finding and evaluating community structure in networks · finding and evaluating community...

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PHYSICAL REVIEW E 69, 026113 ~2004!

Finding and evaluating community structure in networks

M. E. J. Newman1,2 and M. Girvan2,3

1Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109-1122Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA3Department of Physics, Cornell University, Ithaca, New York 14853-2501, USA

~Received 19 August 2003; published 26 February 2004!

We propose and study a set of algorithms for discovering community structure in networks—natural divi-sions of network nodes into densely connected subgroups. Our algorithms all share two definitive features:first, they involve iterative removal of edges from the network to split it into communities, the edges removedbeing identified using any one of a number of possible ‘‘betweenness’’ measures, and second, these measuresare, crucially, recalculated after each removal. We also propose a measure for the strength of the communitystructure found by our algorithms, which gives us an objective metric for choosing the number of communitiesinto which a network should be divided. We demonstrate that our algorithms are highly effective at discoveringcommunity structure in both computer-generated and real-world network data, and show how they can be usedto shed light on the sometimes dauntingly complex structure of networked systems.

DOI: 10.1103/PhysRevE.69.026113 PACS number~s!: 89.75.Hc, 87.23.Ge, 89.20.Hh, 05.10.2a

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I. INTRODUCTION

Empirical studies and theoretical modeling of networhave been the subject of a large body of recent researcstatistical physics and applied mathematics@1–4#. Networkideas have been applied with success to topics as diversthe Internet and the world wide web@5–7#, epidemiology@8–11#, scientific citation and collaboration@12,13#, metabo-lism @14,15#, and ecosystems@16,17#, to name but a few. Aproperty that seems to be common to many networks iscom-munity structure, the division of network nodes into groupwithin which the network connections are dense, buttween which they are sparser—see Fig. 1. The ability to fiand analyze such groups can provide invaluable help inderstanding and visualizing the structure of networks. In tpaper, we show how this can be achieved.

The study of community structure in networks has a lohistory. It is closely related to the ideas of graph partitioniin graph theory and computer science, and hierarchical ctering in sociology@18,19#. Before presenting our own findings, it is worth reviewing some of this preceding workunderstand its achievements and shortcomings.

Graph partitioning is a problem that arises in, for eample, parallel computing. Suppose we have a numbern ofintercommunicating computer processes, which we wishdistribute over a numberg of computer processors. Processdo not necessarily need to communicate with all others,the pattern of required communications can be representea graph or network in which the vertices represent proceand edges join process pairs that need to communicate.problem is to allocate the processes to processors in suway as roughly to balance the load on each processor, wat the same time minimizing the number of edges thatbetween processors, so that the amount of interprocecommunication~which is normally slow! is minimized. Ingeneral, finding an exact solution to a partitioning task of tkind is believed to be an NP-hard problem, making it phibitively difficult to solve exactly for large graphs, butwide variety of heuristic algorithms have been develop

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that give acceptably good solutions in many cases, theknown being perhaps the Kernighan-Lin algorithm@20#,which runs in timeO(n3) on sparse graphs.

A solution to the graph partitioning problem is, howevenot particularly helpful for analyzing and understanding nworks in general. If we merely want to find if and howgiven network breaks down into communities, we probado not know how many such communities there are goingbe, and there is no reason why they should be roughlysame size. Furthermore, the number of intercommunedges need not be strictly minimized either, since more sedges are admissible between large communities thantween small ones.

As far as our goals in this paper are concerned, a museful approach is that taken by social network analysis wthe set of techniques known as hierarchical clustering. Thtechniques are aimed at discovering natural divisions of~so-cial! networks into groups, based on various metrics of simlarity or strength of connection between vertices. They finto two broad classes, agglomerative and divisive@19#, de-pending on whether they focus on the addition or removaedges to or from the network. In an agglomerative methsimilarities are calculated by one method or another betwvertex pairs, and edges are then added to an initially em

FIG. 1. A small network with community structure of the typconsidered in this paper. In this case there are three commundenoted by the dashed circles, which have dense internal linksbetween which there is only a lower density of external links.

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M. E. J. NEWMAN AND M. GIRVAN PHYSICAL REVIEW E 69, 026113 ~2004!

network ~n vertices with no edges! starting with the vertexpairs with highest similarity. The procedure can be haltedany point, and the resulting components in the networktaken to be the communities. Alternatively, the entire pgression of the algorithm from empty graph to complegraph can be represented in the form of a tree ordendrogramsuch as that shown in Fig. 2. Horizontal cuts through therepresent the communities appropriate to different haltpoints.

Agglomerative methods based on a wide variety of simlarity measures have been applied to different netwoSome networks have natural similarity metrics built in. Fexample, in the widely studied network of collaborations btween film actors@21,22#, in which two actors are connecteif they have appeared in the same film, one could quansimilarity by how many films actors have appeared ingether@23#. Other networks have no natural metric, but suable ones can be devised using correlation coefficients,lengths, or matrix methods. A well known example ofagglomerative clustering method is the Concor algorithmBreigeret al. @24#.

Agglomerative methods have their problems, howevOne concern is that they fail with some frequency to findcorrect communities in networks where the communstructure is known, which makes it difficult to place mutrust in them in other cases. Another is their tendency to fionly the cores of communities and leave out the periphThe core nodes in a community often have strong similarand hence are connected early in the agglomerative procbut peripheral nodes that have no strong similarity to othtend to get neglected, leading to structures like that showFig. 3. In this figure, there are a number of peripheral nowhose community membership is obvious to the eye—most cases, they have only a single link to a speccommunity—but agglomerative methods often fail to plasuch nodes correctly.

FIG. 2. A hierarchical tree or dendrogram illustrating the typeoutput generated by the algorithms described here. The circles abottom of the figure represent the individual vertices of the nwork. As we move up the tree, the vertices join together to folarger and larger communities, as indicated by the lines, untilreach the top, where all are joined together in a single commuAlternatively, the dendrogram depicts an initially connected nwork splitting into smaller and smaller communities as we go frtop to bottom. A cross section of the tree at any level, such asindicated by the dotted line, will give the communities at that levThe vertical height of the split points in the tree are indicative oof the order in which the splits~or joins! take place, although it ispossible to construct more elaborate dendrograms in which theights contain other information.

