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Tracking Temporal Community Strength inDynamic Networks

Nan Du, Xiaowei Jia, Jing Gao, Vishrawas Gopalakrishnan, and Aidong Zhang, IEEE Fellow

Abstract—Community formation analysis of dynamic networks has been a hot topic in data mining which has attracted much attention.Recently, there are many studies which focus on discovering communities successively from consecutive snapshots by considering boththe current and historical information. However, these methods cannot provide us with much historical or successive information relatedto the detected communities. Different from previous studies which focus on community detection in dynamic networks, we define anew problem of tracking the progression of the community strength - a novel measure that reflects the community robustness andcoherence throughout the entire observation period. To achieve this goal, we propose a novel framework which formulates the problemas an optimization task. The proposed community strength analysis also provides foundation for a wide variety of related applicationssuch as discovering how the strength of each detected community changes over the entire observation period. To demonstrate thatthe proposed method provides precise and meaningful evolutionary patterns of communities which are not directly obtainable fromtraditional methods, we perform extensive experimental studies on one synthetic and five real datasets: social evolution, tweetinginteraction, actor relationships, bibliography and biological datasets. Experimental results show that the proposed approach is highlyeffective in discovering the progression of community strengths and detecting interesting communities.

Index Terms—Dynamic Networks, Community Analysis, Community Strength

F

1 INTRODUCTION

In recent years, there has been a growing interest in mod-eling and mining various kinds of dynamic networkswhose structures evolve over time, such as biologicalnetworks, social networks, co-authorship networks andco-starring networks. Specifically, people have investi-gated community analysis in dynamic networks [1]–[4]. The focus is on detecting communities successivelyfrom consecutive snapshots by considering the historicalinformation [5]–[7]. Although these methods can give usquite reasonable and robust communities by consideringthe temporal smoothness, few historical and successiveinformation related to these communities are provided.Thus we do not know when these communities wereassembled or when they are going to disband. Aimingto answer these questions, we propose a novel measurecalled community strength, which can reflect a commu-nity’s temporal community robustness and coherencethroughout the entire observation period.

In this paper, we define that a community is with highstrength if it has relatively stronger internal interactionsconnecting its members than the external interactionswith the members to the rest of the world. Dense internalinteractions and weak external interactions guaranteethat the community is under a low risk of memberchange (current members leaving or/and new membersjoining). Intuitively, a friend community is “strong” if itsmembers tie together closely and ignore the temptation

• All of the authors are with the Computer Science and EngineeringDepartment, State University of New York at Buffalo, Buffalo, NY 14260.E-mail:nandu,xiaoweij,jing,vishrawa,[email protected]

from the outside world. On the contrary, a friend com-munity is regarded as a “weak” community if it is likelyto confront a member alteration situation. To illustratethis concept, Fig. 1(a) shows a toy example, where thenodes represented by the same geometric shape belongto the same community, solid lines represent internalinteractions and dash lines represent external interac-tions. The circle community (i.e. nodes A, B, C and D) isconsidered to be stronger than the rectangle community(i.e. nodes E, F, G and H), due to the weaker externalattractions. On the other hand, node H has a closerelationship with the diamond community (i.e. nodes I,J and K), which makes the rectangle community in therisk of losing its members. In other words, the higherstrength score a community obtains, the less possiblemember alternation occurs in it. It is worth noticing thatcommunity strength is a measure which syntheticallyconsiders both the community cohesion (i.e. how closethe members are in a community) and separation (i.e.how distinct a cluster is from the other clusters).

Furthermore, community strength should be a tempo-ral measure whose value may change as the networkevolves. Here’s an example in the real world. A set ofauthors have collaborated closely from 2000 to 2006.During this period, they cooperated frequently amongthemselves and barely with others outside the commu-nity. However, after 2006, because of interest changes,some authors’ attentions have been attracted to someother fields. Thus the internal cooperation decreasedand the external cooperation increased. In this case, thisauthor community’s strength is high and stable during2000-2006, but begins to decrease after 2006. As a toyexample, in Fig. 1(b) (i.e. the network in the 2ndsnapshot

IEEE Transactions on Knowledge and Data Engineering Vol No 99 Year 2015 1

2

) which evolves from Fig. 1(a) (i.e. the network at the1stsnapshot ), the strength of the rectangle commu-nity decreases, because the internal connections becomeweaker and external connections become stronger.

