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Statistical Mechanics of Community Detection

Jorg Reichardt1 and Stefan Bornholdt1

1Institute for Theoretical Physics, University of Bremen, Otto-Hahn-Allee, D-28359 Bremen, Germany

(Dated: February 3, 2008)

Starting from a general ansatz, we show how community detection can be interpreted as finding theground state of an infinite range spin glass. Our approach applies to weighted and directed networksalike. It contains the at hoc introduced quality function from [1] and the modularity Q as definedby Newman and Girvan [2] as special cases. The community structure of the network is interpretedas the spin configuration that minimizes the energy of the spin glass with the spin states being thecommunity indices. We elucidate the properties of the ground state configuration to give a concisedefinition of communities as cohesive subgroups in networks that is adaptive to the specific class ofnetwork under study. Further we show, how hierarchies and overlap in the community structure canbe detected. Computationally effective local update rules for optimization procedures to find theground state are given. We show how the ansatz may be used to discover the community around agiven node without detecting all communities in the full network and we give benchmarks for theperformance of this extension. Finally, we give expectation values for the modularity of randomgraphs, which can be used in the assessment of statistical significance of community structure.

PACS numbers: 89.75.Hc,89.65.-s,05.50.+q,64.60.Cn

INTRODUCTION

The amount of empirical information that scientists fromall disciplines are dealing with is constantly increasing.And so is the need for robust, scalable, and easy to useclustering techniques for data abstraction, dimensionalityreduction, or visualization for many scientists performingexploratory data analysis [3, 4]. A basic objective is togroup objects which are similar together, and dissimilarobjects apart. But already the question of how to mea-sure similarity/dissimilarity is a subject of discussion [4].Two main approaches to clustering are identified in theliterature [4]: On one hand there is hierarchical cluster-ing where the data set is grouped into a hierarchy of clus-ters from single items to the whole data set. Data pointsare either joined successively in an agglomerative mannerstarting from the closest pair of data points or the dataset is recursively partitioned into two parts, an approachwhich is called divisive. On the other hand, there is par-titional clustering, where the data set is directly parti-tioned into k different clusters usually optimizing somequality function. The number of clusters k is either an in-put parameter of the algorithm or found by the clusteringprocedure itself. Transforming the similarity matrix intoa graph by e.g. thresholding, the clustering problem canbe tackled from a graph partitioning point of view. Theseapproaches apply directly to networks or relational datasets where the proximity information is given as a set ofpairwise relations, i.e. the edges of the network. Theproblem is then approached by a min-cut technique thatpartitions a connected graph into two parts minimizingthe number of edges to cut [5, 6, 7]. These approaches,however, suffer greatly from being very skewed as themin-cut is usually found by cutting off only a very smallsubgraph [8]. A number of penalty functions have been

suggested to overcome this problem and balance the sizeof subgraphs resulting from a cut. Among these are ratiocuts [8, 9], normalized cuts [10] or min-max cuts [7].

Though today the development of these methods liesmainly in the realm of computer science, the relations be-tween information theory and statistical physics [11, 12]have brought about a number of such methods that arebased on principles from statistical mechanics or analo-gies with physical models. When using spin models forclustering of multivariate data, the similarity measuresare translated into coupling strengths and either dynam-ical properties such as spin-spin correlations are mea-sured or energies are interpreted as quality functions. Aferromagnetic Potts model has been applied successfullyby Blatt et al. [13]. Bengtsson [14] has used an anti-ferromagnetic Potts model with the number of clustersk as input parameter and the assignment of spins in theground state of the system defines the clustering solution.

In recent years, renewed interest in the graph cluster-ing problem from the physics community has come un-der the term of “community detection”. As communities,one generally understands subsets of nodes that are moredensely interconnected among each other than with therest of the network. Sparked by the work of Girvan andNewman [15], a number of other authors have developednew algorithms for this problem that take very differ-ent approaches. The recent reviews by Newman [16] andDanon et al. [17] may serve as introductory reading andinclude methodological overviews and comparative stud-ies of the performance of different algorithms, includingthe one presented by the authors in [1]. In this article,we intend to set the basis for a unified framework underwhich community detection may be viewed and whichhelps in understanding the underlying properties of theproblem.

2

First, we will show that the problem of community de-tection can be mapped onto finding the ground state ofan infinite ranged Potts spin glass via a simple first prin-ciples ansatz by combining the information from bothpresent and missing links. The energy of the spin systemis equivalent to the quality function of the clustering withthe spins states being the group indices. In the abovetaxonomy of clustering procedures, this corresponds toa partitional method with the number of clusters deter-mined automatically by the algorithm as the number ofoccupied spin states. A single parameter γ relates theweight given to missing and existing links in the qual-ity function and allows for an assessment of overlappingand hierarchical community structures. Thereby, we canbridge the gap between hierarchical and partitional clus-tering and conclude to which extent the cluster structureof the network is hierarchical or not.

In contrast to methods based on dynamical proper-ties of the spin system that measure correlations betweenspins, such as the super-paramagnetic (SPC) Potts clus-tering introduced by Blatt et. al., mapping the problemto a ground state bears several advantages. First, it iscomputationally less demanding, because we do not haveto keep track of an N × N correlation matrix of spinstates. Rather, every spin only carries its most prob-able community index. If a probabilistic extension ofthe method is required, an analysis of the overlap of thecommunity structures in different local minima of theHamiltonian can be performed as done in [1]. Second,the properties of the ground state spin configuration leadto a direct interpretation of the result in terms of graphtheoretical measures, which give an exact definition ofwhat a “community” is in this framework. The interpre-tation of the parameter γ in the evaluation of hierarchyand overlap is much clearer than the interpretation of thetemperature in SPC. Third, the zero temperature energycan be calculated analytically which allows to give expec-tation values of the modularity and assess the clusteringtendency of the graph under study.

For a natural choice of parameters, we recover the“modularity” defined by Girvan and Newman [2] fromour ansatz as well as the ad hoc introduced quality func-tion from [1]. Then we will derive a number of graphstructural properties that define the term “community”from the fact that valid community structures correspond

to minima of the energy landscape of the system. Wecompare this definition to other possibilities from the lit-erature. We then show, how hierarchical and overlappingcommunity structures can be discovered in this frame-work. Even though the quality function resembles an in-finite ranged spin glass with couplings between all pairsof nodes, we show how efficient minimization routinescan be implemented that only need to consider interac-tions along the links in the network and some global bookkeeping. This makes the use of the method feasible evenfor large systems. Furthermore, we show how a methodof finding the community around a given node can be de-veloped in this general framework and give benchmarksfor this method. All clustering procedures will find clus-ters even when applied to random data. Hence, in thelast part of the paper, we focus on the statistical signif-icance of community detection. We show, how commu-nity detection is related to graph partitioning and thatwhen community detection is applied to random graphs,equally sized communities are found. From the knownresults for the cut size of graph partitionings we can cal-culate expectation values for the modularity of randomgraphs which have to be exceeded by any data set thatis to be called truly modular.

