Relative Canonical Network Ensembles -- (Mis)characterizing Small-World Networks
RRelative Canonical Network Ensembles –(Mis)characterizing Small-World Networks
Oskar Pfeffer, Nora Molkenthin, Frank Hellmann
Potsdam Institute for Climate Impact ResearchTU Berlin (Dated: January 26, 2021)What do generic networks that have certain properties look like? We define Relative CanonicalNetwork ensembles as the ensembles that realize a property R while being as indistinguishable aspossible from a generic network ensemble. This allows us to study the most generic features of thenetworks giving rise to the property under investigation. To test the approach we apply it first to thenetwork measure ”small-world-ness”, thought to characterize small-world networks. We find severalphase transitions as we go to less and less generic networks in which cliques and hubs emerge. Suchfeatures are not shared by typical small-world networks, showing that high ”small-world-ness” doesnot characterize small-world networks as they are commonly understood. On the other hand we seethat for embedded networks, the average shortest path length and total Euclidean link length arebetter at characterizing small-world networks, with hubs that emerge as a defining feature at lowgenericity. We expect the overall approach to have wide applicability for understanding networkproperties of real world interest.
I. INTRODUCTION
Network ensembles are sets of networks together witha probability distribution of their occurrence and havebeen successfully used to model a wide range of natu-ral, social and technical systems, in which the interactionstructure is subject to, or the outcome of, stochasticity[1–6]. Typically those ensembles are generated througha heuristic process, thought to capture some aspect ofthe microscopic formation process, which underlies thereal-world system they are trying to model. The result-ing ensemble can then be studied and characterized bymeans of network measures that quantify certain prop-erties of the networks. Examples for this are Watts–Strogatz networks, which are characterized by low aver-age shortest path length and high clustering coefficients[7], and Barabasi–Albert networks, which are character-ized by their power-law degree distribution [8].Here we want to approach network ensembles from theother side. Rather than trying to model real world net-works we ask: What do generic networks that have cer-tain properties look like? Thus, we will define ensemblesthrough a particular property captured by a “propertyfunction” R ( G ) on networks G and a background ensem-ble that defines our notion of generic networks in thegiven context. To this end, we will consider slightly gen-eralized exponential random graphs. Exponential ran-dom graphs have long been a tool in network science,starting with [9–12], see [13] for a recent review, and arealso sometimes known as canonical network ensembles(CNE) [14–16]. We will consider CNEs relative to thebackground ensemble of generic networks. Given someset of networks E on a finite set of vertices, denote theprobability distribution of the background ensemble as q ( G ) for G ∈ E . The relative canonical network ensem-ble (RCNE) of R relative to q is given by the probabilitydistribution proportional to exp( − βR ( G )) q ( G ).We emphasize that our aim is not to model empiri- cally observed network ensembles with certain proper-ties. There is no reason to expect empirical networks,that are the outcome of subtle formation processes, tobe generic. Instead, we will study the properties them-selves, specifically the most generic features that producethem, and whether or not the properties suffice to gener-ically characterize the networks under study. Our aim inthis is to understand properties that are of considerablepractical interest. Companion papers will consider epi-demic thresholds and the vulnerability to failure cascadesin power grids. To introduce our approach, this paperwill focus on well-known and well-established networkmeasures, that are computationally challenging, instead.Specifically, we will consider the notion of ”small-world-ness”.We study two ensembles, the first defined by the small-world-ness , as introduced in [17], the second defined by acombination of Euclidean link length and average short-est path length similar to [18]. To study these ensembleswe sample them using the straightforward Metropolis-Hastings (MH)[19–21] algorithm.In both cases we find phase transitions as we go fromfully generic networks to highly specific ones. At thesephase transitions certain features arise, e.g. hubs andcliques start appearing in the ensemble. Surprisingly wefind that generic networks with high small-world-ness donot resemble small-world networks. Thus, we find thatwhat [17] called small-world-ness does not actually char-acterize small-world networks generically. II. RELATIVE CANONICAL NETWORKENSEMBLE