A

BC

D

E

F

G

H

I

K

J

(a) 1stsnapshot

A

BC

D

E

F

G

I

K

JH

(b) 2ndsnapshot

Fig. 1: A Toy Example Illustrating Community Strength

Discovering the progression of community strengthscan offer significant insights in a variety of applications.It can help us discover some interesting communityinformation which can not be directly obtained fromtraditional community analysis. Interesting examples ofcommunities’ strength progression can be commonlyobserved in real-life scenarios. Here we discuss twospecific cases in detail.

Strengths Progression in Actor Community: As astrong actor community, the cooperation should be morefrequent between the members themselves than be-tween members and non-members. For example, con-sidering the popular and long-running television sitcom‘Friends’1, its six main actors J. Aniston, C. Cox, M. Perry,M. LeBlanc, L. Kudrow and D. Schwimmer collaboratedclosely when this sitcom was aired from 1994 to 2004.Let’s consider each year’s co-starring relationships asone snapshot. We can see that the strength of thiscommunity is very low before 1994 (little cooperationbetween them), and then dramatically increases andkeeps stable from 1994 to 2004 (average 23 episodeseach year). Finally, the strength of this community appar-ently becomes weaker after 2004 (much less cooperationcomparing to the previous years). The progression ofthis actor community’s strengths shows an interestingpattern of cooperation history among these six actors.Learning the strength progression of actor communitieshelps us better understand the entertainment industry.

Strength Progression in Gene Community: In the bi-ological domain, the interactions between genes changegradually in dynamic gene co-expression networks. Thusthe strength of gene communities also changes. Forexample, it has been reported that the expression pro-filing of some key genes will change [8] as the can-cer progresses. In such cases, the corresponding genecommunities’ strength also changes. Discovering thestrengths of gene communities throughout a specificdisease progression can help us find significant clues inthe fields of medicine and biology. For a specific disease,

1. http://www.imdb.com/title/tt0108778/

if a gene community is found strong only at the earlystage, it is very likely to be a crucial trigger for thedisease deterioration.

From the above cases, we can see that discovering theprogression of community strengths helps us understandthe underlying behavior of communities. The initial ideawas published in [9], which covers the basic definitionof community strength and the evolutionary analysis ondynamic networks. By utilizing the community strengthvalue, the consistent communities can be detected andtracked over an observation period. This paper extendsthe original idea to formulate a solid method withbroader applications and provide more supportive andcomprehensive experiments. In this paper, our goal is todetect the temporal strength of each detected communitythroughout all the snapshots so that we can answerthe following questions: How does the strength of eachcommunity change over the observation period? Whatare the top-K strong communities throughout the obser-vation period? How do the communities from adjacentsnapshots influence the strength of each other?

To sum up, our main contributions in this paper areas follows:• We introduce the notion of progression analysis of

community strengths. To the best of our knowledge,this is the first work on analyzing the temporal com-munity quality or structure information consideringboth time and community information.

• We formulate the problem as an optimizationframework that can effectively detect the temporalstrength of communities and track the strength pro-gression pattern.

• Experiments on the synthetic dataset show the pro-posed approach is effective on identifying strongcommunities. On real datasets, interesting andmeaningful communities are detected. Case studiessuggest that the proposed approach can providemore reasonable results.

The organization of the paper is as follows: In Sec-tion 2, we describe the setting of our problem. Section 3presents the analysis and discussions related to theproposed algorithm. This is followed by discussions onthe extensibility of our approach to other applications -Section 4. Extensive experimental studies are reported inSection 5. We then recapitulate related existing work inSection 6 before concluding our work in Section 7.