DERIVATION OF THE HAMILTONIAN

For the term “community” or “cluster” or “cohesive sub-group” a number of different and sometimes conflict-ing definitions exist [17]. All of them have in commonthat communities are understood as groups of denselyinterconnected nodes that are only sparsely connectedwith the rest of the network. Any quality function foran assignment of nodes into communities should there-fore follow the simple principle: group together what islinked, keep apart what is not. From this, we find fourrequirements of such a quality function: it should a.)reward internal edges between nodes of the same group(in the same spin state) and b.) penalize missing edges(non-links) between nodes in the same group. Further,it should c.) penalize existing edges between differentgroups (nodes in different spin state) and d.) rewardnon-links between different groups. This leads to the fol-lowing function:

H ({σ}) = −∑

i6=j

aij Aijδ(σi, σj)︸ ︷︷ ︸

internal links

+∑

i6=j

bij (1 − Aij)δ(σi, σj)︸ ︷︷ ︸

internal non-links

(1)

+∑

i6=j

cij Aij(1 − δ(σi, σj))︸ ︷︷ ︸

external links

−∑

i6=j

dij (1 − Aij)(1 − δ(σi, σj))︸ ︷︷ ︸

external non-links

in which Aij denotes the adjacency matrix of the graphwith Aij = 1, if an edge is present and zero otherwise,

σi ∈ {1, 2, ..., q} denotes the spin state (or group index) of

3

node i in the graph and aij , bij , cij , dij denote the weightsof the individual contributions, respectively. The num-ber of spin states q determines the maximum numberof groups allowed and can, in principle, be as large asN , the number of nodes in the network. Note, that notall group indices have to be used necessarily in the opti-mal assignment of nodes into communities, as some spinstates may remain unpopulated in the ground state. Iflinks and non-links are each weighted equally, regard-less whether they are external or internal, i.e. aij = cij

and bij = dij , then it is enough to consider the internallinks and non-links. It remains to find a sensible choice ofweights aij and bij , preferably such that the contributionof links and non-links can be adjusted through a param-eter. As we will see, a convenient choice is aij = 1− γpij

and bij = γpij , where pij denotes the probability thata link exists between node i and j, normalized, suchthat

∑

i6=j pij = 2M . For γ = 1 this leads to the nat-ural situation that the total amount of energy that canpossibly be contributed by links and non-links is equal:∑

i6=j Aijaij =∑

i6=j(1 − Aij)bij . For weighted networksthis approach is generalized in a straight-forward man-ner by using a weighted adjacency matrix Wij . In caseof a directed network with a non-symmetric adjacencymatrix Aij 6= Aij , one can construct a symmetric repre-

sentation of network introducing Aij = 1/2(Aij + Aji)and pij = 1/2(pij + pji). In this article, we will only dealwith undirected, unweighted adjacency matrices. Ourchoice of the weights allows us to further simplify theHamiltonian (2):

H ({σ}) = −∑

i6=j

(Aij − γpij) δ(σi, σj). (2)

This represents a spin glass with couplings Jij = Aij−pij

between all pairs of nodes: ferromagnetic where linksbetween nodes exist and anti-ferromagnetic where linksare absent.

Depending on the graph under study, one can assumedifferent expressions for pij . The Hamiltonian (2) is ef-fectively comparing the true distribution of links in thegraph under study with the expected distribution givenby a particular null model which defines pij . With thisin mind, we can rewrite (2) in the following two ways:

H ({σ}) = −∑

s

(mss − γ[mss]pij

)= −

∑

s

css (3)

and

H ({σ}) =∑

s<r

(mrs − γ[mrs]pij

)=

∑

s<r

ars. (4)

Here, the sum runs over the q spin states and mrs denotesthe number of edges between spins in group r and s.Consequently, the number of internal edges of group s isdenoted by mss. The symbol [·]pij

denotes an expectation

value under the assumption of a link distribution pij ,given the current assignment of spins.

In equation (3) and (4) we have also introduced thecoefficients of “cohesion” css and “adhesion” ars to ournetwork terminology, which measure the difference be-tween realized and expected internal links or realized andexpected external links, respectively. Note, that both de-pend on the choice of the model of connectivity pij andthe parameter γ. The choice of a particular form of pij al-lows for the adaptation of the quality function to the spe-cific problem under study and hence allows for the com-parison of the quality function for graphs with differenttopology. The only restriction on pij is that the numberof expected edges between and within groups is an exten-sive quantity, i.e. [m13]pij

+ [m23]pij= [m1+2,3]pij

for allchoices of disjoint groups n1, n2 and n3 and [m33]pij

=[m11]pij

+ [m22]pij+ [m12]pij

for all groups 3 with propersubgroups n1 and n2 of empty intersection and union n3.Using these equalities, we can give a relation for the coef-ficient of cohesion of a group of nodes ns and two propersubsets ns1 and ns2 with empty intersection and unionns. It is easy to prove, that

css = c11 + c22 + a12, (5)

where c11 and c22 are the coefficients of cohesion of therespective subsets ns1 and ns2, and a12 is the coefficientof adhesion between ns1 and ns2. Equivalently, we canwrite for the adhesion coefficients with n2 of two groupsnr1 and nr2 with union nr and empty intersection

ars = a1s + a2s. (6)

Two exemplary choices of link distribution models pij

shall illustrate the above. The simplest choice is to as-sume every link equally probable with probability pij = pwhich leads naturally to

[mss]p = pns(ns − 1)

2and [mrs]p = pnrns, (7)

with nr and ns denoting the number of spins in state rand s, respectively. This choice of model leads to theHamiltonian originally quoted in Ref. [1]:

H ({σ}) = −∑

i,j∈E

δ(σi, σj) + γp

q∑

s

ns(ns − 1)

2. (8)

Here, the first sum runs over all edges and only internaledges contribute. Equivalently, we can write (8) in termsof external edges:

H({σ}) =∑

i,j∈E

(1 − δ(σi, σj)) − γp

q∑

r<s

nrns, (9)

where only edges between different groups contribute tothe first sum. We see that both, (8) and (9), compare the

4

actual value of internal or external edges with its respec-tive expectation value under the assumption of equallyprobable links and given community sizes.

A second choice for pij may take into account, thatthe network does exhibit a particular degree distribution.Since links are in principle more probable between nodesof high degree, links between these nodes get a lowerweight. We may write:

pij =kikj

2M, (10)

which takes this fact and the degree distribution intoaccount. Note that it is possible to also include degree-degree correlations or any other form of prior knowledgeabout pij at this point. With these expressions we have:

[mss]pij=

1

2M

K2s

2and [mrs]pij

=1

2MKrKs. (11)

Here, Ks is the sum of degrees of nodes in spin state sand plays the role of the occupation numbers in equa-tion (8). Using these expressions, we can also write theHamiltonian (2) in a form similar to (8):

H ({σ}) = −∑

i,j∈E

δ(σi, σj) +γ

2M

q∑

s

K2s

2. (12)

Again, we give an equivalent formulation in terms of ex-ternal rather than internal edges similar to (9):

H ({σ}) =∑

i,j∈E

(1 − δ(σi, σj)) −γ

2M

q∑

r<s

KrKs. (13)

For γ = 1 and the model pij = kikj/2M , we can derive

2css +∑

r,r 6=s

asr = 0. (14)

Furthermore, the cohesion is negative (css < 0) if ns

consists of only one single node. We see, that there mustalways exist a group of nodes nr, to which this nodehas positive adhesion. Groups of only one node do notexist. We stress that relation (14) and the conclusionsjust drawn do not hold for γ 6= 1 or pij = p.