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In this paper, therefore, we focus on divisive methoThese methods have been relatively little studied in the pvious literature, either in social network theory or elsewhebut, as we will see, they seem to offer a lot of promise. Indivisive method, we start with the network of interest aattempt to find theleast similar connected pairs of verticeand then remove the edges between them. By doingrepeatedly, we divide the network into smaller and smacomponents, and again we can stop the process at anyand take the components at that stage to be the netwcommunities. Again, the process can be represented as adrogram depicting the successive splits of the network ismaller and smaller groups.

The approach we take follows roughly these lines, badopts a somewhat different philosophical viewpoint. Ratthan looking for the most weakly connected vertex pairs,approach will be to look for the edges in the network thatmost ‘‘between’’ other vertices, meaning that the edge issome sense, responsible for connecting many pairs of othSuch edges need not be weak at all in the similarity senHow this idea works out in practice will become clear in tcourse of the presentation.

Briefly then, the outline of this paper is as follows. In SeII we describe the crucial concepts behind our methodsfinding community structure in networks and show hothese concepts can be turned into a concrete prescriptionperforming calculations. In Sec. III we describe in detail timplementation of our methods. In Sec. IV we consider waof determining when a particular division of a network incommunities is a good one, allowing us to quantify the scess of our community-finding algorithms. And in Sec. V wgive a number of applications of our algorithms to particunetworks, both real and artificial. In Sec. VI we give oconclusions. A brief report of some of the work containedthis paper has appeared previously as Ref.@25#.

II. FINDING COMMUNITIES IN A NETWORK

In this paper, we present a class of new algorithmsnetwork clustering, i.e., the discovery of community struture in networks. Our discussion focuses primarily on nworks with only a single type of vertex and a single typeundirected, unweighted edge, although generalizationsmore complicated network types are certainly possible.

There are two central features that distinguish our alrithms from those that have preceded them. First, our a

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FINDING AND EVALUATING COMMUNITY STRUCTURE . . . PHYSICAL REVIEW E 69, 026113 ~2004!

rithms are divisive rather than agglomerative. Divisive algrithms have occasionally been studied in the past, butdiscussed in the Introduction, ours differ in focusing notremoving the edges between vertex pairs with the lowsimilarity, but on finding edges with the highest ‘‘betweeness,’’ where betweenness is some measure that favors ethat lie between communities and disfavors those thatinside communities.

To make things more concrete, we give some examplethe types of betweenness measures we will be looking atof them are based on the same idea. If two communitiesjoined by only a few intercommunity edges, then all pathrough the network from vertices in one community to vtices in the other must pass along one of those few edGiven a suitable set of paths, one can count how manyalong each edge in the graph, and this number we thenpect to be largest for the intercommunity edges, thus proing a method for identifying them. Our different measurcorrespond to various implementations of this idea aslows:

~i! The simplest example of such a betweenness meais that based on shortest~geodesic! paths: we find the shortest paths between all pairs of vertices and count how mrun along each edge. To the best of our knowledge,measure was first introduced by Anthonisse in a nevpublished technical report in 1971@26#. Anthonisse called it‘‘rush,’’ but we prefer the termedge betweenness, since thequantity is a natural generalization to edges of the wknown ~vertex! betweenness measure of Freeman@27#,which was the inspiration for our approach. When we neto distinguish it from the other betweenness measuressidered in this paper, we will refer to it asshortest-path be-tweenness. A fast algorithm for calculating the shortest-pabetweenness is given in Sec. III A.

~ii ! The shortest-path betweenness can be thought oterms of signals traveling through a network. If signals trafrom source to destination along geodesic network paths,all vertices send signals at the same constant rate to allers, then the betweenness is a measure of the rate at wsignals pass along each edge. Suppose, however, that sido not travel along geodesic paths, but instead just perforrandom walk about the network until they reach their denation. This gives us another measure on edges, therandom-walk betweenness: we calculate the expected net numbertimes that a random walk between a particular pair of veces will pass down a particular edge and sum over all vepairs. The random-walk betweenness can be calculated umatrix methods, as described in Sec. III C.

~iii ! Another betweenness measure is motivated by idfrom elementary circuit theory. We consider the circuit cated by placing a unit resistance on each edge of the netwand unit current source and sink at a particular pair of veces. The resulting current flow in the network will travfrom source to sink along a multitude of paths, those wleast resistance carrying the greatest fraction of the currThecurrent-flow betweennessfor an edge we define to be thabsolute value of the current along the edge summed ovesource/sink pairs. It can be calculated using Kirchholaws, as described in Sec. III B. In fact, as we will show, t

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current-flow betweenness turns out to be exactly equal torandom-walk betweenness of the previous paragraph, bunonetheless consider it separately since it leads to a simderivation of the measure.

These measures are only suggestions; many otherspossible and may well be appropriate for specific applitions. Measures~i! and~ii ! are in some sense extremes in tspectrum of possibilities, one corresponding to signals tknow exactly where they are going, and the other to signthat have no idea where they are going. As we will showever, these two measures actually give rather similarsults, indicating that the precise choice of betweenness msure may not, at least for the types of applications considehere, be that important.

The second way in which our methods differ from prevous ones is in the inclusion of a ‘‘recalculation step’’ in thalgorithm. If we were to perform a standard divisive clusting based on edge betweenness, we would calculate thebetweenness for all edges in the network and then remedges in decreasing order of betweenness to produce adrogram like that of Fig. 2, showing the order in which thnetwork split up.

However, once the first edge in the network is removedsuch an algorithm, the betweenness values for the remaiedges will no longer reflect the network as it now is. This cgive rise to unwanted behaviors. For example, if two comunities are joined by two edges, but, for one reasonanother, most paths between the two flow along just onethose edges, then that edge will have a high betweenscore and the other will not. An algorithm that calculatbetweennesses only once and then removed edges intweenness order would remove the first edge early incourse of its operation, but the second might not get remountil much later. Thus the obvious division of the netwointo two parts might not be discovered by the algorithm.the worst case, the two parts themselves might be indivially broken up before the division between the two is maIn practice, problems like this crop up in real networks wsome regularity and render algorithms of this type ineffectfor the discovery of community structure.

