combination. As it is shown in figure 2, for PSO and S / H networks of size N = 10 ,
000 and average degree (cid:104) k (cid:105) = 10 a considerably high ASI can be obtained with all three he inherent community structure of hyperbolic networks
15 10 5 0 5 10 1515105051015 a b
15 10 5 0 5 10 1515105051015 c d
Figure 1. Communities found by the Louvain algorithm in hyperbolic networks. (a) Theobtained communities (colour coded) in a network with N = 1000 number of nodes, generated by thePSO model with parameters m = 5 (corresponding to (cid:104) k (cid:105) = 10), β = 0 . γ = 2 . T = 0 . K = −
1, with the nodesarranged according to their coordinates assigned during the network generation process. The weightedmodularity for the found partition is Q = 0 .
75 and the angular separation index is ASI = 1 .
0. (b) Layoutof the network shown in panel (a) on the Euclidean plane. (c) The detected communities in a networkgenerated by the S / H model with parameters N = 1000, (cid:104) k (cid:105) = 10, γ = 2 .
43 and α = 5 (resulting in anaverage clustering coefficient of 0.71), shown in the native disk representation of the hyperbolic plane ofcurvature K = −
1. The weighted modularity of the shown partition is Q = 0 .
74, the angular separationindex is ASI = 0 . he inherent community structure of hyperbolic networks T − β and α − γ parameter settings.In order to verify that the angularly separated modules detected by the asynchronous labelpropagation, the Louvain and the Infomap algorithms are indeed relevant structural units ofthe networks, we measured the quality of the extracted community partitions by the weightedmodularity Q given in equation (8). In figures 3 and 4 we show the corresponding results fornetworks of size N = 10 ,
000 and expected average degree (cid:104) k (cid:105) = 10, where the weighted modularityis plotted as a function of the model parameters with the help of heat maps. According to figure 3,for a considerably large region in the parameter plane the modularity averaged over 100 networksis larger than 0.65 for the communities found by Infomap (figure 3(c)), larger than 0.75 for thecommunities extracted by asynchronous label propagation (figure 3(a)) and larger than 0.85 forthe communities located by Louvain (figure 3(b)). For Louvain and Infomap, the highest scoresin the modularity are achieved at low T and β parameters, corresponding to networks with a highaverage clustering coefficient and a rather homogeneous degree distribution. The modularity ishigh in this region also for the asynchronous label propagation; however, in this case the highestmodularity values occur for mid-range β values. When β approaches 1, the observed Q seems todecrease for all community finding methods. Nevertheless, Q can still take relatively high valuesat e.g. β = 0 .
6, where the generated network is expected to be scale-free with a degree decayexponent of γ (cid:39) .
67. According to the results displayed in figure 4, the maximum of Q forthe S / H model is in the low-value regime of the 1 /α − / ( γ −
1) parameter plane for all threecommunity finding methods, where the modularity values seem to be higher by a small margincompared to the case of the PSO model, e.g. reaching up to (cid:104) Q (cid:105) = 0 .
99 for the communities foundby Louvain.As mentioned in the Introduction, a large modularity value alone does not always indicatea true modular structure as e.g. both Erd˝os–R´enyi random graphs and Barab´asi–Albert randomgraphs have been shown to display relatively high modularity values under certain circumstances[57, 58]. However, for random graphs generated by the aforementioned two classical models withthe same size and link density as in figures 3 and 4, the modularity can reach up to only about0 .
28, which is significantly smaller compared to the Q values we observed in the studied hyperbolicnetworks. Furthermore, in the present study 2 out of the 3 community finding methods applied arenot based on modularity maximisation, and they still find communities that yield high Q values.In order to examine the significance of the found communities from another aspect, we alsocompared the community partitions obtained with the different methods using the adjusted mutualinformation described in section 2.2.3. The results are displayed in figure 5 with the help of heatmaps, showing the AMI averaged over 100 networks as a function of the model parameters in thestudied parameter planes. According to the figure, the highest similarity values occur between thecommunities found by asynchronous label propagation and Infomap (figures 5(a) and 5(b)). Thesecan reach up to even (cid:104) AMI (cid:105) = 0 .
9, indicating an almost one-to-one correspondence between themodules of the different partitions. On the other hand, the lowest similarity values can be observedfor Louvain and Infomap (figures 5(e) and 5(f)), where the typical value of the AMI is about 0.5.However, this is still in the range of acceptable consistency between the different partitions and isdefinitely way higher than what we would expect e.g. for random partitions. Therefore, based onfigure 5 we can say that in those parameter regions where the communities are characterised by he inherent community structure of hyperbolic networks T β label propagation › A S I fi a /α / ( γ − ) label propagation › A S I fi b T β Louvain › A S I fi c /α / ( γ − ) Louvain › A S I fi d T β Infomap › A S I fi e /α / ( γ − ) Infomap › A S I fi f Figure 2. Angular separation index in the PSO and the S / H models. Theresults for the PSO model are given in the left column (panels (a), (c) and (e)), whereasthe ASI obtained for the S / H model appears in the right column (panels (b), (d) and(f)). The ASI for the communities detected by asynchronous label propagation is givenin the top row (panels (a) and (b)), the ASI regarding the results of Louvain is shown inthe middle row (panels (c) and (d)) and the ASI for the partitions found by Infomap ispresented in the bottom row (panels (e) and (f)). We show the measured ASI (indicatedby the color, averaged over 100 samples) as a function of the model parameters T and β , or 1 /α and 1 / ( γ −
1) for networks of size N = 10 ,
000 and expected average degree (cid:104) k (cid:105) = 10. he inherent community structure of hyperbolic networks T β › Q fi = 0 . label propagation › Q fi a T β › Q fi = 0 . Louvain › Q fi b T β › Q fi = 0 . Infomap › Q fi c T β › Q fi = 0 . best according to Q › Q fi d Figure 3. Modularity in the PSO model . We show the weighted modularity Q (indicated by the color, averaged over 100 samples) as a function of the model parameters T and β for networks of size N = 10 ,
000 and expected average degree (cid:104) k (cid:105) = 10. The panelscorrespond to the results obtained with asynchronous label propagation (a), Louvain (b),Infomap (c), and when the best community partition is taken from the three methodsaccording to Q (d). relatively high modularity scores, the partitions obtained with the different community detectionmethods also show significant consistency with each other. This fact reassures that the moduleswe observe in the studied hyperbolic networks are indeed relevant and apparent structural unitsthat can be detected based on multiple approaches in a consistent way.A basic statistic regarding the revealed community structures is given by the community sizedistribution, which is exemplified by figure 6 for the three examined community finding methods.According to that, the size of the communities found by the asynchronous label propagation followsmore or less a power law for both the PSO model (figure 6(a)) and the S / H model (figure 6(b)).In the regime of small and middle-sized communities, the curve corresponding to Infomap seems tobe close to that; however, towards the larger sizes it decays faster. In contrast, the community sizedistribution yielded by Louvain is quite distinct from the curves obtained with both asynchronouslabel propagation and Infomap, mostly due to a peak at higher community sizes for both thePSO model and the S / H model. This difference between the community size distributions is in he inherent community structure of hyperbolic networks /α / ( γ − ) › Q fi = 0 . label propagation › Q fi a /α / ( γ − ) › Q fi = 0 . Louvain › Q fi b /α / ( γ − ) › Q fi = 0 . Infomap › Q fi c /α / ( γ − ) › Q fi = 0 . best according to Q › Q fi d Figure 4. Modularity in the S / H model . We show the weighted modularity Q (indicated by the color, averaged over 100 samples) as a function of the model parameters1 /α and 1 / ( γ −
1) for networks of size N = 10 ,
000 and expected average degree (cid:104) k (cid:105) = 10.The panels correspond to the results obtained with asynchronous label propagation (a),Louvain (b), Infomap (c), and when the best community partition is taken from the threemethods according to Q (d). correspondence with the results seen for the AMI, where the output of Infomap and asynchronouslabel propagation turned out to be more similar to each other than to Louvain.An interesting question related to the visibly strong community structure obtained with thestudied hyperbolic models is how does it relate to the community structure of such networkswhere the angular distribution of the nodes is non-uniform, as in the case of the hyperbolicnetwork models proposed in Refs. [16–18, 20]. To address this question, here we define a transitionbetween PSO networks with uniform angular node distribution and PSO networks generated withclear angular separation between modules in a similar fashion to the nPSO model introduced inRefs. [17, 18], but with uniform angular distribution within the supposed communities instead ofGaussian distribution. Our related framework begins with generating a PSO network as outlinedin section 2.1.1, and then running a community finding algorithm on the resulting network forlocating its modules (we used Louvain for this purpose). Based on the found communities, we canthen generate PSO networks with equally-sized gaps between the supposed modules by dividing the he inherent community structure of hyperbolic networks T β label propagation Infomap › A M I fi a /α / ( γ − ) label propagation Infomap › A M I fi b T β label propagation Louvain › A M I fi c /α / ( γ − ) label propagation Louvain › A M I fi d T β Louvain Infomap › A M I fi e /α / ( γ − ) Louvain Infomap › A M I fi f Figure 5. Adjusted mutual information between the different communitypartitions.