We see, that even though we are dealing with an in-finite range spin glass with couplings between all pairsof nodes, one only needs to consider the ferromagneticinteractions along the links and the occupation numbersor the sum of node degrees of the individual spin states.This makes it easy to implement an efficient minimiza-tion routine for this Hamiltonian. It should be noted,that both the formulations (8), (9) and (12), (13) areequivalent in case of a network with fixed connectivity.

EQUIVALENCE WITH NEWMAN-GIRVAN

MODULARITY

Comparing the performance in retrieving a known com-munity structure from computer generated test networks

may be used as a benchmark for different communitydetection algorithms. Alternatively, many authors havegiven values of the quality function Q defined by Newmanand Girvan as “modularity” [2] as a global, comparative,objective measure of how good a community structurefound by an algorithm is. Alternative formulations fo-cussing on the local aspects of community structure alsoexist, such as that of “local modularity” introduced byMuff et al. [18]. Newman and Girvans’s modularity mea-sure can be written as [2]:

Q =∑

s

ess − a2s, with as =

∑

r

ers. (15)

Here, ers is the fraction of links that fall between nodesin group r and s, i.e. the probability that a randomlydrawn link connects a node in group r to one in groups. The probability that a link has one end in group sis expressed by as. From this, we expect a fraction ofa2

s links to connect nodes in group s among themselves.Newman’s modularity measure hence compares the ac-tual link density in a community with an expectationvalue. One can write this modularity in a slightly differ-ent way following [19]:

Q =1

2M

∑

i6=j

(

Aij −kikj

2M

)

δ(σi, σj). (16)

This already resembles (2) when pij takes the formkikj/2M . It is now clear, that we can write:

Q = − 1

MH({σ}) (17)

with the Hamiltonian (2) and γ = 1. Therefore, max-imum modularity is reached, when the Hamiltonian (2)with pij = kikj/2M or equivalently (12) or (13) withγ = 1 are minimal. To maximize the modularity of acommunity structure is hence equivalent to finding thespin configuration that minimizes these Hamiltonians.This form of writing the modularity Q is much simplerthan the one given by Guimera et al. [20], which also in-volves 3- and 4-spin interactions. We will see below, thatusing this form, we can give efficient update rules thatallow the direct optimization of the modularity even onvery large networks.

PROPERTIES OF THE HAMILTONIAN AND ITS

GROUND STATE

Having mapped the problem of community finding ontofinding the ground state configuration of a spin glass,we can investigate the properties of this minimum en-ergy spin configuration. These properties will provide uswith a definition of what a community is in the frame-work of maximizing a quality function. These properties

5

will apply to any local minimum of the Hamiltonian aswell, such that we can interpret these local minima asalternative community structures. Inspection of the to-tal energy landscape and comparison of global and lo-cal minima and the respective community structure willthen provide insight into the clustering tendency of thenetwork. Obviously, the more local minima with littleoverlap but energies comparable to the global minimumthere are, the more spin glass like the energy landscapeis, the less the network shows a truly modular structure.

Since the Hamiltonians are all additive with respect tothe different communities, i.e. the numbers of edges andthe corresponding expectation values are extensive, theycan be seen as independent entities and we can treata single community independently from the rest of thenetwork. The configuration space over which the Hamil-tonian is minimized is a discrete space. Once we havedefined a move set that is ergodic in this discrete space,a (local) minimum of the Hamiltonian (with respect tothis move set) is defined as a configuration for which noneof the steps from the move set leads to a lower energy. Itis sufficient to consider only one move: change a groupof nodes n1 from spin state s to spin state r. The changein energy for this move in configuration space is:

∆H = a1,s\1 − a1r. (18)

Here a1,s\1 is the adhesion of n1 with its complement inns and a1r is the adhesion of n1 with nr. If we move n1 toa previously unpopulated spin state, then ∆H = a1,s\1.This move corresponds to dividing group ns. Further-more, if n1 = ns, we have ∆H = −asr, which correspondsto joining groups ns and nr. For a spin configuration tobe a local minimum of the Hamiltonian, there must notexist a move of this type that leads to a lower energy.It is clear that some moves may not change the energyand are hence called neutral moves. In case of equalitya1,s\1 = a1,r and nr being a community itself, we say thatcommunities ns and nr have an overlap of the nodes inn1.

For a community defined as a group of nodes with thesame spin state in a spin configuration that makes theHamiltonian minimal, we then have the following prop-erties:

1. Every proper subset n1 of a community ns hasa maximum coefficient of adhesion with its com-plement in the community compared to the co-efficient of adhesion with any other community(a1,s\1 = max).

2. The coefficient of cohesion is non-negative for allcommunities (css ≥ 0).

3. The coefficient of adhesion between any two com-munities is non-positive (ars ≤ 0).

The first property is proven by contradiction from thefact that we are dealing with a spin configuration that

makes the Hamiltonian minimal. We also see immedi-ately that every proper subset n1 of a community ns

must have a non-negative adhesion with its complementns\1 in the community. In particular this is true forevery single node l in ns (al,s\l ≥ 0). Then we canwrite

∑

l∈nsal,s\l ≥ 0. Since

∑

l∈nsml,s\l = 2mss and

∑

l∈ns[ml,s\l]pij

= 2[mss]pij, this implies css ≥ 0 for all

communities s and proves the second property. The thirdproperty is proven by contradiction again. Again, westress that for γ = 1 and pij = kikj/2M , no communityis formed of a single node due to condition (14). Thelast two properties can be summarized in the followinginequality which provides an intuition about the signifi-cance of the parameter γ:

css ≥ 0 ≥ ars ∀r, s. (19)

Assuming a constant link probability, we can rewrite thisinequality in order to relate the inner link density of acommunity and the outer link density between commu-nities with an average link density:

2mss

ns(ns − 1)≥ γp ≥ mrs

nrns∀r, s. (20)

We see, that γp can be interpreted as a threshold betweeninner and outer link density under the assumption of aconstant link probability. The above definition of whata community is adapts itself to any network, since thespecific network model is encoded in the definition of co-hesion and adhesion. This makes it possible to comparecommunity structures of networks with different topol-ogy.

SIMPLE DIVISIVE AND AGGLOMERATIVE

APPROACHES TO MODULARITY

MAXIMIZATION

Hierarchical clustering techniques can be dichotomizedinto divisive and agglomerative approaches [4]. We willshow, how a simple recursive divisive approach and anagglomerative approach may be implemented and wherethey fail.

In the present framework, a hierarchical divisive algo-rithm would mean to construct the ground state of theq-state Potts model by recursive partitioning the networkinto two parts according to the ground state of a 2-statePotts or Ising system. This procedure would be com-putationally simpler and result directly in a hierarchy ofclusters due to repetition of the procedure on the partsuntil the total energy cannot be lowered anymore. Sucha procedure would be justified, if the ground state of theq-state Potts Hamiltonian and the repeated applicationof the Ising system cut the network along the same edges.We will derive a condition under which this can be en-sured.

6

FIG. 1: Illustration of the problem of recursive bi-partitioning. The ground state of the Hamiltonian with only2 possible spin states, as shown in a.), would cut through oneof the communities that are found when allowing 3 spin statesas shown in b.).

