Spatial Scan Statistics for Graph Clustering

Bei Wang ∗ Jeff M. Phillips † Robert Schreiber ‡ Dennis Wilkinson §

Nina Mishra ¶ Robert Tarjan ‖

Abstract

In this paper, we present a measure associated with detectionand inference of statistically anomalous clusters of a graphbased on the likelihood test of observed and expected edgesin a subgraph. This measure is adapted from spatial scanstatistics for point sets and provides quantitative assessmentfor clusters. We discuss some important properties ofthis statistic and its relation to modularity and Bregmandivergences. We apply a simple clustering algorithm to findclusters with large values of this measure in a variety ofreal-world data sets, and we illustrate its ability to identifystatistically significant clusters of selected granularity.

1 Introduction.

Numerous techniques have been proposed for identifyingclusters in large networks, but it has proven difficult tomeaningfully and quantitatively assess them, especiallyfrom real-world data whose clustering structure is a pri-ori unknown. One of the key challenges encountered byprevious clustering methods is rating or evaluating theresults. In large networks, manual evaluation of theresults is not feasible, and previous studies have thusturned to artificially created graphs with known struc-ture as a test set. However, many methods, especiallythose in which the number of clusters must be speci-fied as an algorithm parameter, give very poor resultswhen applied to real-world graphs, which often have ahighly skewed degree distribution and overlapping, com-plex clustering structure [18, 33].

1.1 Prior Work on Measures.

Modularity. The problem of assessment was par-tially solved by the introduction of modularity [12], aglobal objective function used to evaluate clusters thatrewards existing internal edges and penalizes missinginternal edges. Non-overlapping clusters, or partitionsof a graph, are obtained by maximizing the distancefrom what a random graph would predict, either byextremal optimization [15], fast greedy hierarchical al-

∗Duke University : beiwang@cs.duke.edu†Duke University : jeffp@cs.duke.edu‡HP Labs : rob.schreiber@hp.com§HP Labs : dennis.wilkinson@hp.com¶Visiting Search Labs, Microsoft Research; University of Vir-

ginia : nmishra@cs.virginia.edu‖Princeton University; HP Labs : robert.tarjan@hp.com

gorithms [31, 37], simulated annealing [38] or spectralclustering [32].

However, modularity cannot directly assess howunexpected and thus significant individual clusters are.Additionally it cannot distinguish between clusterings ofdifferent granularity on the same network. For example,comparable overall modularities were reported for hardclusterings of the same scientific citation graph into 44,324, and 647 clusters [37], results which are clearly ofvarying usefulness depending on the application.

Spatial Scan Statistics. Scan statistics [19] mea-sure densities of data points for a sliding window on or-dered data. The densest regions under a fixed size win-dow are considered the most anomalous. This notionof a sliding window has been generalized to neighbor-hoods on directed graphs [36] where the neighborhoodof a vertex is restricted to vertices within some constantnumber of edges in the graph. The number of neighborsis then compared to an expected number of neighborsbased on previous data in a time-marked series.

Spatial scan statistics were introduced by Kull-dorff [23] to find anomalous clusters of points in 2 orgreater dimensions without fixing a window size. Thesestatistics measure the surprise of observing a particu-lar region by computing the log-likelihood of the prob-ability of the most likely model for a cluster versus theprobability of the most likely model for no cluster. Kull-dorff argues that the region with the largest spatial scanstatistic is the most likely to be generated by a differ-ent distribution, and thus is most anomalous. This testwas shown to be the most powerful test [27] for findinga region which demonstrates that the data set is notgenerated from a single distribution. Kulldorff [23] de-rived expressions for the spatial scan statistic under aPoisson and Bernoulli model. Agarwal et.al. [2] general-ized this derivation to 1-parameter exponential families,and Kulldorff has studied various (see [26]) other formsof this statistic. Many techniques exist for computingthe statistic quickly for anomaly detection for point sets[30, 25, 1].

1.2 Previous Algorithmic Work. Clustering iswell-established as an important method of information

extraction from large data sets. Hard clustering dividesdata into disjoint clusters while soft clustering allowsdata elements to belong to more than one cluster. Exist-ing techniques include MCL [41], Ncut [40], graclus [13],MCODE [5], iterative scan [9], k-clique-community [37],spectral clustering [35, 32], simulated annealing [28], orpartitioning using network flow [34], edge centrality [18],and many other techniques e.g. [42].

Several statistically motivated graph clusteringtechniques exist [20, 39, 22]. Itzkovitz et. al. discusseddistributions of subgraphs in random networks with ar-bitrary degree sequence, which have implications for de-tecting network motifs [20]. Sharan et. al. introduceda probabilistic model for protein complexes taking con-servation into consideration [39]. Koyuturk et. al. iden-tified clusters by employing a min-cut algorithm wherea subgraph was considered to be statistically significantif its size exceeded a probabilistic estimation based ona piecewise degree distribution model [22]. These tech-niques are all different from our approach as our modelbased on spatial scan statistics has properties essentialfor detecting statistically significant clusters.

A general clustering framework using Bregman di-vergences as optimization functions has been proposedby Banerjee et.al. [14, 6, 7]. This approach is of notebecause the optimization function we use can be inter-preted as a Bregman divergence, although our theoreti-cal and algorithmic approaches are completely different(as explained in Section 3).

1.3 Our Contribution. Our main contribution isthe generalization of spatial scan statistics from pointsets to graphs. Our statistic, Poisson discrepancy, offersa metric that determines how significant the clustersare using a normalized measure of likelihood. Bycomparison to random graphs with the same expecteddegree sequence as the network under consideration, ap-value based on the Poisson discrepancy is associatedto each cluster found, providing a problem independentmeasure of its statistical significance. The statistic iscorrectly normalized to provide an absolute measureof cluster significance on an individual level, allowingcomparison between clusters from different networks orfrom different clusterings of the same network.