The results for the PSO model are given in the left column (panels (a),(c) and (e)), whereas the AMI obtained for the S / H model appears in the right column(panels (b), (d) and (f)). The AMI between the communities detected by asynchronouslabel propagation and Infomap is given in the top row (panels (a) and (b)), the AMIregarding the results of asynchronous label propagation and Louvain is shown in themiddle row (panels (c) and (d)) and the AMI between the partitions found by Louvainand Infomap is presented in the bottom row (panels (e) and (f)). We show the measuredAMI (indicated by the color, averaged over 100 samples) as a function of the modelparameters T and β , or 1 /α and 1 / ( γ −
1) for networks of size N = 10 ,
000 and expectedaverage degree (cid:104) k (cid:105) = 10. he inherent community structure of hyperbolic networks -1 community size -7 -6 -5 -4 -3 -2 -1 p r o b a b ili t y d e n s i t y f un c t i o n a PSO model label propagationLouvainInfomap -1 community size -8 -7 -6 -5 -4 -3 -2 -1 p r o b a b ili t y d e n s i t y f un c t i o n b / model label propagationLouvainInfomap Figure 6. Community size distributions. (a) The probability density function ofthe community size in the PSO model according to asynchronous label propagation (bluecircles), Louvain (green squares) and Infomap (orange triangles) based on 100 networks ofsize N = 10 , (cid:104) k (cid:105) = 10, temperature T = 0 . β = 0 .
7. (b) The probability density function of the community sizein the S / H model with the same symbol and colour coding as in panel (a), based on100 networks of size N = 10 , (cid:104) k (cid:105) = 10, 1 /α = 0 . / ( γ −
1) = 0 . he inherent community structure of hyperbolic networks g › Q fi a β = 0 . T = 0 . T = 0 . T = 0 . g › Q fi b β = 0 . T = 0 . T = 0 . T = 0 . g › Q fi c β = 0 . T = 0 . T = 0 . T = 0 .
15 10 5 0 5 10 1515105051015 d total gap width: 0° Q =0 .
15 10 5 0 5 10 1515105051015 e total gap width: 90° Q =0 .
15 10 5 0 5 10 1515105051015 f total gap width: 180° Q =0 .
15 10 5 0 5 10 1515105051015 g total gap width: 270° Q =0 . Figure 7. Transition to non-uniform angular node distribution in the case of the PSO model.
The weighted modularity Q averaged over 100 networks of size N = 1000 and expected average degree (cid:104) k (cid:105) = 10 is shown as a function of the relative gap size g between the modules for β = 0 . β = 0 . β = 0 . N = 1000, (cid:104) k (cid:105) = 10, β = 0 . T = 0 .
2, where the colours indicate the communities found by theLouvain algorithm. [0 , π ) interval into subintervals having a width proportional to the size of the given community,where the aggregated width of the subintervals can be expressed as 2 π (1 − g ) when the aggregatedwidth of the gaps is 2 πg . The number of nodes placed in a given subinterval is equal to thenumber of members of the corresponding community, and the angular coordinate of these nodesis distributed uniformly at random within the subinterval. Otherwise, the network generationprocess is identical to that in the original PSO model.In figure 7 we show results obtained from this framework, where the top panels depict themodularity for communities found by the Louvain algorithm as a function of the relative gap size g , and the bottom panels provide layout examples at different values of g . According to the figure,although Q increases as a function of the relative gap size g as expected, this increase is rather mild,except for large β or T parameters. In other words, the modularity in the uniform PSO model canbe quite close to the Q that we obtain for modules with high angular separation, and therefore,the communities we observe in the uniform PSO model can be viewed also as a meaningful limitfor the modular structure of systems where the angular distribution of the nodes is non-uniform.As a closing of this section, we draw the attention to the supplementary materials, listingfurther results on the communities found in PSO and S / H networks at different system sizesand average degree values (see sections S2–S4). In addition, in the supplementary materials ouranalysis is repeated on an extension of the PSO model known as the E-PSO model [15] (describedin section S1), yielding results that are very similar to what we have detailed here. In section S5we also examine what happens in the PSO model if the angular distribution of the nodes is strictly he inherent community structure of hyperbolic networks S / H networks when setting all the linkweights to 1 instead of using the link weights given in equation 9.
4. Discussion and conclusions
Motivated by interesting signs of modules in hyperbolic networks with homogeneous angular nodedistribution reported in Refs. [45–48, 53], here we revisited the question of community structurein the PSO and S / H models in a detailed in-depth study. Although for both of these modelsthe model construction itself lacks any intentionally built-in community structure, the networksgenerated in these approaches still possess apparently strong communities for a wide range ofthe model parameters, as indicated by the high modularity values measured on the results ofthree independent community finding algorithms, namely asynchronous label propagation, Louvainand Infomap. The significance of the found communities is supported by the fact that only1 out of the 3 applied methods is based on modularity optimisation, and that the comparisonbetween the different partitions yielded reasonably high AMI values, indicating a considerableconsistency between the results. Furthermore, the modularity values that can be achieved inErd˝os–R´enyi random graphs or Barab´asi–Albert scale-free networks at the same average degreeare way lower compared to the Q values we observed in the hyperbolic networks. In addition, theASI (corresponding to a quality measure independent of the modularity) was also very high forthe major part of the parameter space.The parameter plane in which we examined the behaviour of the modularity corresponded tothe ( T, β ) ∈ [0 , × (0 ,
1] plane in the PSO model and the analogous ( α , γ − ) ∈ (0 , × (0 ,
1) planein the S / H model. The intuitive meaning of these parameters can be summarised as follows: theaverage clustering coefficient of the generated networks is regulated by the temperature T and itscounterpart 1 /α (lower values result in higher average clustering coefficients), while the power-lawdecay exponent γ of the degree distribution is controlled by the popularity fading parameter β in the case of the PSO model according to the formula γ = 1 + 1 /β and is itself a parameter ofthe S / H model. According to our results, when changing these parameters, the behaviour of themodularity follows a similar pattern for both hyperbolic models and all three community findingalgorithms, except for the PSO model combined with asynchronous label propagation.Putting aside the above-mentioned exception, for increasing T (or 1 /α ), together with adecrease in the average clustering coefficient the modularity also decreases (which is absolutelynatural), and when β (or equivalently, 1 / ( γ − Q decreases again. However, the dependence of the modularity on the modelparameters is not at all linear, instead we can observe a high, slightly decreasing plateau in theparameter plane with the maximum values in the origin and a relatively narrow belt of lower Q values at the feet of the plateau, placed far from the origin. For the communities found byasynchronous label propagation in the networks generated by the PSO model, the behaviour isslightly different: although Q is high close to the origin, for increasing β it shows a slow increasingtendency, reaching its maximum in the medium β range, followed by a drop for high β values,similarly to the results seen for the other combinations between network generation models and he inherent community structure of hyperbolic networks T → , β → /α → , / ( γ − → S / H model), which yield the largest modularity values in mostof the cases, it is important to note that the corresponding networks are homogeneous in termsof the degree (the degree decay exponent γ is large) and do not resemble scale-free real networks.However, when β is increased (or equivalently, γ is decreased), the modularity decreases only by asmall magnitude for quite some range. E.g., at β = 0 .