The solution, luckily, is obvious. We simply recalculaour betweenness measure after the removal of each eThis certainly adds to the computational effort of performithe calculation, but its effect on the results is so desirablewe consider the price worth paying.

Thus the general form of our community structure findialgorithm is as follows:

~i! Calculate betweenness scores for all edges in thework.

~ii ! Find the edge with the highest score and removefrom the network.~If two or more edges tie for highest scorchoose one of them at random and remove that.!

~iii ! Recalculate betweenness for all remaining edges.~iv! Repeat from step~ii !.In fact, it appears that the recalculation step is the m

important feature of the algorithm, as far as getting satisftory results is concerned. As mentioned above, our stuindicate that, once one hits on the idea of using betweennmeasures to weight edges, the exact measure one use

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pears not to influence the results highly. The recalculatstep, on the other hand, is absolutely crucial to the operaof our methods. This step was missing from previoustempts at solving the clustering problem using divisive algrithms, and yet without it the results are very poor indefailing to find known community structure even in the simplest of cases. In Sec. V B we give an example compathe performance of the algorithm on a particular netwowith and without the recalculation step.

In the following sections, we discuss implementation agive examples of our algorithms for finding communistructure. For the reader who merely wants to know walgorithm they should use for their own problem, let us gan immediate answer: for most problems, we recommendalgorithm with betweenness scores calculated usingshortest-path betweenness measure~i! above. This measureappears to work well and is the quickest to calculate—described in Sec. III A, it can be calculated for all edgestime O(mn), wherem is the number of edges in the grapandn is the number of vertices@48#. This is the only versionof the algorithm that we discussed in Ref.@25#. The otherversions we discuss, while being of some pedagogical inest, make greater computational demands, and in praseem to give results no better than the shortest-path met

III. IMPLEMENTATION

In theory, the descriptions of the preceding section copletely define the methods we consider in this paper, bupractice there are a number of subtleties to their implemtation that are important for turning the description intoworkable computer algorithm.

Essentially all of the work in the algorithm is in the caculation of the betweenness scores for the edges; the jofinding and removing the highest-scoring edge is trivial anot computationally demanding. Let us tackle our three sgested betweenness measures in turn.

A. Shortest-path betweenness

At first sight, it appears that calculating the edge betweness measure based on geodesic paths for all edges willO(mn2) operations on a graph withm edges andn vertices:calculating the shortest path between a particular pair oftices can be done using breadth-first search in timeO(m)@28,29#, and there areO(n2) vertex pairs. Recently, howevenew algorithms have been proposed by Newman@30# andindependently by Brandes@31# that can perform the calculation faster than this, finding all betweennesses inO(mn)time. Both Newman and Brandes gave algorithms forstandard Freeman vertex betweenness, but it is triviaadapt their algorithms for edge betweenness. We describresulting method here for the algorithm of Newman.

Breadth-first search can find shortest paths from a sinvertex s to all others in timeO(m). In the simplest casewhen there is only a single shortest path from the souvertex to any other~we will consider other cases in a moment!, the resulting set of paths forms a shortest-path tresee Fig. 4~a!. We can use this tree to calculate the contrib

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tion to betweenness for each edge from this set of pathfollows. We find first the ‘‘leaves’’ of the tree, i.e., thosnodes such that no shortest paths to other nodes pass thrthem, and we assign a score of 1 to the single edgeconnects each to the rest of the tree, as shown in the figThen, starting with those edges that are farthest fromsource vertex on the tree, i.e., lowest in Fig. 4~a!, we workupwards, assigning a score to each edge that is 1 plussum of the scores on the neighboring edges immediatelylow it ~i.e., those edges with which it shares a common vtex!. When we have gone though all edges in the tree,resulting scores are the betweenness counts for the pfrom vertexs. Repeating the process for all possible vertics and summing the scores, we arrive at the full betweennscores for shortest paths between all pairs. The breadthsearch and the process of working up through the tree btake worst-case timeO(m) and there aren vertices total, sothe entire calculation takes timeO(mn) as claimed.

This simple case serves to illustrate the basic princibehind the algorithm. In general, however, it is not the cathat there is only a single shortest path between any pavertices. Most networks have at least some vertex pairstween which there are two or more geodesic paths of eqlength. Figure 4~b! shows a simple example of a shortepath ‘‘tree’’ for a network with this property. The resultinstructure is in fact no longer a tree, and in such cases an estep is required in the algorithm to calculate the betweenncorrectly.

In the traditional definition of vertex betweenness@27#,multiple shortest paths between a pair of vertices are giequal weights summing to 1. For example, if there are thshortest paths, each will be given weight1

3. We adopt thesame definition for our edge betweenness~as did Anthonissein his original work@26#, although other definitions are pos

FIG. 4. Calculation of shortest-path betweenness:~a! Whenthere is only a single shortest path from a source vertexs ~top! to allother reachable vertices, those paths necessarily form a tree, wmakes the calculation of the contribution to betweenness fromset of paths particularly simple, as described in the text.~b! Forcases in which there is more than one shortest path to some verthe calculation is more complex. First we must calculate the numof distinct paths from the sources to each vertex~numbers onvertices!, and then these are used to weight the path countsdescribed in the text. In either case, we can check the resultconfirming that the sum of the betweennesses of the edgesnected to the source vertex is equal to the total number of reachvertices—six in each of the cases illustrated here.

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sible @32#!. Note that the paths may run along the same eor edges for some part of their length, resulting in edges wgreater weight. To calculate correctly what fraction of tpaths flows along each edge in the network, we generathe breadth-first search part of the calculation, as follows

Consider Fig. 4~b! and suppose we are performingbreadth-first search starting at vertexs. We carry out thefollowing steps:

~i! The initial vertexs is given distanceds50 and weightws51.

~ii ! Every vertexi adjacent tos is given distancedi5ds1151 and weightwi5ws51.