We also implement two simple greedy algorithmswhich seek to maximize the Poisson discrepancy. Ourmethod includes a tunable parameter which we showis strongly correlated to the average size of the foundclusters. This parameter allows the user to vary theexpected size of the clusters found by the algorithms,which may be useful for various applications. In addi-tion, because clusters of different size may be meaning-fully compared with each other, our method provides a

way to identify important levels of granularity within anetwork. In fact, our results using real-world data showthat statistically significant clusters of varying sizes canexist in a given neighborhood of a network, implyingthat real-world networks display granularity on a num-ber of different levels. Finally, our method allows datapoints to fit in any number of clusters, including zero.This makes it applicable to real-world networks in whichsome vertices may not have any kind of strong associa-tion.

In Sections 2 and 3 of this paper, we present the the-oretical foundation of our clustering method. The algo-rithm and variations are described in Section 4. Bipar-tite extensions of our algorithm are explained in Section5. In Section 6, we empirically evaluate our objectivefunction and algorithms on real and synthetic datasets.We calculate the algorithms’ runtime and power; weevaluate other clustering algorithms using Poisson dis-crepancy; and we demonstrate that the clusters foundusing Poisson discrepancy with our algorithm are mean-ingful, are scaled reliably with a tuning parameter γ,and can overlap with each other.

2 Definitions.

Graphs. Let G = (V,E) be an undirected graphwith vertex set V and edge set E, such that E is amultiset of elements in [V ]2, where [V ]2 denotes a set of2-element multisets of V . That is [V ]2 = {{u, v}|u, v ∈V }, where u and v are not necessarily distinct. Gallow self-loops and multiple edges between a pair ofvertices. A cluster Z = (VZ , EZ) is a subgraph of Ginduced by subset VZ ⊆ V . The collection of clustersZ ⊆ G is denoted as Z. The subgraph in G not in Z isZ = (V,E \ EZ). Let c(Z) be the number of observededges in cluster Z ⊆ G. c(G) is the total number ofobserved edges in G. c∗(x) is the observed number ofedges between a pair of vertices x = {vi, vj} ∈ [V ]2.The degree ki of a vertex vi is the number of edges thatcontain vi. A self-loop at vi is counted twice in thedegree.

The neighborhood of a vertex set U ⊆ V is the setof vertices {v | {v, u} ∈ E for u ∈ U and v ∈ V \ U},denoted N(U). The size of a set V is denoted |V |. Wedefine the distance between two vertex sets U and V asd(U, V ) = |U | + |V | − 2|U ∩ V |.

Poisson Random Graph Model. Many largereal-world graphs have diverse, non-uniform, degreedistributions [8, 4, 3] that are not accurately describedby the classic Erdos and Renyi random graph models[16]. We consider a Poisson random graph model herethat captures some main characteristics of real-worldgraphs, specifically, allowing vertices to have differentexpected degrees. Notice that this model is different

from models used in [10, 12].The model generates random graphs with a given

expected degree sequence k = 〈k1, k2, ..., k|V |〉 for thevertex set V = 〈v1, v2, ..., v|V |〉, where vertex vi has

expected degree ki. Let kV =∑|V |

i=1 ki and m = kV /2.The model chooses a total of m pairs of vertices as edgesthrough m steps, with replacement. At each step, eachof the two end-points of an edge is chosen among allvertices through a Poisson proces, proportional to theirdegrees. The probability that each end point of a chosenedge contains vertex vi is ki/kV . The expected degreeof vertex vi after this process is m(ki/kV + ki/kV ) =m(ki/2m + ki/2m) = ki. We refer to this model as thePoisson random graph model. It fixes the total numberof edges and allows multiple edges between any pair ofvertices.

A bipartite multigraph example captured by aPoisson random graph model is an internet bulletinboard. There is a set of users who post under differentthreads. A post in a thread would constitute an edge.Of course a single user can post multiple times to asingle thread, and some users post more often and somethreads generate more posts.

3 Spatial Scan Statistics for Graphs.

In this section we generalize the notion of a spatialscan statistic [23] to graphs. We also highlight someimportant properties of this statistic, as well as itsrelation to local modularity and Bregman divergences.It is this spatial scan statistic for graphs that we use toprovide quantitative assessment of significant clusters.

3.1 Spatial Scan Statistics Poisson Model.

Given G = (V,E), its edge set describes a degree se-quence k = 〈k1, k2, ..., k|V |〉 for the vertex set V =〈v1, v2, ..., v|V |〉, where ki is the degree of vertex vi. Let

kV =∑|V |

i=1 ki and c(G) = kV /2.According to the Poisson random graph model with

k as the given expected degree sequence and c(G) asthe total number of expected edges, define µ∗(x) asthe expected number of edges connecting the pair x ={vi, vj} ∈ [V ]2. For i 6= j, µ∗(x) = kikj/2c(G). For i =j, µ∗(x) = k2

i /4c(G). For a subgraph A = (VA, EA) ⊆G, define µ(A) as the expected number of edges inA. Let KVA

=∑

vi∈VAki, µ(A) =

∑

x∈[VA]2 µ∗(x) =

k2VA

/4c(G). Notice that µ(G) =∑

x∈[V ]2 µ∗(x) = c(G).A simple example is shown in Figure 1 where cluster Zis induced by dark vertices in the graph, c(G) = µ(G) =10, c(Z) = 6 and µ(Z) = 132/(4 × 10) = 169/40.

In the spatial scan statistics Poisson model, weassume edges in G are generated by a Poisson process Nwith intensity λ. The probability that there are exactly

Figure 1: Example graph.

n occurrences in the process is denoted as Poi(n, λ) =e−λλn/n!. N assumes the same vertex set and expecteddegree sequence as G and generates edges according tothe Poisson random graph model. Let N (A) be therandom variable describing the number of edges in thesubgraph A ⊆ G under such a process. N (A) follows aPoisson distribution, denoted by N (A) ∼ Poi(n, λ).