6, corresponding to γ (cid:39) .
67, the modularityaveraged over 100 networks can still reach up to (cid:104) Q (cid:105) = 0 .
929 in the PSO model and (cid:104) Q (cid:105) = 0 . S / H model. In other words, when setting the degree decay exponent to moderate valuesoften seen in real systems with the help of β or by directly tuning γ , the networks obtained withthe studied models can still possess a strong community structure if the other parameter ( T or1 /α , controlling the clustering coefficient) is not pushed to extremely high values, meaning thatthe clustering coefficient is not reduced to extremely low values.The regime where Q drops to lower values is on the one hand where β → γ → T → α → γ = 2, it might be a better option to choose the models inRefs. [16–18,20], where the community formation is helped by the non-uniform angular distributionof the nodes. Nevertheless, except the mentioned extreme regimes, the studied ”traditional”hyperbolic models seem to produce a strong enough community structure that can be taken as asimple model for the apparent modular structures often observed in real systems.A remaining interesting question is why do the observed communities arise despite the absenceof any explicit community formation mechanisms built into the construction of the studied models?In short, the same model properties that allow the development of a large clustering coefficientin the generated random graphs on the level of nodes also make the emergence of communitiespossible on a slightly larger scale. Communities are local structures in the sense that membersconnect to each other with a larger link density than to the rest of the system. As mentioned inthe Introduction and as it can be seen in figures 1(a) and 1(c), in hyperbolic networks such unitscorrespond to well-defined angular regions [45–50], with a relatively low number of links acrossthem. Thus, as noted in Ref. [53], the community structure of a network can be also viewed as acoarse version of its layout in the hyperbolic space.In our view, the key element in the formation of communities in the studied models is thatdue to the hyperbolicity of the native disk, for a node newly appearing at the periphery it is mucheasier to connect radially than ”sideways” (i.e. to nodes with similarly large radial coordinate),as indicated by e.g. the distance formula in equation (2). If the angular separation between thepreviously arrived nodes that are placed at smaller radii is large enough, they can become distinctattractive community cores to which the new nodes can connect with only a small interference(cross-links) between the different angular regions. In the PSO model, the condition for a largeenough separation between the inner nodes is that they are pushed outwards (according to thepopularity fading) relatively fast, i.e. β is not large. In parallel, the cutoff in the connectionprobability as a function of the hyperbolic distance must also be sharp enough for localised he inherent community structure of hyperbolic networks T must not be set large either to support community formation. A similarline of arguments holds also for the S / H model. When γ is large, then due to the relativelyrapid decay in the degree distribution, the hidden variables κ i take low values that are mappedto relatively high radial coordinates even for the inner nodes, helping the formation of communitycores. In parallel, a large α parameter in the S / H model has a similar effect to a low T valuein the PSO model, sharpening the cutoff in the connection probability as a function of the metricdistance.We also compared the community structure in the PSO model to the communities in networkswith a non-uniform angular distribution of the nodes in a simple framework, motivated by thefact that the embedding of real networks is often non-homogeneous in terms of the angularcoordinates, similarly to the hyperbolic models with built-in community formation introducedin Refs. [16–18, 20]. Our framework enables a continuous transition between the homogeneousangular node distribution of the PSO model and an angular distribution with empty gaps betweenthe supposed modules, where the angular coordinates are distributed uniformly at random insidethe allowed angular regions. According to our results, the modularity shows only a mild increase asa function of the relative gap size for the majority of the parameter settings. Thus, the modules inthe original PSO model can be quite close in strength to modules occurring in hyperbolic networkswith a non-uniform angular node distribution, and the modular structure of the PSO model as awhole can be treated as a limiting case for those hyperbolic systems where the community structureis accompanied with a non-uniform distribution in the angular coordinates of the nodes.Our findings are also closely related to the community structures observed in networks grownwith the help of simplicial complexes [39, 40] that were also shown to be hyperbolic. Explicitcommunity formation is not built in these models either; however, the simplicial complexes formcomplete subgraphs (cliques), and when aggregating such dense structures, the appearance ofcommunities seems to be more natural compared to the models studied here, where links areintroduced one by one. Nevertheless, the formation of communities observed here deepens furtherthe connection between hyperbolic networks and the models introduced in Refs. [39, 40], that areknown to possess a strong community structure.In conclusion, our study draws the attention to the important but less known fact thatthe PSO and S / H models are capable of generating random graphs that are not just small-world, highly clustered and scale-free, but in addition contain communities as well. Althoughthe advantageous properties of hyperbolic models were already appreciated in the literature, thisrecognition makes them even more suitable for modelling real systems than thought before. Inreal systems, communities provide very important units at an intermediate level of the structuralorganisation of the network. Our detailed study of the behaviour of the community structure as afunction of the model parameters show that modules are formed also in hyperbolic networks in an”automatic” way, simply as a consequence of the connection rules and the nature of the underlyinghyperbolic geometry. These findings add a novel perspective and motivation for the studies andapplications of hyperbolic network models. he inherent community structure of hyperbolic networks Acknowledgments
The authors are grateful for the enlightening discussions with Carlo Vittorio Cannistraci. Theresearch was partially supported by the Hungarian National Research, Development and InnovationOffice (grant no. K 128780, NVKP 16-1-2016-0004) and by the Research Excellence Programmeof the Ministry for Innovation and Technology in Hungary, within the framework of the DigitalBiomarker thematic programme of the Semmelweis University.