In order for this recursive approach to work, we mustensure that the ground state of the 2-state Hamiltoniannever cuts though a community as defined by the q-stateHamiltonian. Assume a network made of three commu-nities n1, n2 and n3 as defined by the ground state ofthe q-state Hamiltonian. For the bi-partitioning, we nowhave two possible scenarios. Without loss of generality,the cut is made either between n2 and n1 + n3 or be-tween n1, n2 and n3 = na + nb, parting the network inton1+na and n2 +nb. Since the former situation should beenergetically lower for the algorithm to work, we arriveat the condition that

mab − [mab]pij+ m1b − [m1b]pij

> m2b − [m2b]pij, (21)

which must be valid for all subgroups na and nb of com-munity n3. Since n3 is a community, we further know,that mab−[mab]pij

> m1b−[m1b]pijand mab−[mab]pij

>m2b − [m2b]pij

. Though mab − [mab]pij> 0, since n3 is

a community, m1b − [m1b]pij< 0 and m2b − [m2b]pij

< 0for the same reason and hence condition (21) is not gen-erally satisfied. Figure 1 illustrates a counter example.Assuming pij = p, the link probability in the network.The upper part a.) of the figure shows the ground stateof the system when using only two spin states. Part b.)of the figure shows the ground state of the system with-out constraints on the number of spin states, resultingin a configuration of 3 communities. We see that the bi-partitioning approach would have cut through one of thecommunities in the network. Recursive bi-partitioningscannot generally lead to an optimal assigment of spinsthat maximizes the modularity.

In [21] Newman has introduced a fast greedy strategyfor modularity maximization. It effectively correspondsto a simple nearest neighbor agglomerative clustering ofthe network where the adhesion coefficient ars is used assimilarity measure between. Newman’s algorithms ini-tially assigns different spin states to every node and then

FIG. 2: Example network for which an agglomerative ap-proach of grouping together nodes of maximal adhesion willfail. Starting from an assignment of different spin states toevery node, the largest adhesion is found for the nodes con-nected by edge x and the nodes connected by x are groupedtogether first by the agglomerative procedure. However, it isclearly seen, that x should lie between different groups.

proceeds by grouping those nodes together that have thehighest coefficient of adhesion. As Figure 2 shows, thisapproach fails, if the links between two communities con-nect nodes of low degree. The network consists of 14nodes and 37 links. Is is clearly seen that in the groundstate, the network consists of two communities and edgex lies between them. However, when initially assigningdifferent spin states to all nodes, the adhesion betweenthe nodes connected by x is largest: a = 1 − 16/2M ,since the product of degrees at this edge is lowest. There-fore, the agglomerative procedure described is misled intogrouping together the nodes connected by x already inthe very first step. Furthermore, it is clear that in a net-work, where all nodes have the same degree initially, alledges connect nodes of the same coefficient of adhesion.In this case, it cannot be decided, which nodes to grouptogether in the first step of the algorithm at all. It wasshown by Newman, that the approach does deliver goodresults in benchmarks using computer generated test net-works. The success of this approach depends of course onwhether or not the misleading situations have a strongeffect on the final outcome of the clustering. In the ex-ample shown, after grouping together the nodes at theend points of x, the algorithm will then proceed to fur-ther adding nodes from only one of the two communitieslinked by x. Hence, the initial mistake persists, but doesnot completely destroy the result of the clustering.

COMPARISON WITH OTHER DEFINITIONS OF

COMMUNITIES

We have defined the term community as a set of nodeshaving properties 1.) through 3.). Compared with themany definitions of community in the sociological liter-ature [22], this definition is most similar to that of an“LS-set”. An LS-set is a set of nodes S in a networks,such that each of its proper subsets has more links to itscomplement in S than to the rest of the network [23].Previously, Radicchi et al. [24] had given a definition ofcommunity “in a strong sense” as a set of nodes V with

7

the condition kini > kout

i , ∀i ∈ V , i.e. every node in thegroup has more links to other members of the group thanto the rest of the network. In the same manner, they de-fine a community in a “weak sense”, as a set of nodes Vfor which

∑

i∈V kini >

∑

i∈V kouti , i.e. the total number

of internal links is larger than half of the number of theexternal links, since the sum of kin

i is twice the numberof internal edges. The similarity with properties 1.) and2.) of our definition is evident, but instead of comparingabsolute numbers, our definition compares absolute num-bers to expectation values for these quantities in form ofthe coefficients of cohesion and adhesion. One of theconsequences of Radicchi et al.’s definitions is that ev-ery union of two communities is also a community. Thisleads to the strange situation that a community in the“strong” or “weak” sense can also be an ensemble of dis-joint groups of nodes. This paradox may only be resolvedif one assumes a priori that there exists a hierarchy ofcommunities. The following considerations and exampleswill show that hierarchies in community structures arepossible, but cannot be taken for granted. The represen-tation of community structures by dendograms, there-fore, cannot always capture the true community struc-ture. Another definition of communities that implies ahierarchy is that given by Palla et al. There, a commu-nity is interpreted as a set of nodes that can be reachedthrough a clique percolation process. This definition isvery strict and focuses more on local structural proper-ties of the graph, whereas the other definitions, includingours, have a link density based interpretation which alsomakes them more robust to in the case of “noisy” datasets.

OVERLAP AND STABILITY OF COMMUNITY

ASSIGNMENTS

One cannot generally assume that a community structureof a network is uniquely defined. There may exist severalbut very different partitions that all have a comparablyhigh value of modularity. Palla et al. [25] have introducedan algorithm to detect overlapping communities by cliquepercolation and Gfeller et al. have introduced the notionof nodes lying “between clusters” [26]. In the frameworkof this article, the overlap of communities is linked tothe degeneracy of the minima of the Hamiltonian. Thisdegeneracy can arise in several ways and we have to dif-ferentiate between two different types of overlap: overlapof community structure and overlap of communities.

We have already seen that it is undecidable whethera group of nodes nt should be member of community ns

or nr, if the coefficients of adhesion are equal for bothof these communities. Formally, we find at,s\t = atr. Inthis situation, we speak of overlapping communities ns

and nr with overlap nt, since the number of communitiesin the network is not affected by this type of degeneracy.

Nodes that do not form part of overlaps will always begrouped together and can be seen as the non-overlappingcores of communities. An example of this can be foundin Figure 3 a.), where communities A and B overlap innode x. The ground state at γ = 1 is twofold degeneratewith node x belonging either to A or B.

On the other hand, it may be undecidable, if twogroups of nodes should be grouped together or apart, ifthe coefficient of adhesion between them is zero, i.e. thereexist as many edges between them as expected from themodel pij . Similarly, it may be undecidable, if a group ofnodes should form its own community or be divided andthe parts joined with different communities, if this can bedone without increasing the energy. In these situations,the number of communities in the ground state is not welldefined and we cannot speak of overlapping communities,since communities do not share nodes in the degeneraterealisations. We will hence refer to such a situation asoverlapping community structures. An example of thiscan be found in Figure 3 d.), where the three nodes ingroups A and B form either one community as in a.) ortwo distinct communities of 2 and 1 node each. In gen-eral, however, both types of overlap may be present in anetwork.

Since the coefficients of adhesion and cohesion dependon the value of γ chosen, one can assess the stability ofcommunity structures under the change of this parame-ter. The network shown in Figure 3 illustrates the changeof the ground state configuration with γ.