Under such a model, consider a single cluster Z =(VZ , EZ) ⊂ G, such that edges in Z are generatedwith rate p, while edges in Z are generated with rateq. The null hypothesis H0 : p = q assumes completespatial randomness [21], that is, edges inside and outsidethe cluster are generated under the same rate. Thealternative hypothesis H1 : p > q assumes that edgesinside the cluster are generated at a higher rate thanthose outside the cluster. µ therefore represents aknown underlying intensity that generates edges underH0 [24]. Therefore, under H0, the number of edgesin any given cluster A ⊆ G is Poisson distributed,N (A) ∼ Poi(n, pµ(A)) for some value p. Under H1,∀A ⊆ G, N (A) ∼ Poi(n, pµ(A ∩ Z) + qµ(A ∩ Z)) [23].

Likelihood Ratio Test. The spatial scan statis-tics for graphs are based on the likelihood ratio testderived from [23].

The spatial scan statistics Poisson Model assumesunder the null hypothesis H0 that all edges in Gare generated at the same rate. That is, under H0,the probability of c(G) edges being observed in Gis Poi(c(G), pµ(G)). The density function f(x) of aspecific pair of vertices x ∈ [V ]2 being an actual edge is

f(x) = pµ∗(x)pµ(G) = µ∗(x)

µ(G) .

Incorporating the density of each edge, the likeli-hood function under H0 is

L0 = maxp=q

Poi(c(G), pµ(G))∏

x∈E

f(x)

= maxp=q

e−pµ(G)(pµ(G))c(G)

c(G)!

∏

x∈E

µ∗(x)

µ(G)

= maxp=q

e−pµ(G)pc(G)

c(G)!

∏

x∈E

µ∗(x)

For a fixed Z, L0 takes its maximum when p = q = c(G)µ(G) .

That is

L0 =e−c(G)

c(G)!

(

c(G)

µ(G)

)c(G)∏

x∈E

µ∗(x).

The alternative hypothesis H1 assumes that for aparticular cluster Z = (VZ , EZ) ⊂ Z, the edges in Zare generated at a different rate than those in Z. UnderH1, the probability of c(G) edges being observed in Gis Poi(c(G), pµ(Z) + q(µ(G) − µ(Z))). For x ∈ EZ , the

density function f(x) is f(x) = pµ∗(x)pµ(Z)+q(µ(G)−µ(Z)) . For

x ∈ E \ EZ , f(x) = qµ∗(x)pµ(Z)+q(µ(G)−µ(Z)) . The likelihood

function under H1 is

L(Z) = maxp>q

Poi(c(G), pµ(Z) + q(µ(G) − µ(Z)))

·∏

x∈EZ

f(x)∏

x∈E\EZ

f(x)

= maxp>q

e−pµ(Z)−q(µ(G)−µ(Z))

c(G)!

· (pµ(Z) + q(µ(G) − µ(Z)))c(G)

·∏

x∈EZ

pµ∗(x)

pµ(Z) + q(µ(G) − µ(Z))

·∏

x∈E\EZ

qµ∗(x)

pµ(Z) + q(µ(G) − µ(Z))

= maxp>q

e−pµ(Z)−q(µ(G)−µ(Z))

c(G)!pc(Z)qc(G)−c(Z)

·∏

x∈E

µ∗(x)

For a fixed Z, L(Z) takes its maximum when p =c(Z)µ(Z) , q = c(G)−c(Z)

µ(G)−µ(Z) . If c(Z)µ(Z) > c(G)−c(Z)

µ(G)−µ(Z) ,

L(Z) =e−c(G)

c(G)!

(

c(Z)

µ(Z)

)c(Z) (

c(G) − c(Z)

µ(G) − µ(Z)

)c(G)−c(Z)

·∏

x∈E

µ∗(x)

otherwise L(Z) = L0.Given a cluster Z ∈ Z, we define the likelihood ratio,

LR, as the ratio of L(Z) and L0. For the Poisson model

c(G) = µ(G), and if c(Z)µ(Z) > c(G)−c(Z)

µ(G)−µ(Z) , we have

LR(Z) =L(Z)

L0

=

(

c(Z)

µ(Z)

)c(Z) (

c(G) − c(Z)

µ(G) − µ(Z)

)c(G)−c(Z)

.

Otherwise LR = 1.We then define the likelihood ratio test statistic, Λ,

as the maximum likelihood ratio over all clusters Z ∈ Z,

Λ = maxZ∈Z

LR(Z).

Poisson Discrepancy. Let r(Z) = c(Z)c(G) and

b(Z) = µ(Z)µ(G) , we define the Poisson discrepancy, dP ,

as

dP (Z) = r(Z) logr(Z)

b(Z)+ (1 − r(Z)) log

1 − r(Z)

1 − b(Z).

Intuitively, r(Z) is the observed edge ratio and b(Z) isthe baseline edge ratio in Z and G.

Since log Λ = c(G)maxZ∈Z dP (Z), for the cluster Zthat maximizes dP , dP (Z) constitutes the test statisticΛ. This means that the likelihood test based onmaxZ∈Z dP (Z) is identical to one based on Λ. Since 0 <r(Z), b(Z) ≤ 1, from this point on, we evaluate clustersbased on the Poisson discrepancy. dP determineshow surprising r(Z) is compared to the rest of thedistribution. Thus clusters with larger values of dP aremore likely to be inherently different from the rest ofthe data.

Point Set Model. It should be noted that thesedefinitions were originally formulated where the set ofall possible edges, [V ]2, was instead a point set, thefunction µ measured a baseline estimate of the popula-tion, and the function c measured reported data, suchas instances of a disease. In this setting, the set of pos-sible clusters is usually restricted to those contained insome geometrically described family of ranges. This setcan often be searched exhaustively in time polynomialin the size of the point set. In our setting we have 2|V |

possible ranges (vertex sets which induce Z) — an in-tractably large number. Hence, Section 4 explains howto explore these ranges effectively.