Exponential random graphs were first introduced in[9–11]. Given the set of simple graphs E N on a set of N vertices, they are defined by the probability distribu-tion over E N , p Rβ ( G ) = Z R ( β ) − exp ( − βR ( G )). That is, a r X i v : . [ phy s i c s . s o c - ph ] J a n they are the Gibbs ensemble at temperature T = 1 /k B β .The use of such network ensembles is sometimes justifiedby the fact that these are maximum entropy ensembleswith a given expectation value for R . However, thereis no a priori reason to expect formation processes thatlead to real world networks to maximize entropy. Forinstance, typical formation processes do not resemble ex-change with an environment at fixed genericity (in anal-ogy to a heat bath). In fact, it was already noted in [12]that the maximum entropy ensembles do not model realworld systems easily and show unexpected structures, in-terpreted there as an “unfortunate pathology”.Instead, we want to understand the most generic fea-tures giving rise to a property R . That is, a feature,that is observed more frequently the more the expecta-tion value of R differs from the value expected for genericnetworks. As mentioned in the introduction, to define ournotion of genericity we specify background ensemble q ( G )(for example an Erd˝os–R´enyi ensemble at a fixed numberof edges). The relative canonical network ensemble of R relative to q is then given by: p Rβ,q ( G ) = 1 Z R,q ( β ) e − βR ( G ) q ( G ) , (1)with normalization/partition function Z R,q ( β ) = (cid:80) G ∈E N e − βR ( G ) q ( G ). This ensemble is characterized bybeing the ensemble of minimum relative entropy D ( p || q )for a fixed expectation value of R . From an information-theoretic perspective it is the ensemble hardest to dis-tinguish from the generic ensemble q while having fixedexpectation value (cid:104) R (cid:105) = R ∗ , for a more detailed discus-sion see Appendix A.The parameter β moderates the trade-off between thegeneric ensemble and highly specific ones peaked on net-works that are high or low in R , see Figure 1. It can rangefrom −∞ to + ∞ with the sign depending on whetherthe expectation value of R is higher or lower than in thegeneric network ensemble given by β = 0. At β → −∞ we have an ensemble concentrated on max( R ), while at β → + ∞ it is min( R ). This, and the fact that inter-pretation of the relative entropy is purely informationtheoretic (rather than thermodynamic), motivates us torefer to β − in this context as the genericity rather thanas a temperature.Of particular note are phase transitions that occur aswe lower the absolute genericity. The structure of theensemble changes at and beyond the phase transition.This change in structure allows us to identify specificfeatures that contribute to property R but are not genericenough to occur before.Throughout the rest of this manuscript we will considercanonical ensembles relative to the Erd˝os–R´enyi ensem-ble at fixed size N and mean degree k , that is, the equidis-tribution over all graphs with vertex set { , ..., N } and kN/ FIG. 1. The inverse genericity β mediates between specificensembles concentrated on maximum R , the generic back-ground ensemble q with (cid:104) R (cid:105) = (cid:104) R (cid:105) q and specific ensemblesconcentrated on minimum R . it might also be appropriate to use maximum entropynull-models as generic ensembles[13].Since exponential random graphs were first introduced,computing capabilities profoundly increased. This meanswe can now use complex, practically relevant networkproperties and analyze what features of networks gener-ically give rise to them. This approach may help in thefuture to gain a better understanding of complex networkmeasures and provide a way to find simpler network mea-sures to act as predictors for the characteristics definingthe ensemble.To study these ensembles we need to sample fromthem. An important property of RCNEs is that theyare well suited for sampling using Metropolis-Hastings(MH) algorithms. To use MH on our relative ensemble,we require a background process that generates proposedsteps compatible with the background distribution q . For q Nk this can be provided simply by considering rewiringof edges. Starting from a system in state x the algorithmproposes rewirings that are accepted with probability P β ( x → y ) = min (cid:18) , p β ( R ( x )) p β ( R ( y )) (cid:19) = min (cid:0) , e − β ∆ R (cid:1) . (2)This algorithm satisfies the detailed balance conditionand the Markov chain defined by it is strongly connected.In the limit of infinite steps the time average for this en-semble converges to the ensemble average of the groundstate which is the relative canonical network ensemble.Unfortunately, there are no guarantees for finite timesamples and we have to resort to heuristics to understandwhether convergence has occurred. To do so we typicallyalso run several chains in parallel from random initialconditions. This further allows us to obtain less corre-lated samples. More details on our sampling approachare provided in the next section.For more general background ensembles it might becomplicated to find step proposals. If q is explicitly a)b) N o r m a li ze d P r o p e r t y F un c t i o n R FIG. 2.
