Complex networks with scale-free nature and hierarchical modularity
aa r X i v : . [ phy s i c s . s o c - ph ] D ec EPJ manuscript No. (will be inserted by the editor)
Complex networks with scale-free nature and hierarchicalmodularity
Snehal M. Shekatkar and G. Ambika
Indian Institute of Science Education and Research, Pune 411008, IndiaReceived: date / Revised version: date
Abstract.
Generative mechanisms which lead to empirically observed structure of networked systems fromdiverse fields like biology, technology and social sciences form a very important part of study of complexnetworks. The structure of many networked systems like biological cell, human society and World Wide Webmarkedly deviate from that of completely random networks indicating the presence of underlying processes.Often the main process involved in their evolution is the addition of links between existing nodes having acommon neighbor. In this context we introduce an important property of the nodes, which we call mediatingcapacity, that is generic to many networks. This capacity decreases rapidly with increase in degree, makinghubs weak mediators of the process. We show that this property of nodes provides an explanation for thesimultaneous occurrence of the observed scale-free structure and hierarchical modularity in many networkedsystems. This also explains the high clustering and small-path length seen in real networks as well as non-zero degree-correlations. Our study also provides insight into the local process which ultimately leads toemergence of preferential attachment and hence is also important in understanding robustness and controlof real networks as well as processes happening on real networks.
PACS.
Complex networks have become a key tool for understand-ing the behavior of complex systems from diverse fieldslike biology [1,2,3], ecology [4], technology [5,6] and socialsciences [7,8]. Such networks are found to have interest-ing properties not possessed by their completely randomcounterparts and this indicates the presence of robust or-ganizing principles behind their growth [9]. The importantfeatures of such networks are existence of scale-free nature[10], high clustering and small average path length [11], hi-erarchical community structure or hierarchical modularity[12,13,14,15] and non-zero degree correlations [16]. Overthe years, various generative models of network growthhave been proposed to explain one or more of these fea-tures [9,17].In scale-free network, the distribution of degrees fol-lows a power law and this is usually attributed to prefer-ential attachment [10]. The scale-free nature also impliessmall value for the average path length of the network [18].Apart from being scale-free, most real networks are highlyclustered as compared to completely random networks likeErd˝os-R´enyi model [9]. This means that in these networks,two nodes which share a neighbor have quite high chanceof themselves being connected to each other. This ten-dency can be quantified using either the global clusteringcoefficient C [19] or the Watts-Strogatz clustering coeffi- cient C W S [11] of the network. Several variants of the basicpreferential attachment model of Barab´asi and Albert (BAmodel) have been proposed to explain this high value ofclustering along with scale-free structure [20]. In additionto scale-free structure, real networks also show modularorganization of clustering. In such networks, there existgroups of nodes such that the nodes in the same groupare very densely connected to each other whereas thesegroups themselves are connected to each other relativelysparsely. Such groups are known as communities in thenetwork and and the network is said to possess commu-nity structure or modular structure [21]. Existence of suchmodular structure implies high value of clustering coeffi-cient C while the reverse is not true. Also, in many ofthese modular networks, modularity is hierarchical. Thismeans that on a global scale, network is divided into com-munities of various sizes and each of those communitiesitself shows a modular structure inside it [12,13,14,15].It is understood that, in real networks, local processesare responsible for the emergence of overall global struc-ture of the network [22]. The present understanding isthat the preferential attachment should be regarded as anoutcome of system-dependent local processes between thenodes, rather than a mechanism in itself [23,24]. For ex-ample it has been suggested that gene duplication could beresponsible for the scale-free structure of protein interac-tion networks [25]. A possible mechanism reported for the Snehal M. Shekatkar, G. Ambika: Complex networks with scale-free nature and hierarchical modularity local interactions which could produce scale-free networkswith high clustering, is triadic closure where new links areadded between the neighbors of a particular node [23,26,20,27]. The models based on this mechanism can also giverise to simple or hierarchical community structure [23,27].In these models of triadic closure only new nodes are con-sidered as sources of triadic closure. However, in a verygeneral sense, in real networks, many new links are beingcreated between existing nodes also. We feel the possibil-ity of such “internal links” also should form an importantpart of the growth of such networks.In the present paper we identify a fundamental prop-erty of hubs of complex networks that is almost systemindependent. We show that this property can