Significance of a Cluster. Although we can con-sider many (or all) subsets and determine the one thatis most anomalous by calculating the Poisson discrep-ancy, this does not determine whether this value is sig-nificant. Even a graph generated according to a Pois-son random graph model will have some cluster whichis most anomalous. For a graph G = (V,E) with de-gree sequence k and for a particular cluster Z ⊆ G wecan compare dP (Z) to the distribution of the values ofthe most anomalous clusters found in a large set of (say1000) random data graphs. To create a random datagraph, we fix V ; then we randomly select c(G) edgesaccording to a Poisson random graph model with ex-pected degree sequence k. If the Poisson discrepancyfor the original cluster is greater than all but a .05 frac-tion of the most anomalous clusters from the random

data sets, then we say it has a p-value of .05. The lowerthe p-value, the more significantly anomalous the rangeis. These high discrepancy clusters are most significantbecause they are the most unlikely compared to what isexpected from our random graph model.

3.2 Properties of Spatial Scan Statistics. Kull-dorff has proved some optimal properties for the likeli-hood ratio test statistic for point sets [23]. In the con-text of graphs, we describe those properties essentialfor detecting statistically anomalous clusters in termsof dP . For details and proofs, see [23, 27]. As a directconsequence of Theorem 1 in [23], we have

Theorem 3.1. Let X = {xi|xi ∈ E}c(G)i=1 be the set

of edges in G = (V,E) where Z = (VZ , EZ) is the

most likely cluster. Let X ′ = {x′i|x

′i ∈ [V ]2}

c(G)i=1 be an

alternative configuration of a graph G′ = (V,E′) where∀xi ∈ EZ , x′

i = xi. If the null hypothesis is rejectedunder X , then it is also rejected under X ′.

Intuitively, as long as the edges within the subgraphconstituting the most likely cluster are fixed, the nullhypothesis is rejected no matter how the rest of theedges are shuffled around [23].

This theorem implies that:

1. dP (Z) does not change as long as its internal struc-ture and the total number of reported edges outsideZ remains the same. Intuitively, clusters definedby other subgraphs do not affect the discrepancyon Z. Formally, dP (Z) is independent of the valueof c∗(x) for any potential edge x ∈ E \EZ , as longas c(Z) remains unchanged.

2. If the null hypothesis is rejected by dP , then wecan identify a specific cluster that is significant andimplies this rejection. This distinguishes betweensaying “there exist significant clusters” and “thecluster Z is a significant cluster,” where dP can dothe latter.

Theorem 3.2. dP is individually most powerful forfinding a single significant cluster: for a fixed falsepositive rate and for a given set of subgraphs tested, itis more likely to detect over-density than any other teststatistic [30].

This is a direct consequence of Theorem 2 in [23]and is paramount for effective cluster detection. (Itspower is also explored in Section 6). It implies that:

3. We can determine the single potential edge x ∈ [V ]2

(or set of edges, such as those described by addinga vertex to the cluster) that will most increase thePoisson discrepancy of a cluster, and thus mostdecrease its p-value.

3.3 Spatial Scan Statistics and Local Modular-

ity. Several local versions of modularity have been usedto discover local community structure [11, 29]. Specifi-cally, local modularity introduced in [38] is used to findthe community structure around a given node. The localmodularity of Z ⊆ G measures the difference betweenthe number of observed edges c(Z) and the number ex-pected, µ(Z),

Mγ(Z) = c(Z) − γµ(Z).

One approach to clustering is to find the cluster Z thatlocally maximizes Mγ . The γ parameter with defaultvalue 1, is a user specified knob [38] that scales theexpected number of edges within Z under a Poissonrandom graph model. We observe that it effectivelytunes the size of the clusters which optimize Mγ(Z). Fora fixed cluster Z, Mγ can be treated as a linear functionof γ, where its intersection with the Y -axis is c(Z), andits slope is −µ(Z). Mγ for all Z ∈ Z forms a familyof linear functions whose upper envelope corresponds toclusters that maximize M as γ varies. It can be observedthat as γ increases, c(Z) is non-increasing and µ(Z) isnon-decreasing for the cluster Z that maximizes Mγ .

It is important to distinguish M from dP . While M

measures the edge distance from the expected randomgraph, dP measures the difference in how likely thetotal number of edges are to occur in a general randomgraph and how likely they are to occur in cluster Z andits complement as separate random graph models. Tosummarize, M calculates the distance, and spatial scanstatistics measure how unexpected this distance is, givenZ.

Several properties are also shared between dP andM. The tuning knob γ can be used in Poisson discrep-ancy to scale the expected number of edges in a clusterZ.

dP,γ(Z) = r(Z) logr(Z)

γb(Z)+ (1 − r(Z)) log

(

1 − r(Z)

1 − b(Z)

)

Technically, the function dP,γ describes the effect ofscaling by γ the expected number of edges in a clusterZ (but not outside the cluster), while not allowing q,the parameter to model the random graph outside thiscluster, to reflect γ. Thus in the same way as withMγ for large γ, clusters need to have significantly moreedges than expected to have a positive dP value. Thefollowing lemma highlights this relationship.

Lemma 3.1. Consider two clusters Z1 and Z2 such thatdP,γ(Z1) = dP,γ(Z2) and that c(Z1) > c(Z2). Then forany δ > 0 we know dP,γ+δ(Z1) < dP,γ+δ(Z2).

The same property holds for Mγ and µ in placeof dP,γ and c, respectively. That is, consider two

clusters Z1 and Z2 such that Mγ(Z1) = Mγ(Z2) andthat µ(Z1) > µ(Z2). Then for any δ > 0 we knowMγ+δ(Z1) < Mγ+δ(Z2).

Proof. We can write

dP,γ(Z) = dP (Z) − r(Z) log γ,

thus as γ increases dP,γ(Z1) will decrease faster thandP,γ(Z2).

We can also write

Mγ(Z) = c(G)(r(Z)−γb(Z)) = M1(Z)−c(G)(γ−1)b(Z).

Thus the same argument applies.

This implies that for the discrepancy measure, weshould expect the size of the optimal clusters to besmaller as we increase γ, as is empirically demonstratedin Section 6.3.