We have already stressed, that properties 1 through 3are also valid for any local minimum of the energy land-scape defined by the Hamiltonian and the graph. Theyonly imply that one cannot jump over energy barriersand move into deeper minima using the suggested moveset. It may therefore be interesting to study also the lo-cal minima and compare them to the ground state. Localminima may be sampled by running greedy optimizationalgorithms using random initial conditions. This allowsfor a probabilistic interpretation of the community struc-ture induced by the minima of the Hamiltonian. For cor-related energy landscapes, it is known that deeper localminima have larger basins of attraction in the configu-ration space. The Hamiltonian (2) induces such a cor-related energy landscape on the graph, since the totalenergy is not drastically affected by single spin changes.We therefore expect that the deep local minima will besampled with higher frequency and that pairs of nodesthat are grouped together in deep minima will have largerentries in a co-appearance matrix Cij that keeps track ofhow frequently node i and j have been grouped togetherin a local minima for multiple runs of a minimizationroutine. A number of examples of co-apperance matri-ces sampling local energy minima at different values of γhave been given in [1].

Here, we shall instead investigate the possible hierar-chies of the community structures directly from the ad-

8

FIG. 3: For different values of γ, different spin configurationsminimize the energy to form ground states. For γ < 16/63,the ground state is ferromagnetic. For 16/63 < γ < 8/7,the two-fold degenerate configuration a.) is the ground state,with node x belonging either to community A or B. For8/7 < γ < 8/3, configuration b.) shows the non-degenerateground state. For γ = 8/3, configurations b.), c.), d.), e.)and f.) all form ground states, but only f.) is ground statefor 8/3 < γ < 4.

jacency matrix. The ordering of the rows and columnscorresponding to nodes of the network is such that be-tween any two nodes that are assigned the same spinstate, there never lies a node of different spin. The in-ternal order among the nodes of the same spin state israndom. The choice of the ordering of the communitiesis arbitrary, but some orderings may be more intuitivethan others. The link density in the adjacency matrixis directly transformed into grey levels. Since the innerlink density of a community is higher than the exter-nal, we can distinguish communities as square blocks ofdarker grey. Different orderings may be combined into aconsensus ordering. That is, starting from a super order-ing given, we reorder the nodes within each communityaccording to a second given sub-ordering, i.e. we onlychange the internal order of the nodes within communi-ties of the super-ordering.

First, we give an example of a completely hierarchicalnetwork. By hierarchical, we mean, that all communitiesfound at a value of γ2 > γ1 are proper sub-communitiesof the communities found at γ1. In our example, wehave constructed a network made of four large commu-nities of 128 nodes each. Each of these nodes have anaverage of 7.5 links to the 127 other members of their

community and 5 links to the remaining 384 nodes in thenetwork. Each of these four communities is composedof four sub-communities of 32 nodes each. Each nodehas an additional 10 links to the 31 other nodes in itssub community. Figure 4 shows the adjacency matrixof this network in different orderings. At γ = 1, theground state is composed of the four large communitiesas shown in the left part of Figure 4. Increasing γ abovea certain threshold makes assigning different spin statesto the 16 sub communities the ground state configura-tion. The middle part of Figure 4 shows an orderingobtained with a value of γ = 2.2. We can see, that someof the these sub-communities are more densely connectedamong each other. Imposing the latter ordering on top ofthe ordering obtained at γ = 1 then allows to display thefull community structure and hierarchy of the network asshown in the right part of Figure 4. Note that we havenot used a recursive approach applying the communitydetection algorithm to separate subgroups. Instead, wehave obtained two independent orderings which are onlycompatible with each other, because the network has ahierarchical structure of dense communities composed ofdenser sub-communities.

In contrast to this situation, Figure 5 shows an exam-ple of a network that is only partially hierarchical. Thenetwork consists of 2 large communities A and B contain-ing 512 nodes, which have on average 12 internal links pernode. Within A and B, a sub-group of 128 nodes exists,which we denote by a and b, respectively. Every nodewithin this sub-group has 6 of its 12 intra-communitylinks with the 127 other members of this sub-group. Thetwo sub-groups a and b have on average 3 links per nodewith each other. Additionally, every node has two linkswith randomly chosen nodes from the network. FromFigure 5, we see that we find the two large communi-ties using γ = 0.5. Maximum modularity, however, isreached at γ = 1 when a and b are joined into a sepa-rate community. Only when using the consensus of theordering obtained at γ = 0.5 and γ = 1, we can under-stand the full community structure with a and b beingsubgroups that are responsible for the majority of linksbetween A and B. It is understood, that this situationcannot be interpreted as a hierarchy, even though a andb are cohesive subgroups in A and B, respectively. Weshall now turn to a real world example to see, whetherthese structural properties can indeed be found outsideof artificially constructed examples.

As a real world example, we study the co-authorshipnetwork [27] of the Los Alamos condensed matterpreprint archive, considering articles published betweenApril 1998 and February 2004. This network has alsobeen analyzed by Palla et al. in [25]. Every article inducesa complete subgraph between the authors in this network.Since articles with a large number of authors induce verylarge cliques, every link induced by a single paper of n au-thors is only given a weight of 1/(n− 1). After summing

9

FIG. 4: Example of an adjacency matrix for a perfectly hierarchical network. The network consists of four communities,each of which is composed of four sub-communities. Using γ = 1, we find the four main communities (left). With γ = 2.2,we find the 16 sub communities (middle). Link density variations in the off diagonal parts of the adjacency matrix alreadyhint at a hierarchy. The consensus ordering (right) shows, that each of the larger communities is indeed composed of foursub-communities each.

FIG. 5: Example of an adjacency matrix for an only partially hierarchical network with overlapping community structure.The network consists of two large communities A and B, each of which contains a sub-community a and b, which are denselylinked with each other, Using γ = 0.5, we find the two large communities (left). With a larger γ = 1, we find the two smallsub-communities a and b grouped together. The consensus ordering (right) shows, that most of the links, that join A and B infact lie between a and b.

the weights for all papers, only links with a weight of 0.1and greater were kept, transforming the network into anon-weighted one. The network consists of 30, 561 nodesconnected by 125, 959 links. There are 668 connectedcomponents, the largest of which has 28, 502 nodes and123, 604 links. We only work with the largest connectedcomponent. The average degree is 〈k〉 = 8.7. We thenminimize the Hamiltonian (2) using pij = kikj/2M andq = 500. Three different values of γ were used. For eachof the values of γ, some of the 500 spin states remainedunpopulated, which makes us confident that we providedenough spin states. Figure 6 shows the adjacency ma-trix of the co-authorship network with rows and columnsordered according to the ground state at γ = 0.5. Wecan distinguish 3 major communities along the diago-nal of the matrix and a large number of smaller com-

munities. Off-diagonal entries in the matrix show, wherecommunities are connected with each other. Figure 7shows the same adjacency matrix, but ordered accordingto the ground state obtained at γ = 1, while Figure 8was obtained ordering the adjacendy matrix accordingto the ground state obtained at γ = 2. We see, howthe increase of γ leads to to a higher number of smallercommunities and a reduction in size of the major commu-nities as expected. In Figure 9, we show the adjacencymatrix in a consensus ordering of the three single order-ings. If the network was hierarchical with respect to γ,i.e. the communities found for larger values of γ are allcomplete sub-communities of those found at smaller γ,we should be able to distinguish this from the adjacencymatrix in the same manner as in Figure 4. From theconsensus ordering, we can see that community A from

10

FIG. 6: Adjacency matrix of the co-author network orderedaccording to the ground state with γ = 0.5.