2 PROBLEM SETTING

In this section, we first introduce the definition of com-munity strength and related notations, and then formallydefine the problem. Before proceeding further, we in-troduce the notation that will

beused in the following

discussion: Let a matrix be represented with uppercaseletter (e.g. D), dij denotes the ij-th entry in D, and di.and d.j denote vectors of i-th row and j-th column ofD, respectively. Now, let us start by introducing thedefinition of the community strength.


3

Community strength: Given a network G = (N,E,W )where N is the set of nodes in this network, E is the setof edges connecting the nodes, and W is a symmetricweight matrix representing the weights on edges. Therehave been some existing work on measuring the strengthof community by considering its internal compactnessor identifying outlier data with probability model [10].In this paper, we propose the measurement for thecommunity strength which very well fits our problem inreal scenarios. The community strength of a communityz can be defined as:

Strength(z) =∑i∈N

∑j∈N

wij∗∑k∈z

∑l∈z

wkl−

(∑k∈z

∑v∈N

wkv

)2

, (1)

where∑

i∈N∑

j∈N wij denotes the sum of all edgeweights in the network,

∑k∈z

∑l∈z wkl denotes the

sum of internal edge weights inside community z, and∑k∈z

∑v∈N wkv denotes the sum of internal and external

edge weights attached to nodes in community z. Theterm

∑i∈N

∑j∈N wij is adopted to guarantee the pos-

itive strength value. We propose Eq. 1 inspired by themodularity definition in [11], where the metric is used tomeasure the quality of the overall network partitioning.The rationale of Eq. 1 is that a strong community shouldsimultaneously obtain dense internal connections andsparse external connections.

Now, our problem can be defined as follows:Input:• A series of undirected networks Gt = (V,Et,W t)

(1 ≤ t ≤ T ), where each network has N nodes (i.e.|V | = N ). For each snapshot t, V is a set of nodes,Et is a set of interactions between these nodes andW t

N×N is a symmetric weight matrix. For vi, vj ∈V , wt

ij indicates the interaction frequency betweennodes vi and vj at snapshot t. Note that edges inEt (1 ≤ t ≤ T ) could be weighted or unweighted.

Output:• Community Pool Matrix: We summarize the com-

munities detected from all the snapshots into anN × K community pool matrix C where K is thenumber of all the unique communities (i.e. K = |C|).In addition, C equals to C1 ∪ C2∪, ...,∪CT whereCt (1 ≤ t ≤ T ) is the temporal community indicatormatrix with respect to a certain snapshot t (moredetails will be introduced later).

• Strength for each community at each snapshot: Leta K × T matrix A denote the temporal strength forall detected communities, where akt refers to thestrength of community k at snapshot t.

To derive the output, the following variables areneeded.

Nuisance Parameters:• Temporal Community Indicator Matrices: At each

snapshot t, the community indicator matrix Ct (1 ≤t ≤ T ) is an N ×Kt matrix where Kt is the numberof communities captured at snapshot t. If node i is

assigned to community k at snapshot t, then Ctik = 1

and 0 otherwise. As we mentioned, all the temporalcommunity indicator matrices Ct (1 ≤ t ≤ T )compose the community pool matrix C. Note thatthe communities represented in this matrix can beeither overlapping or non-overlapping.

• Temporal Community Relationship Matrices: Ateach snapshot t, we denote the community relation-ship matrix St as a Kt × Kt matrix. Note that stijrepresents the similarity between community i andcommunity j that are detected at snapshot t.

Table 1 summarizes the important notations that weuse in this paper.

TABLE 1: Table of Notations

Symbol DefinitionW t

N×N wtij : weight between object i and j at time t

CtN×Kt

ctik: indicator of object i in community k at time tCN×K cik: object i grouped into community kAK×T akt: strength score of community k at time t1, ..., T snapshot indexes1, ..., N object indexes1, ...,Kt community indexes at time t1, ...,K community indexes in the community pool

3 METHODOLOGY

In this section, we present our method for solving theproblem of temporal community strength analysis. Webegin by introducing the method of partitioning thenetwork from each snapshot into communities in Section3.1, and then show the method of tracking the strengthof each community over time in Section 3.2.