potentiallyexplain the simultaneous existence of scale-free structureand hierarchical modularity even with inclusion of inter-nal links. Our model for growth of complex networks isalso based on the concept of triadic closure but with anadditional property for nodes called mediating capacity.This mediating capacity is a degree dependent propertysuch that high degree nodes are weak mediators of tri-adic closure. As we argue in the following sections, hubsin any real network have capability to start and maintaininteractions with diverse types of nodes. This makes thetriadic closure around the hubs less likely and hence re-sults in low value for their clustering coefficients endowingthe network with hierarchical modularity.The rest of the paper is organized as follows. Section2 describes the existence of triadic closure in various realworld networks along with the social networks. In Sec-tion 3 we introduce the concept of mediating capacity.Section 4 describes our model for growth of complex net-works called mediated attachment model. In Section 5, weshow the existence of hierarchical community structure inthis model along with scale-free structure. Section 6 dealswith time/size dependence of various important charac-teristics of the mediated network. Finally, in section 7 wedraw conclusions related to mediated attachment modelsand indicate future directions of study.
The triadic closure is proposed as one of the local pro-cesses that seem to be present in many real networks [20,27]. This involves the formation of links between a pairof nodes of the networks which share one or more neigh-bors in the network (See Fig. 1(a)). We now describe theexistence of mechanisms analogous to triadic closure invarious real networks.In the case of social networks like acquaintance, friend-ships, coauthors or Facebook networks, triadic closure isobviously present since usually people get to know eachother through a common neighbor in the network. Lan-guages are also viewed as a complex network of wordswhich are connected to each other if they frequently oc-cur in the same sentence [28,29]. During the evolution ofa particular language, when words A and B start appear-ing together in a sentence and if the same happens for words B and C , with a fairly high probability words A and C would appear in the same sentence. In the case ofweb pages, most web pages keep evolving by the additionand modification of their content and hence new hyper-links are usually added to the page. Suppose a web pageA has hyperlink to another page B, which itself containshyperlink to page C, then A can know about C throughB and can connect to C. The co-citations also work thesame way where instead of hyperlinks, bibliographies areused to know and connect to second neighbors [23].All the network growth models involving triadic clo-sure models assume that no direct preferential attachmentis present while network is growing and preferential at-tachment is only an outcome of local process of triadicclosure [23,24,27]. Such triadic closure models have beenproved to be extremely successful in producing highly clus-tered scale-free networks. However the models of triadicclosure proposed so far usually consider only new node asthe source of triadic closure [20,23,30,27]. As noted ear-lier, in real networks majority of links are in fact createdby old nodes. So we have to include the formation of suchinternal links in the growth model. This requires the intro-duction of a new property of hubs which makes them weakmediators such that the generated network are scale-freeand hierarchically modular. We now make an important observation about the nodesof all complex networks, especially scale-free networks.Usually hubs in scale-free networks are considered as mostimportant nodes. We claim that the high degree nodes orhubs in complex networks have capability to interact withnodes of a wide range of properties. If we consider Googleweb page which is a hub in world wide web, the web pageswhich link to it are of diverse nature. Similarly, a hub insocial network usually has interest in wide variety of topicsand hence develops the connections to people from highlydiverse fields. As our third example, we consider a networkof words of a particular language. In this network, hub isusually a word that capable of appearing with words be-longing to diverse range of contexts. For example, the word“THE” in English can appear with many different words.In the jargon of network theory, we can thus say thathubs are usually connected to dissimilar nodes. On theother hand, a low degree node (a Wikipedia page about atechnical topic from mathematics) would have links onlyto nodes similar to it and hence usually its neighbors arealso similar with each other.In all these cases the probability that such diverse ordissimilar nodes will get connected is very low. This leadsto the observation that hubs should function as weak me-diators in such networks. For example, a hub in social net-work may introduce two of his contacts with each otherbut those two contacts would usually share different inter-ests and hence the link may not get established betweenthem. This implies that the mediating capacity of a nodedecreases with increase in degree. As we will see, this in- nehal M. Shekatkar, G. Ambika: Complex networks with scale-free nature and hierarchical modularity 3 sight naturally unifies the emergence of hierarchical com-munity structure with scale-free nature in such networks.