FIG. 7: Adjacency matrix of the co-author network orderedaccording to the ground state with γ = 1.

FIG. 8: Adjacency matrix of the co-author network orderedaccording to the ground state with γ = 2.

FIG. 9: Adjacency matrix of the co-author network orderedfirst according to the ground state with γ = 0.5. Within theclusters, the nodes were then ordered again according to theground state with γ = 1 and within these clusters, the nodeswere ordered according to the ground state with γ = 2.

the γ = 0.5 ordering is composed of a number of smallercommunities in a somewhat hierarchical manner, whilecommunity B seems to consist of a dense core and manyadjacent nodes that are gradually removed as γ increases.Community C again is decomposed into several smaller

subgroups by the consensus ordering that seems to showtwo levels of hierarchy. The interpretation of the com-munity structure and its hierarchy in terms of researchfields is beyond the scope of this article and shall not beattempted here. Rather, we intend to show that both

11

hierarchical and overlapping community structure existsin the link patterns of real world networks and how itcan be uncovered.

MINIMIZING THE HAMILTONIAN

After having studied some properties of the groundstate, we now turn to the problem of actually finding it.Though any optimization scheme that can deal with com-binatorial optimization problems may be implemented[28, 29], we show the use of Simulated Annealing [30]for this Potts-model, because it yields high quality re-sults, is very general in its application and very simpleto program. The single spin heat bath update rule attemperature T = 1/β is as follows:

p(σl = α) =exp (−βH({σi6=l, σl = α}))

∑qs=1

exp (−βH({σi6=l, σl = s})) . (22)

That is, the probability of spin l being in state α is pro-portional to the exponential of the energy of the entiresystem with all other spins i 6= l fixed and spin l in stateα. Since this is costly to evaluate, we pretend that weknow the energy of the system with spin l in some ar-bitrarily chosen spin state φ, which we denote by Hφ.Then we can calculate the energy of the system with l instate α as Hφ + ∆H(σl = φ → α). The energy Hφ thenfactors out in (22) and we are left with:

p(σl = α) =exp {−β∆H(σl = φ → α)}

∑qs=1

exp {−β∆H(σl = φ → s)} . (23)

The change in energy ∆H(σl = φ → α, φ 6= α) is easilycalculated for both the models of pij . For the simpler ofthe two with pij = p, we find:

∆H(σl = φ → α, φ 6= α) =∑

j 6=l

(Alj − γp)δ(φ, σj) −∑

j 6=l

(Alj − γp)δ(α, σj) (24)

=∑

j 6=l

Aljδ(φ, σj) − γp(nφ − 1) −∑

j 6=l

Aljδ(α, σj) + γpnα (25)

= alφ − alα. (26)

Here, nφ and nα are the number of nodes in spin state φ and α respectively, i.e. the size of groups φ and α. For themodel with pij = kikj/2M we find the following update rule:

∆H(σl = φ → α, φ 6= α) =∑

j 6=l

(Alj − γklkj

2M)δ(φ, σj) −

∑

j 6=l

(Alj − γklkj

2M)δ(α, σj) (27)

=∑

j 6=l

Aljδ(φ, σj) − γkl

2M(Kφ − kl) −

∑

j 6=l

Aljδ(α, σj) + γkl

2MKα (28)

= alφ − alα. (29)

Here, again, Kφ and Kα denote the sum of degrees ofnodes in states φ and α, respectively. In both cases,comparing the adhesion of spin σl with its present com-munity nφ and all other communities nα the spin statefor which the adhesion is largest is assigned the largestprobability. Only local information about the states ofthe neighbors of a node and some global bookkeeping isnecessary. This makes the implementation of a simulatedannealing or any other optimization algorithm especiallysimple and efficient, even though we are dealing with aninfinite range spin glass which has non-zero couplings be-tween all pairs of nodes.

FINDING THE COMMUNITY AROUND A

GIVEN NODE

Often, it is desirable not to find all communities in anetwork, but to find only the community to which a par-ticular node belongs. This may be especially useful if thenetwork is very large and detecting all communities maybe time consuming [31]. In the framework presented inthis article, we can do this using a fast, greedy algorithm.Starting from the node j we are interested in, we succes-sively add nodes with positive adhesion to the group, aslong as the adhesion of the community we are formingand the rest of the network decreases. Adding a node ifrom the rest of the network r to the community s around

12

the start node, the adhesion between s and r changes by:

∆asr(i → s) = air − ais. (30)

For pij = p, this can be written as

∆asr(i → s) = kir − kis − γp(nr − 1 − ns), (31)

where nr = N − ns is the number of nodes in the rest ofthe network, and ns the number of nodes in the commu-nity. For pij = kikj/2M , the change in adhesion reads:

∆asr(i → s) = kir − kis −γ

2Mki (Kr − ki − Ks) . (32)

Here, Kr and Ks are the sums of degrees of the restof the network and the community under study, respec-tively, and ki is the degree of node i to be moved from rto s, which has kis links connecting it with s and kir linksconnecting it with the rest of the network. It is under-stood that only when the adhesion of i with s is largerthan with r, the total adhesion of s with r decreases.Equivalent expressions can be found for removing a nodei from the community s and rejoining it with r. Forγ = 1 and pij = kikj/2M , we have ais + air + 2cii = 0,and cii < 0 by definition and close to zero for all practi-cal cases. Then, ais and air are either both positive andvery small or have opposite sign. Choosing the node thatgives the smallest ∆ars will then result in adding a nodewith positive coefficient of adhesion to s. It is easy tosee, that this ensures a positive coefficient of cohesion inthe set of nodes around j.

In order to benchmark the performance of this ap-proach, we applied it again to computer generated testnetworks as done for the algorithm on the entire networkin [1]. We used networks of 128 nodes, which are groupedinto four equal sized communities of size 32. Each nodeshas an average degree of 〈k〉 = 16. The average numberof links to members of the same community 〈kin〉 and tomembers of different communities 〈kout〉 is then varied,but always ensuring 〈kin〉+〈kout〉 = 〈k〉. Hence, decreas-ing kin renders the problem of community detection moredifficult. Starting from a particular node, we are inter-ested in the performance of the algorithm in discoveringthe community around it. We measure the percentageof nodes that are correctly identified as belonging to thecommunity around the start node as sensitivity and thepercentage of nodes that are correctly identified as not

belonging to the community as specificity.Figure 10 shows the results obtained for different val-

ues of 〈kin〉 at γ = 1 and using pij = kikj/2M as modelof the connection probability. We note, that this ap-proach performs rather well for a large range of 〈kin〉with good sensitivity and specificity. In contrast to thebenchmarks for running the simulated annealing on theentire network as shown in [1], we obtain a sensitivitythat is generally larger than the specificity. This shows,that running the simulated annealing on the entire net-work tends to mistakenly group things apart, that do not

68101214k

in

0.4

0.6

0.8

1

Sens

itivi

ty

68101214k

in

0.4

0.6

0.8

1

Spec

ific

ity

FIG. 10: Benchmark of the algorithm for discovering the com-munity around a given node in networks with known com-munity structure. We used networks of 128 nodes and fourcommunities. The average degree of the nodes was fixed to16, while the average number of intra-community links 〈kin〉was varied. Sensitivity measures the fraction of nodes cor-rectly assigned to the community around the start node, whilespecificity measures the fraction of nodes correctly kept outof the community around the start node.

belong apart by design, while constructing the commu-nity around a given node, tends to group things togetherthat do not belong together by design. This behavioris understandable, since working on the entire networkamounts to effectively implementing a divisive method,while starting from a single node means implementing anagglomerative method.