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PhysRev E. 2010;81:046106.[59] Rosvall M, Bergstrom CT. Multilevel Compression of Random Walks on Networks Reveals HierarchicalOrganization in Large Integrated Systems. PLOS ONE. 2011 04;6(4):1–10. Available from: https://doi.org/10.1371/journal.pone.0018209 .[60] Raghavan UN, Albert R, Kumara S. Near linear time algorithm to detect community structures in large-scale networks. Phys Rev E. 2007 Sep;76:036106. Available from: https://link.aps.org/doi/10.1103/PhysRevE.76.036106 .[61] Vinh NX, Epps J, Bailey J. Information Theoretic Measures for Clusterings Comparison: Variants, Properties,Normalization and Correction for Chance. Journal of Machine Learning Research. 2010;11(95):2837–2854.Available from: http://jmlr.org/papers/v11/vinh10a.html .[62] Krioukov D, Papadopoulos F, Kitsak M, Vahdat A, Bogu˜n´a M. Hyperbolic geometry of complex networks.Phys Rev E. 2010 Sep;82:036106. Available from: https://link.aps.org/doi/10.1103/PhysRevE.82.036106 .[63] We used the C++ implementation of the S / H model available athttps://github.com/networkgeometry/mercator;. (Accessed: 14/07/2020).[64] Newman MEJ. Analysis of weighted networks. Phys Rev E. 2004 Nov;70:056131. Available from: https://link.aps.org/doi/10.1103/PhysRevE.70.056131 .[65] We calculated the modularity values with the Python function ‘modularity’ available in the ‘net-workx.algorithms.community.quality’ package.;.[66] Danon L, D´ıaz-Guilera A, Duch J, Arenas A. Comparing community structure identification. J Stat Mech.2005;.[67] Lancichinetti A, Fortunato S, Kert´esz J. Detecting the overlapping and hierarchical community structure in he inherent community structure of hyperbolic networks complex networks. New J Phys. 2009;11:033015.[68] McCarthy AD, Matula DW. Normalized mutual information exaggerates community detection performance.In: SIAM Workshop on Network Science 2018; 2018. p. 78–79. Available from: http://cs.jhu.edu/~arya/mccarthy+matula.ns18.pdf .[69] We calculated the adjusted mutual information values with the Python function ‘adjusted mutual info score’available in the ‘sklearn.metrics.cluster’ package.;.[70] We used the Python function ‘asyn lpa communities’, an implementation of the asynchronous label propagationalgorithm available in the ‘networkx.algorithms.community.label propagation’ package.;.[71] Brandes U, Delling D, Gaertler M, Goerke R, Hoefer M, Nikoloski Z, et al. Maximizing Modularity is hard.ArXiv Physics e-prints. 2006 August;.[72] We used the Python implementation of the Louvain algorithm available athttps://github.com/taynaud/python-louvain;. (Accessed: 14/07/2020).[73] We used the Python package for the Infomap algorithm available at https://pypi.org/project/infomap/;.(Accessed: 14/07/2020). UPPORTING INFORMATION
S1. The E-PSO model
In our studies of the community structure of hyperbolic networks, besides the PSO model and the S / H model, we also used the E-PSO model for random graph generation. The results on thecommunities found in this model are presented in sections S2–S4 and section S6, whereas in thepresent section we provide a brief introduction to the model itself.The popularity-similarity optimisation (PSO) model [14] was generalised in the SupplementaryNotes of Ref. [14] by the introduction of so-called internal links: in this generalised PSO model , inaddition to the m number of external links connecting the new node to the already existing nodes,at each time step an L number of further internal connections are created between disconnectedpairs of previously appeared nodes, where the formation of all types of links is determined bythe usual distance-dependent probabilities. It is straightforward to extend this model to negativevalues of the parameter L [44], in which case after connecting the new node to m number of thealready existing nodes, the total number of links between the previously appeared nodes is changedby L <
0, i.e. | L | number of internal links are removed at each time step. To maintain the trendthat mostly hyperbolically close nodes are connected to each other, it is natural to use the sameprobability formula for retaining a link as for the link creation and, accordingly, the complementaryprobability of link creation as the probability of link removal.The E-PSO model [15] is an equivalent of the generalised PSO model using solely externallinks, i.e. all edges of the network are established by the actual new node. Contrary to the originaland the generalised PSO models, in the E-PSO model the number of links created at a time stepis time-dependent. The expected number of links emerging at time i ∈ [1 , N ] can be given as¯ m i = m + ¯ L i (cid:39) m + L · − β )(1 − N − (1 − β ) ) (2 β − (cid:34)(cid:18) Ni (cid:19) β − − (cid:35) (cid:0) − i − (1 − β ) (cid:1) , (S1.1)where ¯ L i is the expected total number of internal links from previous nodes on the node appearingat iteration i at the end of the network generation process in the generalised PSO modelparametrised by the number of nodes N , the number m of external links created at each timestep, the change L in the number of internal links at each time step and the popularity fadingparameter β . Compared to the generalised PSO model, with the E-PSO model one can generateeven large networks relatively fast, since in the generalised PSO model the hyperbolic distanceneeds to be calculated for all node pairs at each time step to update the probabilities of internallink creation or retainment in accordance with the updated node positions, whereas in the E-PSOmodel always only the distances from the newly appeared node need to be determined.Not only the original PSO model, but the generalised PSO model and the analogous E-PSOmodel are capable of producing scale-free networks (with a degree decay exponent γ = 1 + 1 /β )that are highly clustered (in the case of small temperature T ) and have the small-world property. UPPORTING INFORMATION L even it becomes adjustable how the average internal degree of the subgraphs spanningbetween nodes having a degree larger than a certain threshold depends on the degree threshold [44]:for 0 < L the average internal degree increases with the degree threshold and for L = 0(corresponding to the original PSO model) the average internal degree does not depend on thedegree threshold until the degree threshold remains below a value at which the subgraphs becomeextremely small, while for L < (cid:104) k (cid:105) = 2( m + L ) instead of (cid:104) k (cid:105) = 2 m . UPPORTING INFORMATION S2. Quality of the detected communities
We studied the quality of the community structures detected by the asynchronous labelpropagation [60], the Louvain [52] and the Infomap [59] algorithms in PSO [14], E-PSO [15, 44]and S / H [19, 22] networks of various parameter combinations. The isolated nodes emergingin the case of the S / H model and occasionally also in the networks generated by the E-PSOmodel of L < N inputted in these models.Each community detection algorithm was executed once for each network. Figures S2.1–S2.3show the achieved highest weighted modularity averaged over 100 networks of each parametersetting together with the corresponding standard deviations. Figures S2.4–S2.12 present how theperformance of the three different community detection algorithms depends on the parameters ofthe examined network generation models.Figure S2.1 depicts the effect on the achieved highest weighted modularity of changing thenumber of nodes N , the expected average degree (cid:104) k (cid:105) , the popularity fading parameter β and thetemperature T in the PSO model, which corresponds to the E-PSO model with L = 0. For large N and small (cid:104) k (cid:105) most of the nodes have the possibility to create connections only with hyperbolicallyclose nodes, while for small N/ (cid:104) k (cid:105) ratios the nodes are forced more often to connect even withfarther nodes to create all the expected number of links. For this reason, a larger N/ (cid:104) k (cid:105) ratio leadsto connections that are more strongly determined by the hyperbolic distances, and thus to a moreclear separation between the angular regions of the hyperbolic disk, i.e. a community structurewith higher modularity. Besides, by sharpening the cutoff in the connection probability, smallvalues of T also facilitate the localisation of the node-node connections; thus, with the decrease ofthe temperature T the modularity of the detected community structures increases. Furthermore,for smaller values of β the inner nodes drift faster away from each other during the network growth,forming thereby more separated attraction centres for the outer nodes, due to which most of thenetwork nodes can make a more definite choice between the community centres, which leads tocommunities with less external connections, i.e. larger modularity.Figure S2.2 display how the parameters m and L affect the achieved highest weightedmodularity in E-PSO networks. Based on these, not only the expected average degree (cid:104) k (cid:105) =2( m + L ), but even m and L itself has an effect on the community structure of the generatednetworks. According to equation (S1.1), for L < < L the inner nodescreate at their appearance a relatively large number of connections with the previously appearednodes, which – in the absence of enough hyperbolically close candidates – leads to the emergenceof connections between not so close nodes too. Taking into consideration the concept that themore the connections are restricted to hyperbolically close node pairs, the stronger the arisingcommunity structure, we can conclude that for a given expected average degree (cid:104) k (cid:105) the modularitycan be increased by decreasing the parameter L and, at the same time, increasing the parameter UPPORTING INFORMATION m accordingly.According to figure S2.3, the strength of the community structure in S / H networks dependsthe same way on the model parameters as in PSO networks: the achieved highest weightedmodularity is higher for larger number of nodes N , smaller average degree (cid:104) k (cid:105) , larger degreedecay exponent γ (corresponding to smaller popularity fading parameter β in the E-PSO model)and larger α (which is analogous to lower temperature T in the E-PSO model). T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.1. The mean and the standard deviation of the highest weighted modularity Q achievedamong the asynchronous label propagation , the Louvain and the
Infomap algorithms in 100
PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.2. The mean and the standard deviation of the highest weighted modularity Q achievedamong the asynchronous label propagation , the Louvain and the
Infomap algorithms in 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the parameters m and L given in the title of the subplot pairtogether with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 ineach case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S2.3. The mean and the standard deviation of the highest weighted modularity Q achievedamong the asynchronous label propagation , the Louvain and the