3.1 Community Detection at Each SnapshotGiven a series of temporal networks Gt =(V,Et,W t) (1 ≤ t ≤ T ), we first partition eachnetwork independently into Kt communities at eachtimestamp t. Due to the change of network, the valueof Kt may not be the same across different snapshots.Then we store all the detected communities from all thesnapshots in a community pool.

To detect communities from each temporal network,we use Non-negative Matrix Factorization (NMF) tech-nique [12]. There are two major reasons to choose NMF:First, it can be easily applied to both hard clustering(i.e. each object belongs to exactly one community) andsoft clustering (i.e. each object can belong to multiplecommunities). The property of soft clustering very wellfits many real social scenarios. For instance, each userin social network usually participates in more than onediscussion group, as he may have a variety of interestedtopics. Second, it could uncover the underlying inter-community relationships quite accurately, that can beutilized for other related tasks like progression analysis- refer Section 4.1. The details of these advantages arediscussed further in the following discussion of the


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method. Please note that we believe one can opt to useother evolutionary clustering algorithm so long as itprovides a mechanism for soft clustering and also theability to identify inter-community relationships.

In this paper, we mainly focus on the undirected net-work, where the matrix W is symmetric, the clusteringto the rows and columns should be identical. Hencewe propose to symmetrically factorize each temporalnetwork as follows:

minCt≥0,St≥0

∥∥∥W t − CtStCtTrans∥∥∥2 , s.t. CtTrans

Ct = I. (2)

W t is an N × N symmetric matrix that demonstratesthe interactions between objects at time t. Ct is anN×Kt community indicator matrix, each entry of whichrepresents the probability of assigning an object into acommunity. St is a Kt ×Kt matrix, providing the rela-tionship between communities detected at time t. BothCt and St should be non-negative. To solve this problem,we propose the following procedure to iteratively updateCt and St. Specifically, at iteration t, when St is fixed,we update Ct as:

ctij ← ctij2

√(W tTransCtStTrans)ij

(CtCtTransWTransCtStTrans)ij. (3)

Similarly, fixing Ct, we can obtain the update rule for St

as:

stij ← stij2

√(CtTransW tTransCt)ij

(CtTransCtStCtTransCt)ij. (4)

During implementation, we add a small value on thedenominator to make sure it is not zero. We iterativelyupdate Ct and St until convergence. It is worth noticingthat the community indicator matrix Ct can be used toderive both hard clustering and soft clustering. Eachrow of Ct denotes the chance that the correspondingobject belongs to the Kt communities. Thus, to get hardclustering results, each object can be assigned to thecommunity with the largest value in the correspondingrow of the community indicator matrix. As for softclustering, we can simply set up a cut-off threshold forthe row-based normalized community indicator matrixCt so that the object is assigned to the cluster whosevalue in Ct is greater than the threshold.

Moreover, it can be proved that stlk ≈ CtTrans

l. W tCtk. =

1|Ct

l ||Ctk|∑Kt

i=1

∑Kt

j=1Wtij [13]. It can be seen that stlk

demonstrates the relationship between the l-th commu-nity and the k-th community. If the network is well-separated, hard clustering can be applied and then St

is approximately a diagonal matrix where off-diagonalelements are much smaller than diagonal elements.When there exist overlapping between communities, softclustering is more appropriate, and thus the differencebetween diagonal and off-diagonal elements is smallerthan that observed in the hard clustering case. Therefore,St can uncover the underlying relationships between

communities more accurately. It is better than the ap-proach that only compares the memberships betweencommunities using measures such as Jaccard coefficient[14]. Furthermore, St can be used to construct thestrength progression net which will be introduced inSection 4.1. These advantages justify our usage of non-negative matrix factorization to decompose each tempo-ral network.St only captures community relationships at each

snapshot, but we are interested in its evolution. There-fore, to consider the communities from different times-tamps, we propose to put all the detected communitiesinto a community pool CN×K and derive the evolution-ary pattern. This community pool covers a larger can-didate set where all the communities can be compared.Based on this community pool, we plan to find out whichcommunities are grouped closely and consistently overthe entire tracking period and which communities aregrouped temporarily.