EXPECTATION VALUES FOR THE

MODULARITY

In order to assess the statistical significance of the mod-ularities found with any algorithm, it is necessary, tocompare them with expectation values for random net-works. This is of course always possible by rewiring thenetwork randomly [32], keeping the degree distributioninvariant and then running a community detection al-gorithm again, comparing the result to the original net-work. This method, however, can only give an answer towhat a particular community detection algorithm mayfind in a random network and hence depends on the verymethod of community detection used. Much better seemsa method to compare the results of a community detec-tion algorithm with a theoretical result, obtained inde-pendently of any algorithm. We have already seen, thatthe problem of community detection can be mapped ontofinding the ground state of an infinite range spin glass. Inthe limit of large N , the local field distribution of infiniterange spin glasses is Gaussian and can hence be char-acterized by only the first two moments of the couplingdistribution, the mean and the variance. The couplingsused in the study of modularity are Jij = Aij − γpij

which have a mean independent of the particular form ofpij :

J0 = (1 − γ)p (33)

13

which is zero in the case of the “natural partition” atγ = 1. The variance amounts to:

J2 = p − (2γ − γ2)〈p2〉. (34)

Since the mean of the coupling distribution couples tothe magnetization of the ground state, all coupling dis-tributions with zero mean will have zero magnetizationin the ground state. Hence, for a random graph weexpect maximum modularity for an equi-partition. Anumber of well known results exist in the literature forequi-partitions. Fu and Anderson [33] have given resultsfor bi-partitionings and Kanter and Sompolinsky for q-partitionings [34]. With these, we can write immediatelyfor the modularity at γ = 1:

Q = − 1

MHGS =

N3/2

MJ

U(q)

q, (35)

where U(q) is the ground state energy of a q-state Pottsmodel with Gausssian couplings of zero mean and vari-ance J2. For large q, we can approximate U(q) =

√q ln q.

In Table I we give some small values of q obtained byusing the exact formula for calculating U(q) from [34].We see, that maximum modularity is obtained at q = 5,

q 2 3 4 5 6 7 8 9

U(q)/q 0.384 0.464 0.484 0.485 0.479 0.471 0.461 0.452

TABLE I: Values of U(q)/q for various values of q obtainedfrom [34], which can be used to approximate the expectedmodularity with equation (35).

though the value of U(q)/q for q = 4 is not much differ-ent from it. This qualitative behavior, that dense randomgraphs tend to cluster into only a few large communitiesis confirmed by our numerical experiments. By rewrit-ing M = pN2/2 and under the assumption of pij = pas in the case of Erdos Renyi (ER) random graphs [35],we can further simplify equation (35) and write for themaximum value of the modularity of a ER random graphwith connection probability p and N nodes:

Q = 0.97

√1 − p

pN(36)

where we have already made use of the fact, that q = 5makes the modularity maximal. Figure 11 shows thecomparison of equation (36) and experiments where wehave numerically maximized the modularity using a sim-ulated annealing approach as described in an earlier sec-tion. We see, that the prediction fits the data well fordense graphs and that modularity decays as a functionof (pN)−1/2 instead of (2/pN)2/3 as proposed in [20].

While the value of Q for random graphs from the Pottsspin glass is rather close to the actual situation for sparserandom graphs, the number of communities, at whichmaximum modularity is achieved is not. In [20], it had

10 100pN=<k>

0.1

1

Mod

ular

ity Q

ER-GraphsPotts PredictionGuimera et. al.

FIG. 11: Modularity of Erdos Renyi random graphs with av-erage connectivity pN = 〈k〉 compared with the estimationfrom equation (36). For the experiment, random graphs withN = 10000 were used.

already been shown, that the number of communities forwhich the modularity reaches a maximum is

√N for tree-

like networks with 〈k〉 = 2. Unfortunately, no plot wasgiven for the number of communities found in denser net-works. Our numerical experiments on large Erdos Renyirandom graphs also show, that the number of commu-nities found in sparse networks tends to increase as 〈k〉decreases.

Even though we have seen that in general, recursivebi-partitioning will not lead to an optimal communityassignment, we shall still use this approach for randomgraphs. Maximum modularity for random graphs isachieved for equipartitions. Partitioning the network re-cursively until no further improvement of Q is possibleallows us to find the number of communities in a ran-dom graph. The number of cut edges C = C(N, M) inany partition, will be a function of the number of nodes inthe remaining part and the number of connections withinthis remaining part and their distribution. We note, thatthe M connections will be distributed into internal andexternal links per node kin + kout = k. This allows usto write C = N〈kout〉/2 for a bi-partition. After eachpartition, the number of internal connections a node hasdecreases due to the cut. We use these results in orderto approximate the number of cut edges after b recursivebi-partitions which lead to 2b parts:

C =

b∑

t=1

2t−1 N

2t〈kout,t〉 =

b∑

t=1

N

2〈kout,t〉 (37)

where 〈kout,t〉 is the average number of external edgesa node gains after cut t. Since for an Ising-model, theground state energy is −EGS = M − 2C we find:

〈kin〉 =〈k〉2

− EGS(〈k〉) = 〈k〉 − 〈kout〉. (38)

14

This shows, that for any bi-partition, we can, on aver-age, always satisfy more than half of the links of everynode on average. This means also, that any bi-partitionwill satisfy the definition of community given by Radicci[24] at least on average, which further means, that everyrandom graph has - at least on average - a communitystructure, assuming Radicci’s definition of community ina strong sense (kin > kout) for every node of the ran-dom graph. The definition of community in a weak sense∑

i kini >

∑

i kouti can always be fulfilled in a random

graph.From (38) we can then calculate the total number of

edges cut after t recursions according to (37) using resultsof Fu and Anderson [33] again who find for a bi-partition:

C =M

2

[

1 − c

√1 − p

pN

]

. (39)

with a constant of c = 1.5266± 0.0002. We can write

〈kin〉 =pN + c

√

pN(1 − p)

2= pN − 〈kout〉 (40)

from which we can calculate (37) substituting pN withthe appropriate 〈kin〉 in every step of the recursion. Themodularity can then be written:

Q =2b − 1

2b− 1

〈k〉

b∑

t=1

〈kout,t〉. (41)

Now we only need to find the number of recursions bthat maximizes Q. Since the optimal number of recur-sions will depend on pN , we also find an estimation ofthe number of communities in the network. Figure 12shows a comparison between the theoretical predictionof the maximum modularity that can be obtained fromequation (41). The improvement of (41) over (36) mustbe due to the possibility of having larger numbers of com-munities, since (39) also assumes a Gaussian distributionof local fields, which is a rather poor approximation forthe sparse graphs under study. Again, we find that themodularity behaves asymptotically like k−1/2 as alreadypredicted from the Potts spin glass and contrary to theestimation in [20].