Infomap algorithms in 100 S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalentto the popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSOmodel), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. Weused K = − UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.4. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the asynchronous label propagation algorithm in 100 PSO networks of differentparametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of thesubplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.5. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Louvain algorithm in 100
PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T ,with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. Thecurvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.6. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Infomap algorithm in 100
PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T ,with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. Thecurvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.7. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the asynchronous label propagation algorithm in 100 E-PSO networks ofdifferent parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with the parameters m and L given in the title of the subplot pair together with thecorresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. Thecurvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.8. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Louvain algorithm in 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , withthe parameters m and L given in the title of the subplot pair together with the corresponding expected averagedegree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S2.9. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Infomap algorithm in 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , withthe parameters m and L given in the title of the subplot pair together with the corresponding expected averagedegree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S2.10. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the asynchronous label propagation algorithm in 100 S / H networks of differentparametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularityfading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with thenumber of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S2.11. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Louvain algorithm in 100 S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β inthe E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S2.12. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Infomap algorithm in 100 S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β inthe E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION S3. Community size distributions
Here we present the characteristics of the community size distributions obtained with theasynchronous label propagation [60], the Louvain [52] and the Infomap [59] algorithms for thePSO [14], E-PSO [15, 44] and S / H [19, 22] networks of various parameter combinations. Eachcommunity detection algorithm was executed once for each network. The isolated nodes emergingin the case of the S / H model and occasionally also in the networks generated by the E-PSOmodel of L < N inputted in these models.We generated 100 networks with each parametrisation and investigated the sample created byassembling the occurring community sizes from all the 100 networks. Figures S3.1–S3.18 displayhow the mean and the standard deviation of this sample depend on the network generationparameters, as well as some examples for the corresponding community size distributions. FiguresS3.1–S3.6 deal with the E-PSO model with L = 0 (i.e. the PSO model), figures S3.7–S3.12 showthe effect of changing the parameter L , and figures S3.13–S3.18 refers to the S / H model. Thecommunity size distribution is typically bell-shaped according to the Louvain algorithm, whereasrather skewed according to the Infomap and the asynchronous label propagation algorithms. Inthe parameter regime where we observed low Q values, the community finding methods tend tomerge the nodes into large communities of sizes comparable with N . T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S3.1. The mean and the standard deviation of the size of communities detected by the asynchronous label propagation algorithm in 100
PSO networks of different parametrisations.
Eachpair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with thenumber of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvatureof the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION
20 40 60 80 100 community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S3.2. The size distribution of the communities detected by the asynchronous label propagation algorithm in 100
PSO networks of different parametrisations.
The parameters of the network generationare listed in the title for each subplot. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effect of the change in a given network generation parameter: fromtop to bottom, the number of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S3.3. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100
PSO networks of different parametrisations.
Each pair of subplots depicts theeffect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and theexpected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 .
20 40 60 80 100 120 140 160 180 200 community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S3.4. The size distribution of the communities detected by the
Louvain algorithm in 100
PSO networks of different parametrisations.
The parameters of the network generation are listed in the titlefor each subplot. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row ofthe figure demonstrates the effect of the change in a given network generation parameter: from top to bottom, thenumber of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S3.5. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100
PSO networks of different parametrisations.
Each pair of subplots depicts theeffect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and theexpected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S3.6. The size distribution of the communities detected by the
Infomap algorithm in 100
PSO networks of different parametrisations.
The parameters of the network generation are listed in the titlefor each subplot. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row ofthe figure demonstrates the effect of the change in a given network generation parameter: from top to bottom, thenumber of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 3 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 8 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 0 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 5 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = 0 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S3.7. The mean and the standard deviation of the size of communities detected by the asynchronous label propagation algorithm in 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , withthe parameters m and L given in the title of the subplot pair together with the corresponding expected averagedegree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 0 › k fi = 4 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 3 › k fi = 10 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 8 › k fi = 20 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = − › k fi = 4 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 0 › k fi = 10
200 400 600 800 1000 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 5 › k fi = 20 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 4 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 10 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = 0 › k fi = 20 Figure S3.8. The size distribution of the communities detected by the asynchronous label propagation algorithm in 100
E-PSO networks of different parametrisations.
We used ζ = 1, i.e. K = − N was 1000, the popularity fading parameter β was 0.7 andthe temperature T was 0.5 in each case. The parameters m and L are given in the title for each subplot togetherwith the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 3 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 8 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 0 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 5 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = 0 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S3.9. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100
E-PSO networks of different parametrisations.
Each pair of subplots depictsthe effect of changing the popularity fading parameter β and the temperature T , with the parameters m and L given in the title of the subplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). Thenumber of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. weused ζ = 1. UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y m = 2 L = 0 › k fi = 4 community size r e l a t i v e f r e q u e n c y m = 2 L = 3 › k fi = 10
50 100 150 200 250 300 350 community size r e l a t i v e f r e q u e n c y m = 2 L = 8 › k fi = 20 community size r e l a t i v e f r e q u e n c y m = 5 L = − › k fi = 4 community size r e l a t i v e f r e q u e n c y m = 5 L = 0 › k fi = 10 community size r e l a t i v e f r e q u e n c y m = 5 L = 5 › k fi = 20 community size r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 4 community size r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 10 community size r e l a t i v e f r e q u e n c y m = 10 L = 0 › k fi = 20 Figure S3.10. The size distribution of the communities detected by the
Louvain algorithm in 100
E-PSO networks of different parametrisations.
We used ζ = 1, i.e. K = − N was 1000, the popularity fading parameter β was 0.7 and the temperature T was0.5 in each case. The parameters m and L are given in the title for each subplot together with the correspondingexpected average degree (cid:104) k (cid:105) = 2( m + L ). UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 3 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 8 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 0 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 5 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = 0 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S3.11. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100
E-PSO networks of different parametrisations.
Each pair of subplots depictsthe effect of changing the popularity fading parameter β and the temperature T , with the parameters m and L given in the title of the subplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). Thenumber of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. weused ζ = 1. UPPORTING INFORMATION community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 0 › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 3 › k fi = 10 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 8 › k fi = 20 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = − › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 0 › k fi = 10 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 5 › k fi = 20 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 10 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = 0 › k fi = 20 Figure S3.12. The size distribution of the communities detected by the
Infomap algorithm in 100
E-PSO networks of different parametrisations.