Using some machinery developed by Agarwalet.al. [2] we can create an ε-approximation of dP withO( 1

εlog2 |V |) linear functions with parameters r(Z) and

b(Z), in the sense that the upper envelope of this set oflinear functions will be within ε of dP . We can inter-pret Mγ as a linear function whose slope is controlledby the value of γ. Figure 2 shows how M1 and M2,respectively, approximate dP . Thus we can find the op-timal cluster for O( 1

εlog2 |V |) values of γ and let Z be

the corresponding cluster from this set which has thelargest value of dP (Z). Let Z∗ be the cluster that hasthe largest value of dP (Z∗) among all possible clusters.Then dP (Z) + ε ≥ dP (Z∗).

However, a further study of Agarwal et.al. [1]showed that a single linear function (which would beequivalent to γ = 2 for Mγ) approximated dP on averageto within about 95% for a problem using point sets.Note in Figure 2 how M2 seems to approximate dP

better than M1, at least for a large portion of the domaincontaining smaller clusters.

3.4 Spatial Scan Statistics and Bregman Diver-

gences. Many Bregman divergences can be interpretedas spatial scan statistics. The KL-divergence, a Breg-man divergence, when between two 2-point distribu-tions, is equivalent to dP up to a constant factor.

Banerjee et.al. [7] use Bregman divergences in adifferent way than does this paper. In the contextof graph clustering, Bregman hard clustering finds abi-partitioning and a representative for each of thepartitions such that the expected Bregman divergenceof the data points (edges) from their representatives isminimized [7]. For details and derivations, see [7].

Given a graph G = (V,E), let xi be a potential

edge xi ∈ [V ]2. Set X = {xi}c(G)2

i=1 ⊆ R has probability

bS

rS

01

1.0

0.5

1.5

bS

rS

01

1.0

0.5

1.5

Figure 2: Comparison of dP (gridded) to 1m

M1 (trans-parent, upper panel) and 1

mM2 (transparent, lower

panel) over (r(Z), b(Z)) ∈ [0, 1]2 such that r(Z) > b(Z).Recall that r(Z) and b(Z) are the actual and expectedfraction of a graph’s edges which lie in a particular clus-ter; for applications to large networks, a range of say(0, 0.2)2 is most pertinent to clustering. For this range,M2 is shown to approximate dP more closely than M1.

measure µ∗. Cluster Z induces a bi-partition of G, Zand Z. Let µ(Z) and µ(Z) be the induced measureson the partitions, where µ(Z) =

∑

xi∈[VZ ]2 µ∗(xi) and

µ(Z) =∑

xi∈X\[VZ ]2 µ∗(xi). Let ηZ and ηZ denote thepartition representative values.

ηZ =∑

xi∈[VZ ]2

µ∗(xi)

µ(Z)c∗(xi)

ηZ =∑

xi∈X\[VZ ]2

µ∗(xi)

µ(Z)c∗(xi)

Define ηi = ηZ for xi ∈ [VZ ]2, otherwise ηi =ηZ . The Bregman clustering seeks to minimize thedivergence between the two c(G)2-point distributions

{〈c∗(x1), c∗(x2), . . . , c

∗(xc(G)2)〉, 〈η1, η2, . . . ηc(G)2〉}

We, on the other hand, maximize the KL-divergence between the two 2-point distributions

{〈r(Z), 1 − r(Z)〉, 〈b(Z), 1 − b(Z)〉}.

But the methods do not conflict with each other. TheirηZ and ηZ variables are akin to p and q in the derivationof the scan statistic. By minimizing their Bregmandivergence, they are trying to allow ηZ and ηZ to beas close to variables they represent as possible (i.e. ηZ

should be close to each c∗(xi) for xi defined on Z); andby maximizing our discrepancy we are separating p andq as much as possible, thus probabilistically representingthe cluster edges and non-cluster edges more accuratelywith these ratios, respectively.

However, the Bregman divergence used by Banerjeeet.al. [6, 7] typically assumes a less informative, uniformrandom graph model where µ∗(xi) = µ∗(xj) for all i andj. Also when minimizing the KL-divergence, no edge atxi would imply c∗(xi) = 0, thus implying that the cor-

responding term of the KL-divergence, c∗(xi) log c∗(xi)ηi

,is undefined. In their Bregman divergence model mostsimilar to ours, this poses a problem as c∗(xi) can be 0in our model; thus we do not compare the performanceof these algorithms.

4 Algorithms.

In this section, we describe two bottom-up, greedyclustering algorithms. For a graph G = (V,E) there are2|V | possible clusters (subgraphs) induced by subsetsof V . That is, |Z| ≤ 2|V |, where each cluster Z =(VZ , EZ) ∈ Z is induced by a vertex set VZ ⊆ V .Clearly it is intractable to calculate discrepancy forevery possible cluster through exhaustive search, as isoften done with spatial scan statistics. We can, however,hope to find a locally optimal cluster. For an objectivefunction Ψ : 2|V | → R, define a local maximum as asubset U ⊆ V such that adding or removing any vertexwill decrease Ψ(U). For some objective function Ψ andtwo vertex sets U and W , define

∂Ψ(U,W ) =

{

Ψ(U ∪ W ) − Ψ(U) W ⊂ V \ UΨ(U \ W ) − Ψ(U) W ⊂ U

where Ψ(VZ) = dP (Z) for Z = (VZ , EZ) ⊆ Z inducedby VZ . Let U+ (resp. U−) be the set of vertices inN(U) (resp. U) such that ∂Ψ(U, v) > 0 for each vertexv. U+

F (resp. U−F ) denotes the subset of U+ (resp.

U−) that contains the fraction F of vertices with thelargest ∂Ψ(U, v) values. We now are set to describetwo algorithms for refining a given subset U to find alocal maximum in Ψ. Notice that both algorithms canbe used to locally optimize any objective function, notlimited to the Poisson discrepancy used here.