Figure 13 shows the comparison of the number of com-munities estimated from (41) and the numerical experi-ments on random graphs. The good agreement betweenexperiment and prediction is interesting, given the fact,that (41) allows only powers of two as the number ofcommunities. For dense graphs, the Potts limit of only afew communities is recovered. We see, that sparse ran-dom graphs cluster into a large number of communities,while dense random graphs cluster into only a hand fullof large communities. Most importantly, sparse randomgraphs exhibit very large values of modularity. Theselarge values are only due to their sparseness and not dueto small size. We also stress that statistically significant

10 100pN=<k>

0.1

1

Mod

ular

ity Q

ER-GraphsIsing PredictionGuimera et. al.

FIG. 12: Modularity of Erdos Renyi random graphs with av-erage connectivity pN = 〈k〉 compared with the estimationfrom equation 41. For the experiment, random graphs withN = 10000 were used.

10 100pN=<k>

0

20

40

60

80

num

ber

of c

omm

uniti

es

ER-GraphsIsing Prediction

FIG. 13: Number of communities found in Erdos Renyi ran-dom graphs with average connectivity pN = 〈k〉 comparedwith the estimation from equation 41. For the experiment,random graphs with N = 10000 were used.

modularity must exceed the expectation values of mod-ularity obtained from a suitable null model of the graph.If this null model is an Erdos Renyi random graph, thenthere is very little improvement possible over the val-ues of modularity obtained for the null model for sparsegraphs.

CONCLUSION

In this article, we have tried to elucidate some of thegeneral properties of the problem of community detec-tion in complex networks. We have shown, that it can bemapped onto finding the ground state of an infinite rangePotts spin glass from a very simple and general one pa-

15

rameter ansatz, which is also valid for weighted networksand directed networks. We could show that our ansatz

leads to known modularity measures in a natural way.We have introduced the concept of cohesion and adhe-sion into the terminology of networks as a measure of thedegree to which groups of nodes belong together or apartin a community structure. From the properties of theground state as the minimal energy or maximally modu-lar configuration, we could deduce a number of propertiesthat define a community. By studying the ground statestructure and its changes under parameter variation, wecould also show, how hierarchical and overlapping com-munity structures manifest themselves. Comparisons ofour with other definitions of communities were given. Wehave provided efficient update rules for single spin heatbath simulated annealing algorithms that allow to opti-mize the spin configuration of an infinite range systemby using solely sparse local information and some globalbookkeeping. We have extended the algorithm of find-ing the entire community structure of the whole networkto finding only the community around a given node andwe have given benchmarks for the performance of thisextension. Finally, we have summarized known resultsfrom the theory of infinite range spin glasses in order toshed some light on the problem of community detectionin Erdos Renyi random graphs. We have seen, that sparseER random graphs may show very large modularities andthat the expected modularity of an ER random graph de-cays as

√

1/〈k〉 independent of the size of the graph. Fur-ther, we have seen, that sparse ER random graphs tendto cluster into many small communities, while for denserandom graphs, maximum modularity is achieved for avery small number of communities only, which is inde-pendent of the average degree of the network. We stressthe importance of comparing the values of modularityfound in real world networks with expectation values ofappropriate null models in order to assess their statisti-cal significance. Only graphs which lead to modularitieslarger than the expectation value should be called mod-ular. In this respect, it is understood that Erdos Renyirandom graphs contain communities, but this alone doesnot make these graphs modular.

ACKNOWLEDGEMENTS

The authors would like to thank Stefan Braunewell fora careful reading of the manuscript as well as MicheleLeone, Ionas Erb and Andreas Engel for many helpfulhints and discussions. Also, we would like to thank G.Palla for letting us use the co-author data.

[1] J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93,218701 (2004).

[2] M. E. J. Newman and M. Girvan, Phys. Rev. E 69,026113 (2004).

[3] B. Everitt, S. Landau, and M. Leese, Cluster Analysis

(Arnold, 2001), 4th ed.[4] A. K. Jain, M. N. Murty, and P. J. Flynn, ACM Com-

puting Surveys 31 (1999).[5] B. Kernighan and S. Lin, The Bell System Technical

Journal 29, 291 (1970).[6] C. Fiduccia and R. Mattheyses, in Proc. 19th Design Au-

tomation Conf. (1982), pp. 175–181.[7] C. H. Q. Ding, X. He, H. Zha, M. Gu, and H. D. Simon,

in Proceedings of ICDM 2001 (2001), pp. 107–114.[8] C.-K. Cheng and Y. A. Wei, IEEE. Trans. Comp. Aided

Des. Integrated Circuits Syst. 10, 1502 (1991).[9] L. Hagen and A. B. Kahng, IEEE. Trans. on Comp.

Aided Des. 11, 1074 (1992).[10] J. Shi and J. Malik, IEEE. Trans. Pattern Analysis and

Machine Intelligence (2000).[11] E. T. Jaynes, Physical Review 106, 620 (1957).[12] E. T. Jaynes, Physical Review 108, 171 (1957).[13] M. Blatt, S. Wiseman, and E. Domany, Phys. Rev. Lett.

76 (1996).[14] M. Bengtsson and P. Roivainen, International Journal of

Neural Systems 6 (1995).[15] M. Newman and M. Girvan, Proc. Natl. Acad. Sci. USA

99, 7821 (2003).[16] M. E. J. Newman, Eur. Phys. J. B 38 (2004).[17] L. Danon, J. Dutch, A. Arenas, and A. Diaz-Guilera, J.

Stat. Mech. p. P09008 (2005).[18] S. Muff, F. Rao, and A. Caflish, Phys. Rev. E. 72, 056107

(2005).[19] A. Clauset, M. E. J. Newman, and C. Moore, Phys. Rev.

E. (2004).[20] R. Guimera, M. Sales-Prado, and L. N. Amaral, Phys.

Rev. E. 70, 025101(R) (2004).[21] M. Newman, Phys. Rev. E. 69, 066133 (2004).[22] S. Wasserman and K. Faust, Social Network Analysis

(Cambridge University Press, 1994).[23] S. B. Seidmann, Social Networks 5, 97 (1983).[24] F. Radicchi, C. Castellano, F. Cecconi, V. Loreto, and

D. Parisi, Proc. Natl. Acad. Sci. USA 101, 2658 (2004).[25] G. Palla, I. Derenyi, I. Farkas, and T. Vicsek, Nature

435, 814 (2005).[26] D. Gfeller, J.-C. Chappelier, and P. de los Rios, Phys.

Rev. E. 72, 056135 (2005).[27] S. Warner, Library Hi Tech 21, 151 (2003).[28] A. K. Hartmann and H. Rieger, eds., Optimization Algo-

rithms in Physics (Wiley-Vch, 2004).[29] A. K. Hartmann and H. Rieger, eds., New Optimization

Algorithms in Physics (Wiley-Vch, 2004).[30] S. Kirkpatrick, C. G. Jr., and M. Vecchi, Science 220,

671 (1983).[31] F. Wu and B. Huberman, Eur. Phys. J. B 38, 331 (2004).[32] S. Maslov and K. Sneppen, Science 296, 910 (2002).[33] Y. Fu and P. W. Anderson, J. Phys. A: Math. Gen. 19,

1605 (1986).[34] I. Kanter and H. Sompolinsky, J. Phys. A: Math. Gen.

20, L636 (1987).[35] P. Erdos and A. Renyi, Publ. Math. Inst. Hung. Acad.

16

Sci. 5, 17 (1960).