Small-world-ness increases significantly as theproperty function approaches the global minimum. R S -ensemble (circles) and R WL -ensemble (squares) networkswith N = 256 and (cid:104) k (cid:105) = 4 to finite inverse genericity β ∈{ − → } . At the start of the range the ensembles arestatistically indistinguishable from the background ensemble q . a) shows the small-world-ness S and b) the property R S and R WL vs β . The stars on the right identify simulationsfor β → ∞ . Each data point is averaged over 32 realizationswith 2 MCMC steps each. known it can be incorporated into the step acceptanceprobability. If there is no explicit formula for q , for exam-ple because it is implicitly defined by a stochastic growthprocess, it is necessary to make use of the growth processdirectly to generate new proposals. III. SMALL WORLD PROPERTIES
To demonstrate the approach, we analyze two differentsmall-world-ness properties. In particular we look at thefeatures that give rise to them and whether they gener-ically characterize what is commonly known as small-world networks. In the first instance we consider theSmall-world-ness measure S = ( C/L ) ( C ER /L ER ) − , in-troduced in [17], where C is the global clustering coeffi-cient defined as the number of closed triplets divided bythe number of all triplets, L the average shortest pathlength C ER is the average clustering coefficient of anErd˝os–R´enyi network [22] of the same size and L ER is theexpected average shortest path length of an Erd˝os–R´enyinetwork of the same size. Finally, the form of the prop-erty we will consider is: R S = LC . (3)That is, proportional to the inverse of S . Thus, smallvalues of R S indicate high Small-world-ness, and we areinterested in positive β .The second property R RR majorclique hubhub hubhubmajorclique a) b)c) d) R FIG. 3.
At low genericity hub and clique structuresemerge, transforming the degree distribution.
The de-gree distribution in a) shows three peaks for the R S -ensemblewhile the degree distribution of the R WL -ensemble in c) showstwo peaks. b) and d) show a heat-map of degree distributionsat various inverse genericities. The degree distributions arean average of 32 realizations with 2 MCMC steps each. R W L = W L (4)is given in terms of the average shortest path length L and the wiring length W in an embedded network. W is given by the sum of the Euclidean length of all edges.The networks for this ensemble are embedded in a 2Dplane. The introduction of W was inspired by [18], whereit was argued that small-world networks might arise as asecondary feature from a trade-off between maximal con-nectivity and minimal edge lengths. Again small R W L isexpected to yield Small-world networks and we considerpositive β .Both ensembles are taken relative to a randomErd˝os–R´enyi network with N vertices and M = (cid:104) k (cid:105) N / (cid:104) k (cid:105) is the average degree of the network.The positions of the vertices in the embedded networksare initialized randomly on a 2D unit square.The proposal for each Monte Carlo step is generatedby rewiring a single edge, i.e. deleting an existing edge atrandom and connecting two previously unconnected ver-tices chosen at random, thus keeping the number of edgesconstant. The proposal is then accepted with the tran-sition probability given in Eq. (2). Proposals of discon-nected graphs are always rejected since L is infinite. Toensure convergence at low genericity, we use an exponen-tial schedule β − ( t ) = (cid:0) β − α t + β − (cid:1) , similiar to theSimulated Annealing approach [23], where t is the stepparameter, α = 0 .