We present a generative mechanism in the growth of net-works with the property of mediating capacity for thenodes as discussed in the previous section. To incorpo-rate this, we conjecture that the average number of linksmediated by a node with degree k decreases with k . There-fore we take the probability that a node i with degree k i connects two of its neighbors at a given time to be1 /k ni where n is a parameter. For a node of degree k ,maximum k C ∼ k links are possible among its k neigh-bors. Hence a node with degree k will form approximately k (1 /k n ) = (1 /k n − ) links at any given time. Thus wemust have n > n < t we add approximately pN ( t ) numberof nodes to the network where p is the parameter withvalue between 0 and 1 and N ( t ) is the number of nodesin the network at time t . The case of addition of one nodeat a time would then correspond to p value that decreaseswith time.The growth of the network in this model thus involvestwo processes:1. Starting from two connected nodes at time t = 0, ⌈ pN ⌉ number of nodes are added to the network at each dis-crete time t where ⌈⌉ denotes the ceiling function. Eachnew node connects to one of the old nodes randomlywith uniform probability.2. At the same time, every existing node i of degree k i ( t )( >
1) connects every pair of its neighbors with proba-bility: w i ( t ) = Ak i ( t ) n (1)where A and n are adjustable parameters with n > w i ( t ) reflects the mediating capacity ofnode i at time t .A small network generated under these processes isshown in Fig. 1(b).It can be easily seen that since each node is allowed toconnect its neighbors with each other, the number of newlinks added to the network is an increasing function of thenetwork size. If the rate of increase of network size is low,this would produce a highly dense network whose averagedegree < k > would increase with time. However, in thepresent model, number of nodes also increases quite fast Fig. 1. (a) A graphical representation of triadic closure pro-cess: node i acts as a mediator for nodes j and l and triesto connect them. (b) Mediated network at time t = 25 with p = 0 . A = 8 and n = 3. The sizes of nodes are proportionalto their degree values and the color of each node is gradedaccording to its clustering coefficient, the darkest being thelowest. and as we will see below, these two factors can stabilize < k > of the network.For the case of n < < k > as a function of time and find thatit increases monotonically. This implies high link densityand non-stationarity of the degree distribution. In general < k > of the network must stabilize if the degree distribu-tion is to be stationary asymptotically. In Fig. 2, < k > isplotted as a function of time for different values of n . Thevalues for n = 1 . n < < k > is a very rapidly increasingfunction of time and hence the degree distribution is notstationary in this case whereas for n >
2, it always stabi-lizes leading to a stationary degree distribution.In Fig. 2 we show < k > as a function of time for dif-ferent values of n for networks grown for 75 units of time.With p = 0 .
1, the results shown are for 100 realizations ofthe growth process. Fig. 3 shows the behavior of averagedegree as a function of parameter n at two different times.The rapid increase in average degree with time (and hencethe existence of non-stationary degree distribution) is ev-ident. This behavior is understandable because for small n , the process of creation of links overtakes the process ofcreation of nodes. For the mediating capacity chosen as inEq.(1) from n = 2 onwards these two effects balance eachother stabilizing < k > .Also when n <
2, the network is so dense that thereare too many nodes with very high degree and hence thenetwork does not exhibit a scale-free structure. To see this,we plot the degree distributions of the network for n = 1 . n = 2 . n = 1 .
5, there exists a peak inthe tail and hence the network in this case is not scale-free.