Greedy Nibble. The Greedy Nibble algorithm(Algorithm 1) alternates between an expansion phaseand a contraction phase until the objective function can-not be improved. During expansion (resp. contraction)we iteratively add (resp. remove) the vertex that mostimproves the objective function until this phase can nolonger improve the objective function.

Algorithm 1 Greedy-Nibble(U)

repeat

expand = false; contract = false

v+ = arg maxv∈N(U) ∂Ψ(U, v).while ∂Ψ(U, v+) > 0 do

expand = true

U = U ∪ v+.v+ = arg maxv∈N(U) ∂Ψ(U, v).

v− = arg maxv∈U ∂Ψ(U, v).while ∂Ψ(U, v−) > 0 do

contract = true

U = U \ v−.v− = arg maxv∈U ∂Ψ(U, v).

until expand = false and contract = false

Greedy Chomp. The Greedy Chomp algorithm(Algorithm 2) is a more aggressive and faster versionof the Greedy Nibble algorithm. Each phase adds afraction F of the vertices which individually increase theΨ value. If adding these F |U+| vertices simultaneouslydoes not increase the overall Ψ value, then the fractionF is halved, unless F |U+| ≤ 1. Similar to simulatedannealing, this algorithm makes very large changes tothe subset at the beginning but becomes more gradualas it approaches a local optimum.

Theorem 4.1. Both the Greedy Nibble algorithm andthe Greedy Chomp algorithm converge to a local maxi-mum for Ψ.

Proof. The algorithms increase the value of Ψ at eachstep, and there is a finite number of subsets, so theymust terminate. By definition the result of terminationis a local maximum.

4.1 Variations. There are many possible heuristicvariations on the above algorithms. We use GreedyNibble because of its simplicity and Greedy Chompbecause it performs similarly but faster.

In terms of initial seed selection, when time is not anissue, we recommend using each vertex as a seed. Thisensures that every interesting cluster contains at leastone seed. For larger graphs, randomly sampling somevertices as seeds should work comparably [38]. Clusterstend to be larger in this case, so most of them will still

Algorithm 2 Greedy-Chomp(U)

repeat

expand = false; F = 1Calculate U+

F

while(

∂Ψ(U,U+F ) < 0 and F |U+| ≥ 1

)

do

F = F/2; Update U+F

while(

∂Ψ(U,U+F ) > 0

)

do

expand = true

U = U ∪ U+F .

Calculate U+F ; F = 1

while(

∂Ψ(U,U+F ) < 0 and F |U+| ≥ 1

)

do

F = F/2; Update U+F

contract = false; F = 1Calculate U−

F

while(

∂Ψ(U,U−F ) < 0 and F |U−| ≥ 1

)

do

F = F/2; Update U−F

while(

∂Ψ(U,U−F ) > 0

)

do

contract = true

U = U \ U−F .

Calculate U−F ; F = 1

while(

∂Ψ(U,U−F ) < 0 and F |U−| ≥ 1

)

do

F = F/2; Update U−F

until (expand = false and contract = false)

contain some seed. Alternatively, we could run anotherclustering algorithm to generate an initial seed and justuse our greedy algorithms as a refinement.

In general, we use dP as the objective function, butit is more prone to getting stuck in local maxima thanis M. Thus we enhance each initial seed by runningthe expansion phase of the algorithm with M2 since itclosely approximates dP as shown in Figure 2.

Since our emphasis is on the discrepancy measure-ment rather than clustering technique, we focus on il-lustrating that these simple clustering techniques basedon Poisson discrepancy find locally significant clusters.

4.2 Complexity. It is difficult to analyze our algo-rithms precisely because they may alternate betweenthe expansion and contraction phases many times. ButTheorem 4.1 shows that this process is finite, and wenotice that relatively few contraction steps are ever per-formed. Hence we focus on analyzing the worst case ofthe expansion phase in both algorithms.

Both algorithms are output dependent, where theruntime depends on the size of the final subset |VZ | andthe size of its neighborhood |N(VZ)|.

For Greedy Nibble we can maintain N(VZ) andcalculate v+ in O(|N(VZ)|) time since dP only dependson the number of edges and KVZ

. Thus the algorithmtakes O(|VZ |·|N(VZ)|) time for each seed since v+ needs

to be calculated each iteration.The Greedy Chomp algorithm could revert to the

Greedy Nibble algorithm if F is immediately reducedto 1/|U+| at every iteration. So worst case it is nofaster than Greedy Nibble. In fact, each iteration takesO(|N(VZ)| log |N(VZ)|) time because the ∂Ψ(U, v) val-ues are sorted for all v ∈ U+. However, in practice,a much smaller number of iterations are required be-cause a large fraction of vertices are added at eachiteration. If F were fixed throughout the algorithm,then we can loosely bound the runtime as O(log |VZ | ·|N(VZ)| log |N(VZ)|). Since F is generally large whenmost of the vertices are added, this is a fair estimate ofthe asymptotics.

This analysis is further evaluated empirically inSection 6.

5 Bipartite Extensions.

Many data sets which are applicable to this model (seeSection 6) are bipartite graphs. Thus we derive here thekey bipartite extensions of section 2 and section 3.

An undirected graph G = (V,E) is bipartite if thereis a partition V = X ∪ Y with X and Y disjoint andE ⊆ {{u, v}|u ∈ X, v ∈ Y } . A cluster is defined as asubgraph Z = (VZ , EZ) = (XZ∪YZ , EZ). The bipartiteversion of the Poisson random graph model does notallow self-loops and the total number of observed edgesin G is c(G) =

∑

vi∈X ki =∑

vj∈Y kj . The bipartiteversion of the local modularity and Poisson discrepancyfollow naturally.

6 Analysis.

This section focuses on empirically exploring four as-pects of this work. First, we investigate the power andruntime of our algorithms. Second, we use Poisson dis-crepancy as a tool to evaluate and compare differentclustering algorithms. Third, we investigate proper-ties of the clusters found by our algorithms maximizingPoisson discrepancy. Specifically, we show that vary-ing the γ parameter can give reliable estimates of thesize of clusters and we examine cases when distinct rel-evant clusters overlap. Finally, throughout this analysiswe demonstrate that maximizing Poisson discrepancyreveals interesting and relevant clusters in real-worldgraphs.