99 is a simulation parameter. β − and β − are the start and final genericities. We gen-erated ensembles with (128 , , , , , ,
16) net-works of size N = (8 , , , , , , (cid:104) k (cid:105) = 4. All the simulations ap-peared to converge, allowing qualitative evaluation of thesimulations. Throughout the manuscript, genericity was hubmajorcliquehub a) b) c) d)e) f) g) FIG. 4.
Examples of the various phases for N = 128 networks. The R S ensemble starts as a random network withoutany recognizable structure in a), then a first large clique emerges in b) and finally a central hub that lowers the average shortestpath length in c), which is close to the network where the small-world-ness is maximal in d). The R WL ensemble starts froma random phase in e), then a central hub with long range connections emerges in f). The network minimizing R WL resemblesa random geometric network with a central hub. decreased over 2 equally long periods, each containing2 MCMC steps for a total of 2 MCMC steps. Notethat while the properties we consider here are concep-tually simple, the presence of the average shortest pathlength, which needs to be recomputed for every proposedstep, renders them computationally expensive. We fur-ther note that achieving convergence for the R S ensemblewas considerably harder than for R W L . Thus, they doconstitute a real check of the ability of the approach tostudy complex network properties.As shown in Fig. 2, the Small-world-ness S increasesfor both network ensembles as they become less generic.For the R S -ensemble, for which it is mathematicallyguaranteed that the expectation value of S increases fordecreasing genericity, this is an important sanity checkon our sampling. In the R W L -ensemble this arises as asecondary effect as Euclidean and network distances arereduced, showing that generic R W L networks do indeedhave high small-world-ness, as anticipated in [18].As a common and simple network measure, we nowlook at the degree distribution. Fig. 3 shows the degreedistributions of the two ensembles for decreasing gener-icity. The shift from generic (poisson distributed) to spe-cific networks is evident. The extremal β → ∞ case (sim-ulated with the same exponential schedule as above untilwe observed convergence) is shown explicitly in Fig. 3 a)and c), and we see highly pronounced features in the de-gree distribution. Example of networks at this state areshown in Fig. 4 d) and g), looking at these allows us toidentify the features in the degree distribution as major cliques and hubs. Note that the degree distribution ofthe R S ensemble in particular does not resemble that ofthe WS-ensemble.The R S example network (Fig. 4 d) looks almost star-shaped with a very highly connected central node anda few fully connected branches. This indicates that thetwo components of the property, namely average shortestpath length and clustering are optimized in specializedareas of the network. The star graph, which is the small-est possible sparse graph, is thereby combined with manynodes in fully connected cliques. The R W L network (Fig.4 g) on the other hand looks like a sparse geometric net-work with star-shaped shortcuts, making it much closerin spirit to the WS-ensemble and its two-dimensional rel-atives.As a result, the nodes in the R S -networks can be cate-gorized into hub-nodes, clique-nodes, and the rest, wherehub-nodes have very high degrees ( k ≈ − k ≈ −
15) and the resthas low degrees. This can be seen in Fig. 3b) as 3 majorpeaks.To understand how these extremal cases come about,we consider the degree distributions over the whole pa-rameter space in (Fig. 3 b and d). Here we see severalabrupt transitions. For the R S ensemble the major cliquestarts forming at β ≈ − , while the hub only emergesat high inverse genericities of around β ≈ .In the embedded networks, nodes fall into two cate-gories: regional nodes and inter-regional hubs. This canbe seen in Fig. 3 c), where regional nodes fall into the nor-mal degree distribution of a (slightly sparser) geometricnetwork and hubs have higher degrees of k ≈ − β ≈ . IV. GENERICITY PHASE TRANSITION