~iii ! For each vertexj adjacent to one ofthoseverticesi,we do one of three things:~a! If j has not yet been assignea distance, it is assigned distancedj5di11 and weightwj5wi ; ~b! if j has already been assigned a distance anddj5di11, then the vertex’s weight is increased bywi , that is,wj←wj1wi ; and ~c! if j has already been assigned a dtance anddj,di11, we do nothing.

~iv! Repeat from step~iii ! until no vertices remain thahave assigned distances but whose neighbors do notassigned distances.

In practice, this algorithm can be implemented most eciently using a queue or first-in/first-out buffer to store tvertices that have been assigned a distance, just as instandard breadth-first search.

Physically, the weight on a vertexi represents the numbeof distinct paths from the source vertex toi. These weightsare precisely what we need to calculate our edge betwnesses, because if two verticesi and j are connected, withjfarther thani from the sources, then the fraction of a geodesic path fromj throughi to s is given bywi /wj . Thus, tocalculate the contribution to edge betweenness fromshortest paths starting ats, we need only carry out the following steps:

~i! Find every ‘‘leaf’’ vertex t, i.e., a vertex such that npaths froms to other vertices go thought.

~ii ! For each vertexi neighboringt, assign a score to thedge fromt to i of wi /wt .

~iii ! Now, starting with the edges that are farthest fromsource vertexs—lower down in a diagram such as Fig4~b!—work up towardss. To the edge from vertexi to vertexj, with j being farther froms than i, assign a score that isplus the sum of the scores on the neighboring edges imdiately below it~i.e., those with which it shares a commovertex!, all multiplied bywi /wj .

~iv! Repeat from step~iii ! until vertexs is reached.Now repeating this process for alln source verticess and

summing the resulting scores on the edges gives us thebetweenness for all edges in timeO(mn).

We have to repeat this calculation for each edge remofrom the network, of which there arem, and hence the complete community structure algorithm based on shortest-pbetweenness operates in worst-case timeO(m2n), or O(n3)time on a sparse graph. In our experience, this typicmakes it tractable for networks of up to aboutn510 000vertices, with current~circa 2003! desktop computers. Insome special cases one can do better. In particular, wethat the removal of an edge only affects the betweennes

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other edges that fall in the same component, and hencewe need only recalculate betweennesses in that componNetworks with strong community structure often break apinto separate components quite early in the progress ofalgorithm, substantially reducing the amount of work thneeds to be done on subsequent steps. Whether this resua change in the computational complexity of the algorithfor any commonly occurring classes of graphs is an opquestion, but it certainly gives a substantial speed boosmany of the calculations described in this paper.

Some networks are directed, i.e., their edges run indirection only. The world wide web is an example; linksthe web point in one direction only from one web pageanother. One could imagine a generalization of the shortpath betweenness that allowed for directed edges by coing only those paths that travel in the forward direction aloedges. Such a calculation is a trivial variation on the odescribed above. However, we have found that in many cait is better to ignore the directed nature of a network in cculating community structure. Often an edge acts simplyan indication of a connection between two nodes, anddirection is unimportant. For example, in Ref.@25# we ap-plied our algorithm to a food web of predator-prey interations between marine species. Predator-prey interactionsclearly directed—one species may eat another, but it islikely that the reverse is simultaneously true. However, asas community structure goes, we want to know only whspecies have interactions with which others. We find, thefore, that our algorithm applied to the undirected versionthe food web works well at picking out the community struture, and no special algorithm is needed for the directed cWe give another example of our method applied to a direcgraph in Sec. V D.

B. Resistor networks

As examples of betweenness measures that take mthan just shortest paths into account, we proposed in Semeasures based on random walks and on current flowresistor networks. In fact, there are well known mathematconnections between random walks and resistor netwo@33#, and the properties of one can often be calculatedconsidering the other. This turns out to be the case hereand, as we now show, when appropriately defined,random-walk and current-flow betweenness measuresprecisely the same. Here we derive the current-flow measfirst, since it turns out to be simpler; in the following sectiowe derive the random-walk measure and show that theare equivalent.

Consider the network created by placing a unit resistaon every edge of our network, a unit current source at vers, and a unit current sink at vertext ~see Fig. 5!. Clearly, thecurrent betweens andt will flow primarily along short paths,but some will flow along longer ones, roughly in inversproportion to their length. We will use the absolute magtude of the current flow along an edge, summed oversource/sink pairs, as our betweenness score.

The current flows in the network are governed by Kirchoff’s laws. To solve them, we proceed as follows for ea

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separate component of the graph. LetVi be the voltage atvertex i, measured relative to any convenient point. Thenall i we have

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whereAi j is the ij element of the adjacency matrix of thgraph, i.e.,Ai j 51 if i and j are connected by an edge anAi j 50 otherwise. The left-hand side of Eq.~1! represents thenet current flow out of vertexi along edges of the networkand the right-hand side represents the source and sink.fining ki5( jAi j , which is the vertex degree, and creatingdiagonal matrixD with these degrees on the diagonalDii5ki , this equation can be written in matrix form as (D2A)•V5s, where the source vectors has components

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We cannot directly invert the matrixD2A to get the volt-age vectorV, because the matrix~which is just the graphLaplacian! is singular. This is equivalent to saying that theis one undetermined degree of freedom corresponding tochoice of reference potential for measuring the voltages.can add any constant to a solution for the vertex voltagesget another solution—only the voltage differences matterchoosing the reference potential, we fix this degree of frdom, leaving onlyn21 more to be determined. In mathematical terms, once anyn21 of the equations in our matrixformulation are satisfied, the remaining one is also automcally satisfied so long as current is conserved in the netwas a whole, i.e., so long as( isi50, which is clearly true inthis case.

Choosing any vertexv to be the reference point, therefore, we remove the row and column corresponding to tvertex fromD andA before inverting. Denoting the resultin(n21)3(n21) matricesDv andAv , we can then write

V5~Dv2Av!21•s. ~3!

Calculation of the currents in the network thus involvinverting Dv2Av once for any convenient choice ofv, and

FIG. 5. An example of the type of resistor network considehere, in which a unit resistance is placed on each edge andcurrent flows into and out of the source and sink vertices.