We used ζ = 1, i.e. K = − N was 1000, the popularity fading parameter β was 0.7 and the temperature T was0.5 in each case. The parameters m and L are given in the title for each subplot together with the correspondingexpected average degree (cid:104) k (cid:105) = 2( m + L ). UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s Figure S3.13. The mean and the standard deviation of the size of communities detected by the asynchronous label propagation algorithm in 100 S / H networks of different parametrisations. Eachpair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in theE-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N andthe expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . Figure S3.14. The size distribution of the communities detected by the asynchronous labelpropagation algorithm in 100 S / H networks of different parametrisations. The parameters of thenetwork generation are listed in the title for each subplot. We used K = − N , the expected average degree (cid:104) k (cid:105) , 1 / ( γ −
1) (equivalent tothe popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSOmodel). UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s Figure S3.15. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100 S / H networks of different parametrisations. Each pair of subplots depictsthe effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected averagedegree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . Figure S3.16. The size distribution of the communities detected by the
Louvain algorithm in 100 S / H networks of different parametrisations. The parameters of the network generation are listed in the titlefor each subplot. We used K = − N , the expected average degree (cid:104) k (cid:105) , 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSOmodel) and 1 /α (analogous to the temperature T in the E-PSO model). UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s Figure S3.17. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100 S / H networks of different parametrisations. Each pair of subplots depictsthe effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected averagedegree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . Figure S3.18. The size distribution of the communities detected by the
Infomap algorithm in 100 S / H networks of different parametrisations. The parameters of the network generation are listed in the titlefor each subplot. We used K = − N , the expected average degree (cid:104) k (cid:105) , 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSOmodel) and 1 /α (analogous to the temperature T in the E-PSO model). UPPORTING INFORMATION S4. Adjusted mutual information of the community structures found by differentcommunity detection algorithms
We compared the community structures found by the asynchronous label propagation [60], theLouvain [52] and the Infomap [59] algorithms in the PSO [14], E-PSO [15, 44] and S / H [19, 22]networks of various parameter combinations. Each community detection algorithm was executedonce for each network. The isolated nodes emerging in the case of the S / H model and occasionallyalso in the networks generated by the E-PSO model of L < N inputted in these models. We generated 100 networks with each parametersetting and calculated the adjusted mutual information (AMI) [61, 68] of the resulted 3 partitionsfor each network. Figures S4.1–S4.3 display the average and the standard deviation of the AMIbetween the community structures obtained with asynchronous label propagation and Louvain,figures S4.4–S4.6 compare the result of asynchronous label propagation with the result of Infomap,while the consistency between the community structures detected by Louvain and Infomap isexamined in figures S4.7–S4.9.According to these figures, asynchronous label propagation and Infomap produce the mostsimilar partitions, while the most different is the result of Louvain and Infomap. In most of thecases, the AMI depends similarly on the network generation parameters as the weighted modularity Q . For the high modularity regions of the parameter space (large number of nodes N , smallaverage degree (cid:104) k (cid:105) , small popularity fading parameter β or large degree decay exponent γ andlow temperature T or large α ) the AMI is relatively large for each pair of community detectionmethods, indicating in these parameter regions the emergence of really apparent communities thatare detectable for all the 3 investigated algorithms alike. UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S4.1. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithmsin 100
PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changingthe popularity fading parameter β and the temperature T , with the number of nodes N and the expected averagedegree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β m = 2 L = 0 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 3 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 8 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 0 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 5 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = 0 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S4.2. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithmsin 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changingthe popularity fading parameter β and the temperature T , with the parameters m and L given in the title of thesubplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I Figure S4.3. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithms in100 S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ − β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of thesubplot pair. We used K = − T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S4.4. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100
PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changingthe popularity fading parameter β and the temperature T , with the number of nodes N and the expected averagedegree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 3 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 8 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 0 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 5 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = 0 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S4.5. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changingthe popularity fading parameter β and the temperature T , with the parameters m and L given in the title of thesubplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I Figure S4.6. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100 S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous to thetemperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given inthe title of the subplot pair. We used K = − UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S4.7. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the
Louvain and the
Infomap algorithms in 100
PSO networks ofdifferent parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the titleof the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β m = 2 L = 0 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 3 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 8 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 0 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 5 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = 0 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S4.8. The mean and the standard deviation of the adjusted mutual information of thetwo community structures detected by the
Louvain and the
Infomap algorithms in 100
E-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the parameters m and L given in the title of the subplot pairtogether with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 ineach case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I Figure S4.9. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the
Louvain and the
Infomap algorithms in 100 S / H networksof different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent tothe popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSOmodel), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. Weused K = − UPPORTING INFORMATION S5. Equidistant angular node arrangement
In order to exclude the possibility that the emergence of communities is a result of theinhomogeneities in the angular node arrangement arising inevitably when a finite number of angularcoordinates is sampled from a uniform distribution in [0 , π ), we modified the PSO model [14] touse strictly equidistant angular arrangement, i.e. instead of sampling the angular coordinatesuniformly randomly, assign the coordinates θ i = ( i − · πN , i = 1 , , ..., N to the network nodes ina randomly chosen order. According to figures S5.1–S5.4, the networks generated by this modifiedPSO model possess a community structure with similarly high weighted modularity as the usualPSO networks (see figures S2.1 and S2.4-S2.6). Hence, we can conclude that the emergence ofcommunities is not just an effect of the finite network size, but the inherent property of the studiedhyperbolic network models. Figures S5.5–S5.10 show similar results for the networks generatedby the modified PSO model with regard to the community size distributions as figures S3.1-S3.6in the case of the original PSO model. Lastly, the adjusted mutual information (AMI) [61, 68]between the community structures detected by asynchronous label propagation [60], Louvain [52]and Infomap [59] behaves the same way for the modified PSO model (figures S5.11–S5.13) as forthe original PSO model (figures S4.1, S4.4 and S4.7). T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S5.1. The mean and the standard deviation of the highest weighted modularity Q achievedamong the asynchronous label propagation , the Louvain and the
Infomap algorithms in 100
PSO networks of different parametrisations with strictly equidistant angular arrangement . Each pair ofsubplots depicts the effect of changing the popularity fading parameter β and the temperature T , with the numberof nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of thehyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S5.2. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the asynchronous label propagation algorithm in 100 PSO networks of differentparametrisations with strictly equidistant angular arrangement . Each pair of subplots depicts the effect ofchanging the popularity fading parameter β and the temperature T , with the number of nodes N and the expectedaverage degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was alwaysset to −
1, i.e. we used ζ = 1. T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S5.3. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Louvain algorithm in 100
PSO networks of different parametrisations withstrictly equidistant angular arrangement . Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S5.4. The mean and the standard deviation of the weighted modularity Q of the communitystructure detected by the Infomap algorithm in 100
PSO networks of different parametrisations withstrictly equidistant angular arrangement . Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S5.5. The mean and the standard deviation of the size of communities detected by the asynchronous label propagation algorithm in 100
PSO networks of different parametrisations withstrictly equidistant angular arrangement . Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION
40 50 60 70 80 90 100 110 community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S5.6. The size distribution of the communities detected by the asynchronous label propagation algorithm in 100
PSO networks of different parametrisations with strictly equidistant angulararrangement . The parameters of the network generation are listed in the title for each subplot. The curvature ofthe hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effectof the change in a given network generation parameter: from top to bottom, the number of nodes N , the expectedaverage degree (cid:104) k (cid:105) = 2 m , the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S5.7. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100
PSO networks of different parametrisations with strictly equidistant angulararrangement . Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of thesubplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 .