Fig. 4 shows various examples of networks taken fromdifferent genericity phases. We can now study the tran-sition between these phases in more detail.As seen in Fig.3 b and d, both the R S and R W L ensem-bles show a qualitative change in the degree distribution.At high genericity we have essentially random graphs inboth cases with the expected Poisson degree distribu-tion. At low genericity both ensembles show multiplepeaks. In case of the R W L -ensemble this comes as a sud-den appearance of a second peak at β = 2 . In case ofthe R S -ensemble this transition appears to be less clearcut with a structure that resembles branching at β ≈ and almost merging again, while another peak appearsat β ≈ .To better understand these transitions we analyze themean largest eigenvalues λ of the adjacency matrix, sizesof the largest non-empty k -core and maximum degree asfunctions of the genericity for network sizes from N = 2 to N = 2 . The results are shown in Fig. 5. Both en-sembles show a phase transition in the largest Eigenvaluebetween a low λ state and a high λ state. This tran-sition is located at β ≈ for the R S -ensemble and at β ≈ for the R W L -ensemble.This transition is mirrored by the maximum k-core incase of the R S -ensemble (see Fig. 5 c)). This indicatesthat here the formation of the first dense region in thegraph is responsible for the phase transition. This isclearly not the case for the R W L -ensemble, where we findno consistent transition genericity for the largest non-empty k-core, but a shift in its rise depending on thenetwork size as shown in Fig. 5 f).Instead, the largest Eigenvalue transition in the R W L -ensemble is mirrored by the maximum degree in the net-work, as displayed in Fig. 5 e). As expected from the de-gree distributions shown above, the changes of the max-imum degree in the R S -ensemble hint at two transitions,one at β ≈ , in which the first dense region forms andone at β ≈ , at which the central hub forms. The phasetransition giving birth to the first dense region found at β ≈ , can be interpreted as similar to [24], where afirst order phase transition was analytically found forStrauss’s model of clustering [11].These results show that certain discrete features sud-denly emerge at certain genericities. The transitionsbecome qualitatively visible in the degree distribution,clearly appear in their graphical representations and canbe quantified in various network measures, where thelargest eigenvalue is a good first indicator and are moredetailed in the maximum k-core and degree. These phasetransitions and the emergence of hubs and cliques are adriving element in the increase of the small-world-ness property. V. DISCUSSION AND CONCLUSION
Here we introduced the concept of relative canonicalnetwork ensembles of arbitrary network properties, asa means to study what the most generic networks withthese properties look like. These ensembles are amenableto Metropolis-Hastings and MCMC methods, providinga simple and straightforward (if potentially computation-ally expensive) way of sampling from non-trivial networkensembles defined through network measures of practicalinterest.To challenge the method we studied two properties tra-ditionally expected to characterize small-world networks.Surprisingly we found that generic networks with a highsmall-world-ness index S in the sense of [17]. Insteadwe find that as S increases the most generic networkswith high S contain first cliques and then hubs, neitherof which occur in the WS-ensemble. An alternative prop-erty defined as the product of wiring and shortest pathlength fared better, here also hubs arise for the leastgeneric networks, but the system appears to resemblesmall world networks more closely. This indicates that atleast for some networks, spatial embedding may actuallybe the defining feature, from which high small-world-nessarises as a secondary effect.The transition from highly generic to very specific en-sembles in both cases is characterized by well definedphase transitions. These are visible in a number of net-work measures. Notably in both cases we have a riseof the largest eigenvalue of the adjacency matrix. This,however corresponds to the growth of the first dense re-gion in the R S -ensemble and to the emergence of an inter-regional hub in the R W L -ensemble.It is somewhat surprising that new things are still tolearn on properties thought to characterize small-worldnetworks. The fact that our perspective on relativecanonical network ensembles could discover novel fea-tures is a promising sign for the study of properties ofgreater practical interest. In companion papers we areconsidering epidemic thresholds, and the vulnerabilityto cascading failures. More generally this method is ofgreat interest wherever we want to understand and de-sign topologies that fulfill certain functions, rather thandescribe empirical networks.
Code and Data availability
All code and data used in this work will be made avail-able at https://doi.org/10.5281/zenodo.4462634 . a) b) c)d) e) f) I III II
I II
I II III I II III
I II III
FIG. 5.
The phase transition is characterized by a rise in the largest Eigenvalue.