Snehal M. Shekatkar, G. Ambika: Complex networks with scale-free nature and hierarchical modularity < k > t n = 1.5n = 2n = 2.5n = 3n = 3.5n = 4 Fig. 2. (Color online) Average degree of the network as afunction of time for various values of n with A = 4 and p = 0 . n = 1 . n < < k > is arapidly increasing function of time and hence leads to a non-stationary degree distribution. All cases with n ≥ N = 9525 and results are averaged over 100 realizations. < k > n time = 50time = 70 Fig. 3. (Color online) Average degree < k > as a functionof parameter n at two different times t = 50 (a continuousline) and t = 70 (a solid line). It can be seen that when n < < k > has tendency to increase rapidly while for n > Many real networks are both scale-free and hierarchicallymodular. The simultaneous occurrence of scale-free struc-ture and hierarchical modularity is a less explored topicin the context of generative network models. As we nowshow, the mediated attachment model based on the con-jecture that the hubs are weak mediators naturally givesrise to networks with both these properties. To establishthe scale-free structure of the resulting network, in Fig. 5 -6 -5 -4 -3 -2 -1 p ( k ) k n = 1.5n = 2.5 Fig. 4. (Color online) Comparison of degree distributionsfor mediated network with n < n >
2. Here for n < n = 1 . N = 9525 and results areaveraged over 100 realizations. -5 -4 -3 -2 -1 p ( k ) k n = 2.5 10 -5 -4 -3 -2 -1 p ( k ) k n = 310 -5 -4 -3 -2 -1 p ( k ) k n = 3.5 10 -5 -4 -3 -2 -1 p ( k ) k n = 4 Fig. 5.
Degree distributions of mediated network of size N =9525 for different values of n with A = 8 and p = 0 .
1. Alogarithmic binning is used to plot the histograms. The dottedline in each plot has slope − we plot the degree distributions of the network for vari-ous values of n with n > ,
4) [9,17] and hence the model canbe tuned to produce the actually observed distributionssatisfactorily.A necessary (but not sufficient) indicator of existenceof hierarchical modularity in the given network is the de-pendence of local clustering coefficient on the degree. Alocal clustering of the node i is given as: nehal M. Shekatkar, G. Ambika: Complex networks with scale-free nature and hierarchical modularity 5 -1 c ( k ) k n = 2.5n = 3n = 3.5n = 4 Fig. 6. (Color online) Local clustering coefficient c as a func-tion of degree k for mediated network. The parameters aresame as for Fig. 5. The black dotted line has slope −
1. Resultsare averaged over 1000 random realizations. c i = E ik C (2)where k is the degree of node and E i is the number of linkspresent among the neighbors of node i . Its value alwayslies between 0 and 1. For the networks with hierarchicalmodularity, c i shows a systematic decrease with degree k with c ( k ) ∼ k − [13]. For the model presented here, wecompute the local clustering coefficient for different val-ues of n and plot it as a function of k in Fig. 6. Thedecrease of c ( k n >
2, the mediated network is hierarchically modu-lar but the number of modular levels depends on the valueof n and in fact is the increasing function of n . Thus, forthe network size N = 9525 used here, for n = 2 . n = 4, the networkhas 8 to 9 levels of hierarchy. Now we look at the various important network charac-teristics like clustering coefficient, path length and degreecorrelations for the mediated network.
The density of triangles in the network is quantified bythe quantity called clustering coefficient. There exist twodefinitions of the clustering coefficient in the literature: C W S t n = 2.5n = 3n = 3.5n = 4 Fig. 7. (Color online) Watts-Strogatz clustering coefficient C WS of the network as a function of time for various values ofparameter n . All the parameter values are same as in Fig. 6 global clustering coefficient and Watts-Strogatz cluster-ing coefficient. In the present case, we use Watts-Strogatzclustering coefficient C W S (also called as average cluster-ing coefficient) as the quantifier for the clustering in thenetwork. It is defined as the average of the local clusteringcoefficients over all the nodes of the network: C W S = 1 N N X i =1 c i (3)Fig. 7 shows the Watts-Strogatz clustering coefficientof the mediated network as a function of time for vari-ous values of n . It can be seen that all C W S values settleasymptotically to values in the range (0 . , . C W S [31]. This agreement with the em-pirical observations further supports the model presentedhere.