20 40 60 80 100 120 140 160 180 200 community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S5.8. The size distribution of the communities detected by the
Louvain algorithm in 100
PSO networks of different parametrisations with strictly equidistant angular arrangement . The parametersof the network generation are listed in the title for each subplot. The curvature of the hyperbolic plane K wasalways set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effect of the change in a given networkgeneration parameter: from top to bottom, the number of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , thepopularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S5.9. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100
PSO networks of different parametrisations with strictly equidistantangular arrangement . Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the titleof the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S5.10. The size distribution of the communities detected by the
Infomap algorithm in100
PSO networks of different parametrisations with strictly equidistant angular arrangement . Theparameters of the network generation are listed in the title for each subplot. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effect of the change in a givennetwork generation parameter: from top to bottom, the number of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m ,the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S5.11. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithmsin 100
PSO networks of different parametrisations with strictly equidistant angular arrangement . Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T ,with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. Thecurvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S5.12. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100
PSO networks of different parametrisations with strictly equidistant angular arrangement . Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T ,with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. Thecurvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S5.13. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the
Louvain and the
Infomap algorithms in 100
PSO networks ofdifferent parametrisations with strictly equidistant angular arrangement . Each pair of subplots depictsthe effect of changing the popularity fading parameter β and the temperature T , with the number of nodes N andthe expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION S6. Community detection on unweighted hyperbolic networks
We studied the quality of the community structures detected by the asynchronous labelpropagation [60], the Louvain [52] and the Infomap [59] algorithms in PSO [14], E-PSO [15, 44]and S / H [19, 22] networks of various parameter combinations. The isolated nodes emergingin the case of the S / H model and occasionally also in the networks generated by the E-PSOmodel of L < N inputted in these models.Each community detection algorithm was executed once for each network. This section presentsthe results obtained in the case of setting each link weight in the previously examined syntheticnetworks to 1, independently of the hyperbolic distance between the connected nodes. Similarlyto figures S2.4–S2.12, figures S6.1–S6.9 show how the unweighted modularity [51] achieved by thethree different community detection algorithms depends on the network generation parameters.As in section S3, figures S6.10–S6.27 display how the mean and the standard deviation of thecommunity sizes depend on the network generation parameters, as well as some examples for thecorresponding community size distributions. Figures S6.28–S6.36 are analogous to the figures ofsection S4, showing the similarity between the community structures detected by the differentmethods in the unweighted networks by means of the adjusted mutual information (AMI) [61, 68].Lastly, figure 2 of the main article depicting the angular separation index (ASI) of the detectedcommunities [50] is repeated for the unweighted case in figure S6.37. T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S6.1. The mean and the standard deviation of the unweighted modularity Q of the communitystructure detected by the asynchronous label propagation algorithm in 100 unweighted PSO networksof different parametrisations. Each pair of subplots depicts the effect of changing the popularity fadingparameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S6.2. The mean and the standard deviation of the unweighted modularity Q of thecommunity structure detected by the Louvain algorithm in 100 unweighted PSO networks of differentparametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of thesubplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β N = 100 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 100 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 1000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β N = 10000 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S6.3. The mean and the standard deviation of the unweighted modularity Q of thecommunity structure detected by the Infomap algorithm in 100 unweighted PSO networks ofdifferent parametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the titleof the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S6.4. The mean and the standard deviation of the unweighted modularity Q of the communitystructure detected by the asynchronous label propagation algorithm in 100 unweighted E-PSO networks of different parametrisations. Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the parameters m and L given in the title of the subplot pairtogether with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 ineach case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S6.5. The mean and the standard deviation of the unweighted modularity Q of the communitystructure detected by the Louvain algorithm in 100 unweighted E-PSO networks of differentparametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the parameters m and L given in the title of the subplot pair together with the correspondingexpected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of thehyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 3 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 2 L = 8 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 0 › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 5 L = 5 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 4 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = − › k fi = 10 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q T β m = 10 L = 0 › k fi = 20 a v e r a g e o f Q T β s t a n d a r d d e v i a t i o n o f Q Figure S6.6. The mean and the standard deviation of the unweighted modularity Q of the communitystructure detected by the Infomap algorithm in 100 unweighted E-PSO networks of differentparametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the parameters m and L given in the title of the subplot pair together with the correspondingexpected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of thehyperbolic plane K was always set to −
1, i.e. we used ζ = 1. /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S6.7. The mean and the standard deviation of the unweighted modularity Q of thecommunity structure detected by the asynchronous label propagation algorithm in 100 unweighted S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ − β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of thesubplot pair. We used K = − UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S6.8. The mean and the standard deviation of the unweighted modularity Q of thecommunity structure detected by the Louvain algorithm in 100 unweighted S / H networks ofdifferent parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to thepopularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model),with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f Q /α / ( γ − ) s t a n d a r d d e v i a t i o n o f Q Figure S6.9. The mean and the standard deviation of the unweighted modularity Q of thecommunity structure detected by the Infomap algorithm in 100 unweighted S / H networks ofdifferent parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to thepopularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model),with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S6.10. The mean and the standard deviation of the size of communities detected bythe asynchronous label propagation algorithm in 100 unweighted PSO networks of differentparametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of thesubplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S6.11. The size distribution of the communities detected by the asynchronous labelpropagation algorithm in 100 unweighted PSO networks of different parametrisations.
The parametersof the network generation are listed in the title for each subplot. The curvature of the hyperbolic plane K wasalways set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effect of the change in a given networkgeneration parameter: from top to bottom, the number of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , thepopularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S6.12. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100 unweighted PSO networks of different parametrisations.
Each pair of subplotsdepicts the effect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolicplane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S6.13. The size distribution of the communities detected by the
Louvain algorithm in 100 unweighted PSO networks of different parametrisations.
The parameters of the network generation arelisted in the title for each subplot. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effect of the change in a given network generation parameter: fromtop to bottom, the number of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S6.14. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100 unweighted PSO networks of different parametrisations.
Each pair of subplotsdepicts the effect of changing the popularity fading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolicplane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 β = 0 . T = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 β = 0 . T = 0 . Figure S6.15. The size distribution of the communities detected by the
Infomap algorithm in 100 unweighted PSO networks of different parametrisations.
The parameters of the network generation arelisted in the title for each subplot. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. Each row of the figure demonstrates the effect of the change in a given network generation parameter: fromtop to bottom, the number of nodes N , the expected average degree (cid:104) k (cid:105) = 2 m , the popularity fading parameter β and the temperature T . UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 3 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 8 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 0 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 5 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = 0 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S6.16. The mean and the standard deviation of the size of communities detected bythe asynchronous label propagation algorithm in 100 unweighted E-PSO networks of differentparametrisations.
Each pair of subplots depicts the effect of changing the popularity fading parameter β and thetemperature T , with the parameters m and L given in the title of the subplot pair together with the correspondingexpected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of thehyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 0 › k fi = 4 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 3 › k fi = 10 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 8 › k fi = 20 community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = − › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 0 › k fi = 10 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 5 › k fi = 20 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 10 community size -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = 0 › k fi = 20 Figure S6.17. The size distribution of the communities detected by the asynchronous labelpropagation algorithm in 100 unweighted E-PSO networks of different parametrisations.
We used ζ = 1, i.e. K = − N was 1000, the popularityfading parameter β was 0.7 and the temperature T was 0.5 in each case. The parameters m and L are given in thetitle for each subplot together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 3 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 8 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 0 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 5 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = 0 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S6.18. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100 unweighted E-PSO networks of different parametrisations.
Each pairof subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with theparameters m and L given in the title of the subplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K wasalways set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y m = 2 L = 0 › k fi = 4 community size r e l a t i v e f r e q u e n c y m = 2 L = 3 › k fi = 10 community size r e l a t i v e f r e q u e n c y m = 2 L = 8 › k fi = 20 community size r e l a t i v e f r e q u e n c y m = 5 L = − › k fi = 4 community size r e l a t i v e f r e q u e n c y m = 5 L = 0 › k fi = 10 community size r e l a t i v e f r e q u e n c y m = 5 L = 5 › k fi = 20 community size r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 4 community size r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 10 community size r e l a t i v e f r e q u e n c y m = 10 L = 0 › k fi = 20 Figure S6.19. The size distribution of the communities detected by the
Louvain algorithm in 100 unweighted E-PSO networks of different parametrisations.