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taking the differences of pairs of columns to get the voltavectorV for each possible source/sink pair.~The voltage forthe one missing vertexv is always zero, by hypothesis.! Theabsolute magnitudes of the differences of voltages aleach edge give us betweenness scores for the given soand sink. Summing over all sources and sinks, we thenour complete betweenness score.

The matrix inversion takes timeO(n3) in the worst case,while the subsequent calculation of betweennesses takesO(mn2), where as beforem is the number of edges andn thenumber of vertices in the graph. Thus, the entire communstructure algorithm, including the recalculation step, wtakeO„(n1m)mn2

… time to complete, orO(n4) on a sparsegraph. Although, as we will see, the algorithm is goodfinding community structure, this poor performance makepractical only for smaller graphs; a few hundreds of verticis the most that we have been able to do. It is for this reathat we recommend using the shortest-path betweennesgorithm in most cases, which gives results about as goobetter with considerably less effort.

C. Random walks

The random-walk betweenness described in Sec. IIquires us to calculate how often on average random wastarting at vertexs will pass down a particular edge fromvertexv to vertexw ~or vice versa! before finding their wayto a given target vertext. To calculate this quantity, we proceed as follows for each separate component of the grap

As before, letAi j be an element of the adjacency matrsuch thatAi j 51 if verticesi and j are connected by an edgandAi j 50 otherwise. Consider a random walk that on eastep decides uniformly between the neighbors of the curvertex j and takes a step to one of them. The numberneighbors is just the degree of the vertexkj5( iAi j , and theprobability for the transition fromj to i is Ai j /kj , which wecan regard as an element of the matrixM5A•D21, whereDis the diagonal matrix withDii 5ki .

We are interested in walks that terminate when they rethe targett, so thatt is an absorbing state. The most convnient way to represent this is just to remove entirely tvertex t from the graph, so that no walk ever reaches aother vertex fromt. Thus letM t5At•Dt

21 be the matrixMwith the tth row and column removed~and similarly forAtandDt).

Now the probability that a walk starts ats, takesn steps,and ends up at some other vertex~not t! is given by theiselement of M t

n , which we denote@M tn# is . In particular,

walks end up atv and w with probabilities @M tn#vs and

@M tn#ws , and of those a fraction 1/kv and 1/kw , respectively,

then pass along the edge (v,w) in one direction or the otherassuming such an edge exists.~Note that they may also havpassed along this edge an arbitrary number of times bereaching this point.! Summing over alln, the mean numberof times that a walk of any length traverses the edge fromvto w is kv

21@(I2M t)21#vs , and similarly for walks that go

from w to v.To highlight the similarity with the current-flow between

ness of Sec. III B, let us denote these two numbersVv and

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Now we define our random-walk betweenness foredge (v,w) to be the absolute value of thedifferenceof thetwo probabilitiesVv and Vw , i.e., the net number of timethe walk passes along the edge in one direction. This sea natural definition—it makes little sense to accord an ehigh betweenness simply because a walk went back andalong it many times. It is the difference between the numbof times the edge is traversed in either direction that mat@49#.

But now we see that this method is very similar to tresistor network calculation of Sec. III B. In that calculatiowe also evaluated (Dt2At)

21•s for a suitable source vecto

and then took differences of the resulting numbers. The odifference is that in the current-flow calculation we hadsink term ins as well as a source. Purely for the purposesmathematical convenience, we can add such a sink inpresent case at the target vertext—this makes no differenceto the solution forV since thetth row has been removefrom the equations anyway. By doing this, however, we tuthe equations into precisely the form of the current-flow cculation, and hence it becomes clear that the two measare numerically identical, although their derivation is qudifferent. ~It also immediately follows that we can removany row or column and still get the same answer—it doeshave to be row and columnt, although physically this choicemakes the most sense.!

IV. QUANTIFYING THE STRENGTHOF COMMUNITY STRUCTURE

As we show in Sec. V, our community structure algrithms do an excellent job of recovering known communitboth in artificially generated random networks and in reworld examples. However, in practical situations the algrithms will normally be used on networks for which the communities are not known ahead of time. This raises a nproblem: how do we know when the communities foundthe algorithm are good ones? Our algorithms always prodsomedivision of the network into communities, even in completely random networks that have no meaningful commnity structure, so it would be useful to have some waysaying how good the structure found is. Furthermore,algorithms’ output is in the form of a dendrogram whicrepresents an entire nested hierarchy of possible commudivisions for the network. We would like to know which othese divisions are the best ones for a given network—whwe should cut the dendrogram to get a sensible divisionthe network.

To answer these questions, we now define a measurthe quality of a particular division of a network, which wcall the modularity. This measure is based on a previomeasure of assortative mixing proposed by Newman@34#.

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Consider a particular division of a network intok communi-ties. Let us define ak3k symmetric matrixe whose elementei j is the fraction of all edges in the network that link verces in communityi to vertices in communityj @50#. ~Here weconsider all edges in the original network—even after edhave been removed by the community structure algorithour modularity measure is calculated using the full networ!

The trace of this matrix Tre5( ieii gives the fraction ofedges in the network that connect vertices in the same cmunity, and clearly a good division into communities shouhave a high value of this trace. The trace on its own, hoever, is not a good indicator of the quality of the divisiosince, for example, placing all vertices in a single communwould give the maximal value of Tre51 while giving noinformation about community structure at all.

So we further define the row~or column! sums ai5( jei j , which represent the fraction of edges that connto vertices in communityi. In a network in which edges falbetween vertices without regard for the communities thbelong to, we would haveei j 5aiaj . Thus we can define amodularity measure by

Q5(i

~eii 2ai2!5Tr e2ie2i , ~5!

whereixi indicates the sum of the elements of the matrixx.This quantity measures the fraction of the edges in thework that connect vertices of the same type~i.e., within-community edges! minus the expected value of the samquantity in a network with the same community divisions brandom connections between the vertices. If the numbewithin-community edges is no better than random, we wgetQ50. Values approachingQ51, which is the maximum,indicate networks with strong community structure@51#. Inpractice, values for such networks typically fall in the ranfrom about 0.3 to 0.7. Higher values are rare.

The expected error onQ can be calculated by treatineach edge in the network as an independent measuremethe contributions to the elements of the matrixe. A simplejackknife procedure works well@34,35#.