3.2 Temporal Community Strength Analysis

Now, we propose an integrated optimization frameworkthat conducts community strength estimation acrosssnapshots. A naive approach for this task is to calculatethe strength of each community individually at eachsnapshot and track the evolution. However, this ap-proach does not take historical information into accountwhen deriving community strengths and the communi-ties derived across snapshots are not easily comparable.In contrast, we propose the following framework basedon the smoothness assumption in which both currentand historical networks contribute to the communitystrength detection. Moreover, in the proposed frame-work, communities across snapshots are brought intoalignment so that we can easily compare them.

Based on Eq. 1, the strength of community z can befurther reformulated in terms of the community poolmatrix C as follows:

Strength(z) =

n∑i,j=1

wij

n∑i,j=1

wij ciz cjz −

n∑i,j=1

wij ciz

2

= sum(W )cTransz. Wcz. − (

n∑i=1

diciz)2

= cTransz. (W −D)cz.,

(5)

where sum(W ) denotes the sum of weights for networkW and W equals to sum(W ) ∗W . D equals to ddTrans

where d is an N × 1 vector such that each di is thedegree of vertex i. Employing Eq. 5, we formulate thetask of estimating a particular community z’s strengthat snapshot t (referred as azt) as the following objectivefunction:


5

mina.t

J(a.t) = α

K∑z=1

log(1

azt)[cTransz. (W t −Dt)cz.

]+(1− α)

K∑z=1

log(1

azt)[cTransz. (W t−1 −Dt−1)cz.

]s.t.

K∑z=1

azt ≤ µt, azt ≥ 0,

(6)

where log( 1azt

) determines the z-th community’s strengthweight corresponding to the snapshot t and µt is the es-timated sum of community strengths with respect to thecurrent snapshot. For the sake of simplicity, we set µt as1 in the experiments. We propose the objective functionEq. 6 based on Eq. 1 to combine the historical informa-tion and the current snapshot. The temporal communitystrength is derived from the optimization frameworkwith the assumption of smoothness. In real scenarios, thenetwork is usually expected to evolve gradually ratherthan abruptly. Hence we include the second term in Eq. 6to combine the strength at previous timestamp. log( 1

azt)

is used to capture community strength weight due tothe following reasons. First, the logarithm function helpsrescaling the strength values by converting them into asmall range. Second, it makes the optimization functioneasier to solve as negative logarithm is a convex function.The stronger a community z at snapshot t, the higherazt will be and hence log( 1

azt) will be lower. Therefore,

to optimize the function, the higher weight will beassociated with the community that is stronger at thecurrent snapshot.

Importantly, smoothness is considered in the ob-jective function. Note that in Eq. 6, the first termα∑K

z=1 log(1


]measures the cost

of all the detected communities in the communitypool with respect to the current snapshot’s net-work, where a high cost means the communitiesin this snapshot are weak. The second term (1 −α)∑K

z=1 log(1


]denotes the

temporal smoothness in terms of the goodness of thecurrent clustering result with respect to the previousnetwork, where a higher temporal cost means that thesmoothness assumption is violated and inconsistencyis observed across snapshots. Therefore, this objectivefunction can better capture the strength calculation ateach snapshot and across snapshots.Temporal Smoothness

In many real-world dynamic network applications,networks are expected to change gradually and stably.Examples include geometric networks [15] and genenetworks [16]. As a consequence, we expect a cer-tain level of temporal smoothness between communitystrengths in successive snapshots. The temporal commu-nity strength should depend on the current network, andit should not deviate too dramatically from the previ-

ous snapshot’s network. Actually, temporal smoothnessassumption has been adopted in many previous evolu-tionary clustering work [6], [7], [17]. However, insteadof applying the smoothness among the clusters detectedin adjacent timestamps as previous work did, we haveapplied it on the temporal community strength. In Eq. 6,the overall cost of the objective function is representedas the linear combination of the cost of communitystrength fitting to the current snapshot and the cost ofcommunity strength fitting to the previous snapshot.Thus α (0 ≤ α ≤ 1) is a predefined parameter toreflect users’ emphasis on the smoothness assumption.Usually, α could be assigned a relatively large valuewhen the networks are stable and evolve slowly (e.g.,social networks). α should be assigned a relatively smallvalue when the target networks include noise and arelikely to evolve swiftly.PACS Algorithm Procedure