The largest Eigenvalue is plottedin a) and d) for R S and R WL respectively. b) and e) show the dependence of the largest degree on the genericity and c) and f)show the maximum k-core over genericity for network sizes from N = { , , , , , , } and average degree (cid:104) k (cid:105) = 4.Simulation details: MCMC steps = 2 each, generated ensemble size (in order of network size) = { , , , , , , } .The realizations for R S , N = 512 , log β ≥ Appendix A: Relative Entropy
The minimization of the relative entropy has an infor-mation theoretic interpretation. Given a distribution q ,the asymptotic probability to obtain a sample that lookslike p goes as the exponential of the negative relative en-tropy D ( p || q ). This result of Chernoff [25] is known asStein’s Lemma (for a modern account phrased in terms ofrelative entropy see e.g. [26] Theorem 4.12) and forms themathematical basis for the interpretation of the relativeentropy as a measure of distinguishability of probabilitydistributions. Our ensembles thus have an informationtheoretic interpretation as being the ensembles that arehardest to distinguish from the generic ensemble q . Inparticular we do not presuppose that real network for-mation processes maximize entropy subject to some con-straints, and do not interpret the resulting ensembles asmodeling real networks that have the property R .For completeness, we recall here the standard argu-ment that the relative entropy, or the Kullback-Leiblerdivergence, is minimized by the exponential ensemble.We are looking for p ∗ = arg min p (cid:104) R (cid:105) = R ∗ D ( p || q )= arg min p (cid:104) R (cid:105) = R ∗ (cid:88) i p i ln (cid:18) p i q i (cid:19) (A1)First, note that this formula diverges to positive infin- ity if p has support outside the support of q . We thus onlyconsider p whose support is contained in that of q . Then,by introducing Lagrange multipliers for the expectationvalue of R as well as for the normalization condition onthe distribution p we can rewrite the constrained mini-mization above as a free minimization: p ∗ ( β n , β R ) = arg min p (cid:88) i p i ln (cid:18) p i q i (cid:19) ++ β n (cid:32)(cid:88) i p i − (cid:33) + β R (cid:32)(cid:88) i p i R i − R ∗ (cid:33) (A2) R ∗ = (cid:88) i R i p ∗ i ( β n , β R )1 = (cid:88) i p ∗ i ( β n , β R )Now the variation in the direction p j produces the fol-lowing condition:0 = ∂∂p j (cid:34)(cid:88) i p i (ln( p i ) − ln( q i )) ++ β n (cid:32)(cid:88) i p i − (cid:33) + β R (cid:32)(cid:88) i p i R i − R ∗ (cid:33)(cid:35) = ln( p j ) − ln( q j ) + 1 + β n + β R R j (A3)From which we can conclude p ∗ j = exp(ln( q j ) − − β n − β R R j )= 1 Z e − β R R j q j (A4)with Z = e β n = (cid:80) i e − β R R i q i fixed by the condition (cid:80) i p ∗ i = 1 and β R determined implicitly by the condition R ∗ = (cid:80) i R i p ∗ i . Note that ∂∂β R (cid:104) R (cid:105) = − Z (cid:88) i R i e − β R R i q i − ∂Z∂β R Z (cid:88) i R i e − β R R i q i = −(cid:104) R (cid:105) + (cid:104) R (cid:105) = − Var( R ) ≤ . (A5)Further, for β R = −∞ we have the distribution peakedcompletely on the global maxima: (cid:104) R (cid:105) = R max and for β R = + ∞ we have the minima instead (cid:104) R (cid:105) = R min . For β R = 0 we have exactly the expectation value of R in thegeneric background ensemble q . [1] P. Schultz, J. Heitzig, and J. Kurths, A random growthmodel for power grids and other spatially embedded in-frastructure networks, The European Physical JournalSpecial Topics , 2593 (2014).[2] D. W. Opitz and J. W. Shavlik, Generating accurate anddiverse members of a neural-network ensemble, in Ad-vances in neural information processing systems (1996)pp. 535–541.[3] J. H. Snoeijer, T. J. 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