6.1 Performance of Our Algorithms.

Runtime on Real-world Datasets. We demon-strate the effectiveness of our algorithm on a variety ofreal world datasets, the sizes of which are summarizedin Table 1.

The DBR dataset describes connections betweenthreads (the set X) and users (the set Y ) of the

Duke Basketball Report message board from 2.18.07to 2.21.07. Other datasets include Web1 which linkswebsites and users, Firm which links AP articles andbusiness firms, Movie which links reviewers and moviesthrough positive reviews, and Gene which links genesand PubMed articles. In each case, high discrepancyclusters can be used to either provide advertising focusonto a social group or insight into the structure of thedataset.

Dataset |X| |Y | |E| Nibble Chomp

DBR 68 97 410 0.025 0.018Web 1023 1008 4230 0.179 0.049Firm 4950 7355 30168 9.377 0.251Movie 1556 57153 1270473 - 32.91Gene 6806 595036 1578537 - 242.7

Table 1: Sizes of real-world datasets and the averageruntime in seconds for the Greedy Nibble and GreedyChomp algorithms starting with singleton seeds. Run-times for Web and Firm were generated with 100 randomsamples. Runtimes for Movie and Gene were generatedwith 50 random samples.

Power tests. The power of the test is the prob-ability that the test statistic exceeds a critical valueunder some alternative hypothesis compared to somenull hypothesis [19]. To calculate the power of our algo-rithm, we synthetically insert significant clusters into100 random graphs and report the fraction of thesegraphs where our algorithm found the injected cluster.

In particular, we generate bipartite graphs using thePoisson random graph model such that |X| = |Y | = 100and |E| = 500 where the expected degrees of verticesvary between 3 and 7. To inject a significant cluster,we choose a random set of vertices VZ = XZ ∪ YZ ,where XZ ⊂ X, YZ ⊂ Y , and |XZ | = |YZ | = 15. Weincrease the probability that an edge is chosen betweentwo vertices in XZ and YZ by a factor of ρ. We scalethe probabilities on the other pairs of vertices in thegraph so that each vertex retains its original expecteddegree. By choosing an appropriate value of ρ, we cangenerate graphs with the same expected degree sequenceand whose injected cluster is expected to be significant.We repeat this process until we generate 100 graphswhose injected clusters have a p-value less than 0.005.

We run Greedy Nibble and Greedy Chomp usingeach vertex as a seed. We say that we successfully foundthe injected cluster Z induced by VZ = XZ ∪ YZ if thealgorithm returns a cluster Z = XZ ∪ YZ such thatd(XZ ,XZ) ≤ |XZ | and d(YZ , YZ) ≤ |YZ | and it either

1We thank Neilsen//Netratings, Inc., who provided the WEBdataset to us, for permission to use its data in this investigation.

has p-value less than 0.05 or is among the top 5 clustersfound.

We report the power of the algorithms in Table 2.It shows that 85% of the time Greedy Chomp locatesthe injected clusters. Note that we have used a relaxedcriteria to determine when an injected cluster is foundby our algorithm; a tighter qualification would reducethis power measurement.

Algorithm Nibble ChompPower 0.83 0.85

Table 2: Power for Greedy Nibble and Greedy Chomptested on graphs of size 100×100 with an injected clusterof size 15 × 15 with p-value at most 0.005.

6.2 Algorithm Comparison. Poisson discrepancyprovides an absolute measure of cluster significance.This allows comparison between different clusteringsof the same graph. We can evaluate the effectivenessof existing clustering algorithms by calculating thediscrepancy of the clusters they find. Furthermore,we can enhance these clusters using Greedy Nibbleor Greedy Chomp to maximize their discrepancy andevaluate how far from the local optimum these clustersused to be. We illustrate this by running the MCLalgorithm [41] and the Ncut algorithm [40] on DBR.MCL generated 35 clusters and we fixed the numberof clusters in Ncut to be 10. We report the top 4clusters with the highest dP value in Table 3. Ncutseems to find clusters with higher discrepancy. We thenuse clusters found by MCL and DBR as seed sets inthe Greedy Chomp, further refining them in terms oftheir discrepancy. Ncut tends to do better than MCLin finding clusters within closer proximity of discrepancylocal maxima.

MCL 0.0376 0.0248 0.0223 0.0211MCL+Chomp 0.0667 0.0790 0.0620 0.0698

Ncut 0.0692 0.0529 0.0527 0.0473Ncut+Chomp 0.0757 0.0688 0.0635 0.0713

Table 3: dP values of top 4 clusters found with MCL andNcut on DBR and the dP values after their refinementwith Greedy Chomp.

6.3 Cluster Properties.

Cluster Overlap Analysis. Many graph cluster-ing methods partition the data into disjoint subsets, es-sentially making each a cluster. Our approach findsclusters which may overlap, and it considers the rest ofthe graph uninteresting instead of forcing it to be a clus-

ter. We examine the top 6 clusters found from GreedyNibble on DBR in an overlap matrix (Table 4). Weuse each vertex in X as a singleton seed set. The 1st,2nd, 3rd and 5th clusters are very similar, representinga consistent set of about 13 threads on topics discussingthe performance of players and strategy. The 4th clus-ter contains 14 threads which were posted by users whoseem more interested in the site as a community and aremore gossiping than analyzing. The 6th cluster containsan overlap of the above two types of topics and users:users who are interested in the community, but also takepart in the analysis. The rest of the threads (about 60)deal with a wider and less focused array of topics.

C 1 2 3 4 5 6 dP p-value1 13 12 12 1 12 8 0.0783 0.0092 12 12 11 1 12 8 0.0764 0.0193 12 11 14 0 11 7 0.0754 0.0204 1 1 0 14 1 6 0.0749 0.0205 12 12 11 1 13 8 0.0718 0.0226 8 8 7 6 8 16 0.0703 0.077

Table 4: Overlap of threads among the top 6 clusters forDBR with their dP and p-values found with the GreedyNibble algorithm.