Typically, we will calculateQ for each split of a networkinto communities as we move down the dendrogram, alook for local peaks in its value, which indicate particularsatisfactory splits. Usually we find that there are only onetwo such peaks, and, as we will show in the next sectioncases where the community structure is known beforehby some means, we find that the positions of these pecorrespond closely to the expected divisions. The heightpeak is a measure of the strength of the community divisi

V. APPLICATIONS

In this section, we give a number of applications of oalgorithms to particular problems, illustrating their operatiand their use in understanding the structure of complexworks.

3-7

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FIG. 6. Plot of the modularity and dendrogram for a 64-vertex random community-structured graph generated as described iwith, in this case,zin56 andzout52. The shapes at the bottom denote the four communities in the graph and, as we can see, the pemodularity ~dotted line! corresponds to a perfect identification of the communities.

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A. Tests on computer-generated networks

First, as a controlled test of how well our algorithms pform, we have generated networks with known communstructure, to see if the algorithms can recognize and extthis structure.

We have generated a large number of graphs withn5128 vertices, divided into four communities of 32 verticeach. Edges were placed independently at random betwvertex pairs with probabilitypin for an edge to fall betweenvertices in the same community andpout to fall between ver-tices in different communities. The values ofpin and poutwere chosen to make the expected degree of each veequal to 16. In Fig. 6, we show a typical dendrogram frothe analysis of such a graph using the shortest-path betwness version of our algorithm.~In fact, for the sake of clarity,the figure is for a 64-node version of the graph.! Results forthe random-walk version are similar. At the right of the fiure we also show the modularity, Eq.~5!, for the same cal-culation, plotted as a function of position in the dendrograThat is, the plot is aligned with the dendrogram so that ocan read off modularity values for different divisions of thnetwork directly. As we can see, the modularity has a sinclear peak at the point where the network breaks into fcommunities, as we would expect. The peak value is aro0.5, which is typical.

In Fig. 7, we show the fraction of vertices in oucomputer-generated network sample classified correctlythe four communities by our algorithms, as a function of tmean numberzout of edges from each vertex to verticesother communities. As the figure shows, both the shortpath and random-walk versions of the algorithm performcellently, with more than 90% of all vertices classified corectly from zout50 all the way to aroundzout56. Only forzout*6 does the classification begin to deteriorate markeIn other words, our algorithm correctly identifies the community structure in the network almost all the way to tpoint zout58 at which each vertex has on average the sa

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number of connections to vertices outside its community adoes to those inside.

The shortest-path version of the algorithm does, howeperform noticeably better than the random-walk version,pecially for the more difficult cases wherezout is large. Giventhat the random-walk algorithm is also more computationademanding, there seems little reason to use it rather thanshortest-path algorithm, and hence, as discussed previowe recommend the latter for most applications.~To be fair,the random-walk algorithm does slightly outperform tshortest-path algorithm in the example addressed in thelowing section, although, being only a single case, it is h

FIG. 7. The fraction of vertices correctly identified by our algrithms in the computer-generated graphs described in the text.two curves show results for the shortest-path~circles! and random-walk ~squares! versions of the algorithm as a function of the number of edges the vertices have to others outside their own comnity. The pointzout58 at the rightmost edge of the plot representhe point at which vertices have as many connections outsideown community as inside it. Each data point is an average overgraphs.

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to know whether this is significant.!

B. Zachary’s karate club network

We now turn to applications of our methods to real-wonetwork data. Our first such example is taken from one ofclassic studies in social network analysis. Over the courstwo years in the early 1970s, Wayne Zachary observed sointeractions between the members of a karate club aAmerican university@36#. He constructed networks of tiebetween members of the club based on their social intetions both within the club and outside it. By chance, a dpute arose during the course of his study between the cladministrator and its principal karate teacher over whetheraise club fees, and as a result the club eventually splitwo, forming two smaller clubs, centered around the admistrator and the teacher.

In Fig. 8, we show a consensus network structuretracted from Zachary’s observations before the split. Feedthis network into our algorithms, we find the results shownFig. 9. In the leftmost two panels, we show the dendrogragenerated by the shortest-path and random-walk versionour algorithm, along with the modularity measures for tsame. As we see, both algorithms give reasonably highues for the modularity when the network is split into twcommunities—around 0.4 in each case—indicating that this a strong natural division at this level. What is more, tdivisions in question correspond almost perfectly to thetual divisions in the club revealed by which group each cmember joined after the club split up.~The shapes of thevertices representing the two factions are the same as tof Fig. 8.! Only one vertex, vertex 3, is misclassified by tshortest-path version of the method, and none are misclafied by the random-walk version—the latter gets a perfscore on this test.~On the other hand, the two-communisplit fails to produce a local maximum in the modularity fthe random-walk method, unlike the shortest-path methfor which there is a local maximum precisely at this poin!

FIG. 8. The network of friendships between individuals in tkarate club study of Zachary@36#. The administrator and the instructor are represented by nodes 1 and 33, respectively. Shsquares represent individuals who ended up aligning with the cladministrator after the fission of the club, open circles those waligned with the instructor.

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In the last panel of Fig. 9, we show the dendrogram amodularity for an algorithm based on shortest-path betweness but without the crucial recalculation step discusseSec. II. As the figure shows, without this step, the algorithfails to find the division of the network into the two knowgroups. Furthermore, the modularity does not reach nesuch high values as in the first two panels, indicating thatdivisions suggested are much poorer than in the casesthe recalculation.

C. Collaboration network

For our next example, we look at a collaboration netwoof scientists. Figure 10~a! shows the largest component ofnetwork of collaborations between physicists who condresearch on networks.~The authors of the present paper, finstance, are among the nodes in this network.! This network~which appeared previously in Ref.@37#! was constructed bytaking names of authors appearing in the lengthy bibliogphy of Ref.@4# and cross-referencing with the Physics e-prArchive at arxiv.org, specifically the condensed-matter stion of the archive, where, for historical reasons, most papon networks have appeared. Authors appearing in both wadded to the network as vertices, and edges between tindicate coauthorship of one or more papers appearing inarchive. Thus the collaborative ties represented in the figare not limited to papers on topics concerning networks—were interested primarily in whether people know one aother, and collaboration on any topic is a reasonable indtor of acquaintance.