We used ζ = 1, i.e. K = − N was 1000, the popularity fading parameter β was 0.7 and thetemperature T was 0.5 in each case. The parameters m and L are given in the title for each subplot together withthe corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 3 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 2 L = 8 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 0 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 5 L = 5 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = − › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s T β m = 10 L = 0 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s T β S D o f c o mm un i t y s i z e s Figure S6.20. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100 unweighted E-PSO networks of different parametrisations.
Each pairof subplots depicts the effect of changing the popularity fading parameter β and the temperature T , with theparameters m and L given in the title of the subplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K wasalways set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 0 › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 3 › k fi = 10 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 2 L = 8 › k fi = 20 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = − › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 0 › k fi = 10 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 5 L = 5 › k fi = 20 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 4 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = − › k fi = 10 community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y m = 10 L = 0 › k fi = 20 Figure S6.21. The size distribution of the communities detected by the
Infomap algorithm in 100 unweighted E-PSO networks of different parametrisations.
We used ζ = 1, i.e. K = − N was 1000, the popularity fading parameter β was 0.7 and thetemperature T was 0.5 in each case. The parameters m and L are given in the title for each subplot together withthe corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s Figure S6.22. The mean and the standard deviation of the size of communities detected bythe asynchronous label propagation algorithm in 100 unweighted S / H networks of differentparametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularityfading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with thenumber of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . Figure S6.23. The size distribution of the communities detected by the asynchronous labelpropagation algorithm in 100 unweighted S / H networks of different parametrisations. The parametersof the network generation are listed in the title for each subplot. We used K = − N , the expected average degree (cid:104) k (cid:105) , 1 / ( γ −
1) (equivalent tothe popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSOmodel). UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s Figure S6.24. The mean and the standard deviation of the size of communities detected by the
Louvain algorithm in 100 unweighted S / H networks of different parametrisations. Each pair ofsubplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSOmodel) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and theexpected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION community size r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . Figure S6.25. The size distribution of the communities detected by the
Louvain algorithm in 100 unweighted S / H networks of different parametrisations. The parameters of the network generation arelisted in the title for each subplot. We used K = − N , the expected average degree (cid:104) k (cid:105) , 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model). UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f c o mm un i t y s i z e s /α / ( γ − ) S D o f c o mm un i t y s i z e s Figure S6.26. The mean and the standard deviation of the size of communities detected by the
Infomap algorithm in 100 unweighted S / H networks of different parametrisations. Each pair ofsubplots depicts the effect of changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSOmodel) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and theexpected average degree (cid:104) k (cid:105) given in the title of the subplot pair. We used K = − UPPORTING INFORMATION community size -3 -2 -1 r e l a t i v e f r e q u e n c y N = 100 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 10000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -5 -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 4 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 20 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . community size -4 -3 -2 -1 r e l a t i v e f r e q u e n c y N = 1000 › k fi = 10 / ( γ −
1) = 0 . /α = 0 . Figure S6.27. The size distribution of the communities detected by the
Infomap algorithm in 100 unweighted S / H networks of different parametrisations. The parameters of the network generation arelisted in the title for each subplot. We used K = − N , the expected average degree (cid:104) k (cid:105) , 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model). UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S6.28. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithmsin 100 unweighted PSO networks of different parametrisations.
Each pair of subplots depicts the effect ofchanging the popularity fading parameter β and the temperature T , with the number of nodes N and the expectedaverage degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was alwaysset to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 3 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 8 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 0 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 5 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = 0 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S6.29. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithmsin 100 unweighted E-PSO networks of different parametrisations.
Each pair of subplots depicts the effectof changing the popularity fading parameter β and the temperature T , with the parameters m and L given in thetitle of the subplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number ofnodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I Figure S6.30. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Louvain algorithmsin 100 unweighted S / H networks of different parametrisations. Each pair of subplots depicts the effectof changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous tothe temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) givenin the title of the subplot pair. We used K = − UPPORTING INFORMATION T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S6.31. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100 unweighted PSO networks of different parametrisations.
Each pair of subplots depicts the effect ofchanging the popularity fading parameter β and the temperature T , with the number of nodes N and the expectedaverage degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was alwaysset to −
1, i.e. we used ζ = 1. T β m = 2 L = 0 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 3 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 8 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 0 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 5 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = 0 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S6.32. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100 unweighted E-PSO networks of different parametrisations.
Each pair of subplots depicts the effectof changing the popularity fading parameter β and the temperature T , with the parameters m and L given in thetitle of the subplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number ofnodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I Figure S6.33. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the asynchronous label propagation and the
Infomap algorithmsin 100 unweighted S / H networks of different parametrisations. Each pair of subplots depicts the effectof changing 1 / ( γ −
1) (equivalent to the popularity fading parameter β in the E-PSO model) and 1 /α (analogous tothe temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) givenin the title of the subplot pair. We used K = − T β N = 100 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 100 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 1000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β N = 10000 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S6.34. The mean and the standard deviation of the adjusted mutual information of the twocommunity structures detected by the
Louvain and the
Infomap algorithms in 100 unweighted PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing the popularityfading parameter β and the temperature T , with the number of nodes N and the expected average degree (cid:104) k (cid:105) = 2 m given in the title of the subplot pair. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. UPPORTING INFORMATION T β m = 2 L = 0 › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 3 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 2 L = 8 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 0 › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 5 L = 5 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 4 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = − › k fi = 10 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I T β m = 10 L = 0 › k fi = 20 a v e r a g e o f A M I T β s t a n d a r d d e v i a t i o n o f A M I Figure S6.35. The mean and the standard deviation of the adjusted mutual information of thetwo community structures detected by the
Louvain and the
Infomap algorithms in 100 unweightedE-PSO networks of different parametrisations.
Each pair of subplots depicts the effect of changing thepopularity fading parameter β and the temperature T , with the parameters m and L given in the title of thesubplot pair together with the corresponding expected average degree (cid:104) k (cid:105) = 2( m + L ). The number of nodes N was 1000 in each case. The curvature of the hyperbolic plane K was always set to −
1, i.e. we used ζ = 1. /α / ( γ − ) N = 100 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 100 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 1000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 4 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 10 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I /α / ( γ − ) N = 10000 › k fi = 20 a v e r a g e o f A M I /α / ( γ − ) s t a n d a r d d e v i a t i o n o f A M I Figure S6.36. The mean and the standard deviation of the adjusted mutual information of thetwo community structures detected by the
Louvain and the
Infomap algorithms in 100 unweighted S / H networks of different parametrisations. Each pair of subplots depicts the effect of changing 1 / ( γ − β in the E-PSO model) and 1 /α (analogous to the temperature T in the E-PSO model), with the number of nodes N and the expected average degree (cid:104) k (cid:105) given in the title of thesubplot pair. We used K = − UPPORTING INFORMATION T β label propagation › A S I fi a /α / ( γ − ) label propagation › A S I fi b T β Louvain › A S I fi c /α / ( γ − ) Louvain › A S I fi d T β Infomap › A S I fi e /α / ( γ − ) Infomap › A S I fi f Figure S6.37. Angular separation index in the unweighted
PSO and S / H models. The results for the PSO model are given in the left column (panels (a), (c) and (e)), whereas theASI obtained for the S / H model appears in the right column (panels (b), (d) and (f)). The ASIfor the communities detected by asynchronous label propagation is given in the top row (panels(a) and (b)), the ASI regarding the results of Louvain is shown in the middle row (panels (c) and(d)) and the ASI for the partitions found by Infomap is presented in the bottom row (panels (e)and (f)). We show the measured ASI (indicated by the color, averaged over 100 samples) as afunction of the model parameters T and β , or 1 /α and 1 / ( γ −
1) for networks of size N = 10 , (cid:104) k (cid:105)(cid:105)