γ as a Resolution Scale. For our algorithm, as γvaries, we observe an inverse linear correlation betweenγ and the average cluster size (Figure 3). We also showthat as γ varies, our algorithm locates clusters that arestatistically significant on different scales, and that theircontents remain meaningful.

This near-linear correlation makes γ a reliable res-olution scale for our clustering algorithm. As γ goes to0, the algorithm produces the whole graph as a cluster.As γ goes to infinity, the algorithm produces trivial sin-gletons. The flexibility to modify γ allows a user to bal-ance the importance of the statistical significance of theclusters found, maximized by dP,γ , and their preferredsize weighted by γ. This helps resolve issues (previ-ously noted about modularity by [17]) about the pre-ferred size of clusters which optimize dP . For instancewhen searching for more focused clusters of smaller size,a reasonable γ weight can be easily inferred.

Cluster content and γ. Manual evaluation of theresults show that the contents of the clusters remainmeaningful and useful as γ is varied. For example, thetop clusters found on the Movie dataset with γ = 200 areshown to be popular box office movies in the 90’s as theyare consistently reviewed favorably by various reviewers.The top two clusters found on the Gene dataset with γ =200 are genes in the UDP glucuronosyltransferase 1 and2 family. The 4th ranked cluster consists of genes suchas MLH and PMS, both are related to DNA mismatch

0 0.1 0.2 0.3 0.4 0.50

50

100

150inverse gamma vs. average cluster size X

aver

age

clus

ter

size

X

inverse gamma0 0.02 0.04 0.06 0.08 0.1

0

20

40

60

80

100inverse gamma vs. average cluster size X

aver

age

clus

ter

size

X

inverse gamma

0 0.005 0.01 0.0150

5

10

15

20

25

30inverse gamma vs. average cluster size X

aver

age

clus

ter

size

X

inverse gamma0 1 2 3 4 5

x 10−3

5

10

15

20

25

30inverse gamma vs. average cluster size X

aver

age

clus

ter

size

X

inverse gamma

Figure 3: Plot of 1/γ vs. average cluster size on Web

(top left), Firm (top right), Movie (bottom left), andGene (bottom right).

repair. The 8th ranked cluster for γ = 200 persistsas the top ranked cluster when γ = 600; it consists ofseveral genes for the zona pellucida glycoprotein, i.e ZP1and ZP3A.

As γ increases, the nontrivial clusters with a highdP,γ-discrepancy should generally be much denser inter-nally, since the ratio between the actual internal edgesand expected edges should be greater than a given γ. Onthe other hand, clusters which persist as the top rankedclusters as γ increases are those that are most statisti-cal significant in a dynamic setting. As γ increases, wewould expect the number of such extremely anomalousclusters to decrease. For example, as shown in Figure 4for the Web data set, as γ increases, the number of out-lier clusters with comparatively very large discrepancydecreases. For γ = 4, many clusters seem to be sig-nificantly larger than the large component, while withγ = 6 and γ = 8 there are very few. Finally, with γ = 10all clusters are basically in the same component.

The identification of clusters of varying size butconsistently high statistical significance suggests thatreal-world networks are characterized by many differentlevels of granularity. This result is consistent with, e.g,the contrasting findings of [37] and [31], where clustersof vastly different sizes but comparable modularities aredetected in the same data set. This finding calls intoquestion the wide variety of clustering methods whichare only designed to detect one cluster for a given regionor group of nodes, and a further study would be ofinterest.

0 0.005 0.01 0.015 0.02 0.0250

50

100

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250cluster rank vs. cluster discrepancy

clus

ter

rank

cluster discrepancy0 0.005 0.01 0.015

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250cluster rank vs. cluster discrepancy

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cluster discrepancy

0 0.002 0.004 0.006 0.008 0.01 0.0120

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250cluster rank vs. cluster discrepancy

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ter

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cluster discrepancy0 2 4 6 8

x 10−3

0

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250cluster rank vs. cluster discrepancy

clus

ter

rank

cluster discrepancy

Figure 4: Web: cluster rank vs. cluster discrepancywith each X vertex used as a singleton seed set, γ = 4(top left), γ = 6 (top right), γ = 8 (bottom left),γ = 10 (bottom right). Top ranked clusters appear atthe bottom right of each figure.

7 Conclusions.

The main contribution of this paper is the introductionof a quantitative and meaningful measure, Poisson dis-crepancy, for clusters in graphs, derived from spatialscan statistics on point sets. According to our defini-tion, the higher the discrepancy, the better the cluster.We identify interesting relations between Poisson dis-crepancy, local modularity, and Bregman divergences.

To illustrate the usefulness of this statistic, we de-scribe and demonstrate two simple algorithms whichfind local optima with respect to the spatial scan statis-tic. In the context of real-world and synthetic datasetsthat are naturally represented as bipartite graphs, thismethod has identified individual clusters of vertices thatare statistically the most significant. These clusters arethe least likely to occur under a random graph modeland thus best identify closely-related groups within thenetwork. Our model places no restrictions on overlap-ping of clusters, thus allowing a data point to be clas-sified into two or more groups to which it belongs. Asour greedy algorithms are the simplest and most intu-itive approach, it remains an open problem to find moreeffective algorithms to explore the space of potentialsubgraphs to maximize the Poisson discrepancy. Noticethat Poisson discrepancy can also detect regions thatare significantly under-populated by requiring p < q inthe alternative hypothesis.

Similarly the spatial scan statistic Bernoulli model

for graph clustering can be derived from the correspond-ing model for point sets. However, this model requiresthat each potential edge be chosen with equal probabil-ities, regardless of the degree of a vertex. Also, underthis model each pair of vertices can have at most oneedge.

In summary, we argue that a graph cluster shouldbe statistically justifiable, and a quantitative justifica-tion comes from a generalization of spatial scan statis-tics on graphs, such as the Poisson discrepancy.

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