Apart from the properties discussed so far, many realnetworks show non-zero degree correlations. Networks inwhich high degree nodes tend to connect to low degreenodes (i.e. networks for which negative degree correlationsexist) are known as dissortative networks while those inwhich similar degree nodes tend to connect to each other(i.e. networks for which positive degree correlations exist)are known as assortative networks.Assortative or dissortative nature of networks can bequantified by calculating the assortativity coefficient forthe network which is given by the following expression[17]:
Snehal M. Shekatkar, G. Ambika: Complex networks with scale-free nature and hierarchical modularity -0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0 10 20 30 40 50 60 70 r t n = 2.5n = 3n = 3.5n = 4 Fig. 8. (Color online) Assortativity coefficient r of the me-diated network as a function of time for different values ofparameter n . It can be seen that r stabilizes to positive valuesasymptotically and hence the mediated network is assortative.All the parameter values are as in Fig. 6 r = P ij ( B ij − k i k j / m ) k i k j P ij ( k i δ ij − k i k j / m ) k i k j (4)Here, B ij is ( i, j ) th element of an adjacency matrix ofthe network, m is the number of links in the network, k i isthe degree of node i and δ ij is the Kronecker delta whosevalue is 1 when i = j and is 0 otherwise.Interestingly, social networks are always seen to pos-sess positive degree correlations i.e they are assortative[16]. In Fig. 8 we show the assortativity coefficient of themediated network as a function of time for different n . Itis clear that our model with n > The real networks are found to have a small-world prop-erty which means that the average path length < l > ofthese networks varies very slowly with size and also theyhave high clustering coefficient [11]. We also calculate the < l > of mediated network as a function of its size fordifferent n . It is clear that < l > increases logarithmi-cally with size (see Fig. 9). Such logarithmic scaling withsize has been reported earlier for many real networks [9].This along with the observed high clustering, makes themediated network a small-world network. Understanding the physical mechanisms which lead to theemergence of various structural properties seen in realnetworks is an important part of the network science.Since the discovery of small-world nature and scale-free < l > N n = 2.5n = 3n = 3.5n = 4 Fig. 9. (Color online) Average path length < l > of the net-work as a function of size N ( t ) of the network for different n .The parameter values are as given in Fig. 6. Each curve in theplot follows a logarithmic increase. nature of various real networks, better and better mod-els of network formation have been proposed over theyears. It has been observed that many real networks ex-hibit both scale-free nature and hierarchical modularity.However, the generic mechanisms which lead to their si-multaneous occurrence in real networks are not yet veryclear. In the present work we showed that the weakness ofhubs in mediating the connections between their neighborsproduces both of these structures successfully. The weak-ness of hubs can be understood from a more fundamentalfact that hubs are usually connected to dissimilar nodesin the network. We feel that a more careful use of thisidea in the context of specific networks will lead to manynovel insights into the structure and function of variousreal networks.The model presented here comes with tunable param-eters p, A and n that can be adjusted to get desired val-ues of network characteristics. For n > A and n .Apart from scale-free nature and hierarchical modularity,we also showed that the mediated network is highly clus-tered and its path length scales logarithmically with size.This shows that a small-world nature is also an emergingproperty of this model. Finally, the network shows positivedegree correlations as observed in case of social networks.We anticipate that detailed investigations to understandthe intrinsic mechanisms that generate this property indifferent cases, can reveal interesting processes underlyingthe complexity of such systems.One of the authors (S.M.S.) would like to thank Uni-versity Grants Commission, New Delhi, India for financialassistance in the form of Senior Research Fellowship. nehal M. Shekatkar, G. Ambika: Complex networks with scale-free nature and hierarchical modularity 7 References
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