The network as presented in Fig. 10~a! is difficult to in-terpret. Given the names of the scientists, knowledgeareaders with too much time on their hands could, no doupick out known groupings, for instance at particular instittions, from the general confusion. But were this a netwoabout which we had noa priori knowledge, we would behard pressed to understand its underlying structure.

Applying the shortest-path version of our algorithm to thnetwork, we find that the modularity, Eq.~5!, has a strongpeak at 13 communities with a value ofQ50.7260.02. Ex-tracting the communities from the corresponding dendgram, we have indicated them with colors in Fig. 10~b!. Theknowledgeable reader will again be able to discern knogroups of scientists in this rendering, and more easily nwith the help of the colors. Still, however, the structure of tnetwork as a whole and of the interactions between groupquite unclear.

In Fig. 10~c!, we have reduced the network toonly thegroups. In this panel, we have drawn each group as a cirwith size varying roughly with the number of individuals ithe group. The lines between groups indicate collaboratibetween group members, with the thickness of the linvarying in proportion to the number of pairs of scientiswho have collaborated. Now the overall structure of the nwork becomes easy to see. The network is centered arothe large group in the middle~which consists of researcherprimarily in southern Europe!, with a knot of intercommu-nity collaborations going on between the groups on the lowright of the picture~mostly Boston University physicists an

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FIG. 9. Community structure in the karate club network. Left: the dendrogram extracted by the shortest-path betweenness versmethod and the resulting modularity. The modularity has two maxima~dotted lines! corresponding to splits into two communities~whichmatch closely the real-world split of the club, as denoted by the shapes of the vertices! and five communities~though one of those fivecontains only one individual!. Only one individual, number 3, is incorrectly classified in the two-community split. Center: the dendrofor the random-walk version of our method. This version classifies all 34 vertices correctly into the factions that they actually split in~firstdotted line!, although the split into four communities gets a higher modularity score~second dotted line!. Right: the dendrogram for theshortest-path algorithm without recalculation of betweennesses after each edge removal. This version of the calculation fails to fininto the two factions.

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their intellectual descendants!. Other groups~including theauthors’ own! are scattered further out and more loosely conected to one another.

One of the problems created by the sudden availabilityrecent years of large network data sets has been our lactools for visualizing their structure@4#. In the early days ofnetwork analysis, particularly in the social sciences, it wusually enough simply to draw a picture of a network to swhat was going on. Networks in those days had tentwenty nodes, not 140 as here, or several billion as inworld wide web. We believe that methods like the one psented here, of using community structure algorithmsmake a meaningful ‘‘coarse graining’’ of a network, therereducing its level of complexity to one that can be intepreted readily by the human eye, will be invaluable in heing us to understand the large-scale structure of thesenetwork data.

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D. Other examples

In this section, we briefly describe example applicatioof our methods to three further networks. The first is a nohuman social network, a network of dolphins, the seconnetwork of fictional characters, and the third not a socnetwork at all, but a network of web pages and the linbetween them.

In Fig. 11, we show the social network of a community62 bottlenose dolphins living in Doubtful Sound, NeZealand. The network was compiled by Lusseau@38# fromseven years of field studies of the dolphins, with ties betwdolphin pairs being established by observation of statisticsignificant frequent association. The network splits naturainto two large groups, represented by the circles and squin the figure, and the larger of the two also splits into fosmaller subgroups. The modularity isQ50.3860.08 for thesplit into two groups, and peaks at 0.5260.03 when the sub-

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FIG. 10. Illustration of the use of the community-structure algorithm to make sense of a complex network.~a! The initial network is anetwork of coauthorships between physicists who have published on topics related to networks. The figure shows only the largest cof the network, which contains 145 scientists. There are 90 more scientists in smaller components, which are not shown.~b! Application ofthe shortest-path betweenness version of the community-structure algorithm produces the communities indicated by the shavertices.~c! A coarse-graining of the network in which each community is represented by a single node, with edges representingrations between communities. The thickness of the edges is proportional to the number of pairs of collaborators between comClearly panel~c! reveals much that is not easily seen in the original network of panel~a!.

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group splitting is included also.The split into two groups appears to correspond to

known division of the dolphin community@39#. Lusseau re-ports that for a period of about two years during observatof the dolphins they separated into two groups alonglines found by our analysis, apparently because of the dispearance of individuals on the boundary between the groWhen some of these individuals later reappeared, thehalves of the network joined together once more. As Lusspoints out, developments of this kind illustrate that the dphin network is not merely a scientific curiosity but, likhuman social networks, is closely tied to the evolution of tcommunity. The subgroupings within the larger half of tnetwork also seem to correspond to real divisions amonganimals: the largest subgroup consists almost of entirelyfemales and the others almost entirely of males, and iconjectured that the split between the male groups is gerned by matrilineage@D. Lusseau~personal communication!#.

Figure 12 shows the community structure of the netwoof interactions between major characters in Victor Hugsprawling novel of crime and redemption in post-restorat

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FIG. 12. The network of interactions between major characters in the novelLes Miserables by Victor Hugo. The greatest modularitachieved in the shortest-path version of our algorithm isQ50.54 and corresponds to the 11 communities shown.
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France,Les Miserables. Using the list of character appeaances by scene compiled by Knuth@40#, the network wasconstructed in which the vertices represent characters anedge between two vertices represents coappearance ocorresponding characters in one or more scenes. The opcommunity split of the resulting graph has a strong moduity of Q50.5460.02, and gives 11 communities as shownthe figure. The communities clearly reflect the subplot strture of the book: unsurprisingly, the protagonist, Jean Vjean, and his nemesis, the police officer Javert, are centrthe network and form the hubs of communities composedtheir respective adherents. Other subplots centeredMarius, Cosette, Fantine, and the bishop Myriel are apicked out.

Finally, as an example of the application of our methoda nonsocial network, we have looked at a web graphnetwork in which the vertices and edges represent web pand the links between them. The graph in question repres180 pages from the web site of a large corporation@52#.Figure 13 shows the network and the communities foundit by the shortest-path version of our algorithm. This netwohas one of the strongest modularity values of the examstudied here, atQ50.6560.02. The links between wepages are directed, as indicated by the arrows in the figbut, as discussed in Sec. III A, for the purposes of findingcommunities, we ignore direction and treat the networkundirected.

Certainly it might be useful to know the communities inweb network; an algorithm that can pick out communitcould reveal which pages cover related topics or the sostructure of links between pages maintained by differentdividuals. Ideas along these lines have been pursued byexample, Flakeet al. @41# and Adamic and Adar@42#.

VI. CONCLUSIONS

In this paper, we have described a new class of algorithfor performing network clustering, the task of extracting tnatural community structure from networks of vertices a

FIG. 13. Pages on a web site and the hyperlinks between thThe different shades denote the optimal division into communifound by the shortest-path version of our algorithm.

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edges. This is a problem long studied in computer scienapplied mathematics, and the social sciences, but itlacked a satisfactory solution. We believe the methodsscribed here give such a solution. They are simple, intuitiand demonstrably give excellent results on networkswhich we know the community structure ahead of time. Omethods are defined by two crucial features. First, we us‘‘divisive’’ technique that iteratively removes edges from thnetwork, thereby breaking it up into communities. The edgto be removed are identified using one of a set of edgetweenness measures, of which the simplest is a generation to edges of the standard shortest-path betweennesFreeman@27#. Second, our algorithms include a recalculatistep in which betweenness scores are reevaluated afteremoval of every edge. This step, which was missing frprevious algorithms, turns out to be of primary importancethe success of ours. Without it, the algorithms fail miseraat even the simplest clustering tasks.

We have demonstrated the efficacy and utility of omethods with a number of examples. We have shownour algorithms can reliably and sensitively extract commnity structure from artificially generated networks wiknown communities. We have also applied them to reworld networks with known community structure and agathey extract that structure without difficulty. And we havgiven examples of how our algorithms can be used to alyze networks whose structure is otherwise difficult to coprehend. The networks studied include a collaboration nwork of scientists, in which our methods allow us to generschematic depictions of the overall structure of the netwand collaborations taking place within and between commnities, other social networks of people and of animals, annetwork of links between pages on a corporate web site.

The primary remaining difficulty with our algorithms ithe relatively high computational demands they make. Tfastest of them, the one based on shortest-path betweenoperates inO(n3) time on a sparse graph, which makesusable for networks up to about 10 000 vertices, butlarger systems it becomes intractable. Although the evimproving speed of computers will certainly raise this limin coming years, it would be more satisfactory if a fasversion of the method could be discovered. One possibilitparallelization: the betweenness calculation involves a sover source vertices and the elements of that sum candistributed over different processors, making the calculattrivially parallelizable on a distributed-memory machinHowever, a better approach would be to find some improment in the algorithm itself to decrease its computatiocomplexity.

Since the publication of our first paper on this topic@25#,several other authors have made use of the shortest-pathsion of our algorithm. Holmeet al. @43# have applied it to anumber of metabolic networks for different organisms, fining communities that correspond to functional units withthe networks, while Wilkinson and Huberman@44# have ap-plied it to a network of relations between genes, as eslished by the co-occurrence of names of genes in publisresearch articles. An interesting application to social nworks is the study by Gleiser and Danon@45# of the collabo-

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ration network of early jazz musicians. They found, amoother things, that the network split into two communitialong lines of race, with black musicians in one group awhite musicians in the other. Guimera` et al. @46# have ap-plied the method to a network of email messages pasbetween users at a university, and found communitiesreflect both formal and informal levels of organization. Tylet al. @47# have also applied the algorithm to an email nwork, in their case at a large company, finding that thesulting communities correspond closely to organizatiounits. The latter work is interesting also in that it suggestmethod for improving the speed of the algorithm. Tyleret al.calculate betweenness for only a subset, randomly chosepossible source vertices in the network, rather than summover all sources. The size of the subset is decided on theby sampling source vertices until the betweenness of at lone edge in the network exceeds a predetermined thresThis technique reduces the running time of the calculatconsiderably, although the resulting estimate of betweenn

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necessarily suffers from the statistical fluctuations inherenrandom sampling methods. This idea, or a variation ofmight provide a solution to the problems mentioned abovethe high computational demands of our algorithms.

We are, of course, delighted to see our methods appliesuch a variety of problems. Combined with the algorithmand measures described in this paper, we hope to see mmore applications in the future.

ACKNOWLEDGMENTS

The authors thank Steven Borgatti, Ulrik Brandes, LintFreeman, David Lusseau, Mason Porter, and Douglas Wfor useful comments. Thanks also to Oliver Boisseau, PHaase, David Lusseau, and Karsten Schneider for providthe data for the dolphin network and to Douglas Whitethe karate club data. This work was funded in part byNational Science Foundation under Grant No. DM0234188 and by the Santa Fe Institute.

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@48# Following the publication of Ref.@25#, the algorithm has beenimplemented in the software packagesUCINET and NETDRAW

and in the open-source network library JUNG.~See http://www.analytictech.com/ and http://jung.sourceforge.net/.!

@49# In fact, we have tried counting each traversal separately,this method gives extremely poor results, confirming our inition that this would not be a good betweenness measure.

@50# As discussed in@34#, it is crucial to make sure each edgecounted only once in the matrixei j —the same edge should no

02611

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appear both above and below the diagonal. Alternatively,edge linking communitiesi and j can be split, half-and-half,between theij and ji elements, which has the advantagemaking the matrix symmetric. Either way, there are a numof factors of 2 in the calculation that must be watched cafully, lest they escape one’s attention and make mischief.

@51# In Ref. @34#, the measure was normalized by dividing by ivalue on a network with perfect mixing, so that we always g1 for such a network. We find, however, that doing this in tpresent case masks some of the useful information to be gafrom the value ofQ, and hence that it is better to use thunnormalized measure. In general, this unnormalized meawill not reach a value of 1, even on a perfectly mixed netwo

@52# The graph is one of the test graphs from the graph drawcompetition held in conjunction with the Symposium on GraDrawing, Berkeley, California, September 18–20, 1996.

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