Now, we derive the solution for the communitystrength scores azt for objective function shown in Eq.6. Using the method of Lagrangian Multipliers, we canrewrite Eq. 6 as follows:

mina.t

J(a.t) = α

K∑z=1

log(1


]+(1− α)

K∑z=1

log(1


]+γ

(K∑

z=1

azt − µt

),

(7)

where γ is a Lagrangian multiplier. Taking the partialderivative of Eq. 7 with respect to azt and setting thederivative to 0, we obtain Eq. 8 and Eq. 9.

azt =αcTrans

z. (W t −Dt)cz. + (1− α)cTransz. (W t−1 −Dt−1)cz.

γ(8)

γ =

∑Kz=1

[αcTrans

z. (W t −Dt)cz. + (1− α)cTransz. (W t−1 −Dt−1)cz.

]µt

(9)

Plugging Eq. 9 into Eq. 8, we obtain the solutionfor azt which is shown as Eq. 10. In this equation, thenumerator measures the strength of the community zat the snapshot t and integrates both the current andhistorical information. The denominator represents theoverall strength across all the communities at snapshott, which serves as a normalization factor. Furthermore,µt, as mentioned before, controls the overall communitystrength at a specific snapshot. The intuition behind Eq.10 is that the communities which have compact structureat the current snapshot will be assigned a larger strengthscore; while the community whose structures are loosewill receive a lower value. The algorithm is summarized


6

in Algorithm 1, and we name our method as PACS(Progression Analysis of Community Strength).

azt=

[αcTrans

z. (W t −Dt)cz.+(1− α)cTransz. (W t−1−Dt−1)cz.

]µt∑K

z=1

[αcTrans

z. (W t−Dt)cz.+(1− α)cTransz. (W t−1−Dt−1)cz.

](10)

Algorithm 1 The PACS AlgorithmInput: A series of temporal networks W t

N×N (1 ≤ t ≤ T ),a series of estimated sum of community strength for eachsnapshot µt (1 ≤ t ≤ T ), community pool matrix CK×Nand a temporal smoothness parameter αOutput: Estimated Community Strength matrix AK×T

1: t← 1;2: begin3: Detect the communities Ct with respect to each snapshot;4: Generate the community pool C;5: repeat6: Estimate a.t using Eq. 10;7: t← t+ 1;8: until t > T9: Output A;

10: end

In the case with directed networks, the communitydetection can be implemented with the revised methodbased on cuts, spectral clustering, random walks ormarkov chains [18]. Also we can generalize our modelto the directed graph with the asymmetric NMF.

4 EXTENSIBILITY TO OTHER APPLICATIONSBy formulating the problem as the task of measuring thecommunity strength over an observation period, we canextend our method to perform some additional tasks. Inthis section, we explain the extensibility of Algorithm 1to: (1) Measure the impact and consequently the changein community strength based on immediate precedingtimestamps, and (2) Identify top-k strongest and weakestcommunities.

4.1 Community Strength Progression NetThe output of Algorithm 1 provides information on howall the communities’ strength evolve over time. In addi-tion to that, we also want to know how the communitiesfrom immediate preceding snapshots (i.e. Ct−1 and Ct)influence the strength of each other. To illustrate theserelationships, we construct a bipartite network that rep-resents the relationship between communities detectedat snapshot t-1 and communities detected at snapshott. In such a network, the nodes on the left representthe communities detected at previous timestamp, thenodes on the right represent the communities detectedat the current timestamp and the edges connecting thenodes denote the influence transmission between thecommunities.

The relationship matrix PKt−1×Kt that represents therelationships between communities captured at adjacentsnapshots (t-1 and t) can be calculated as:

P = D−1St−1Ct−1CtT StT , (11)

where D is a diagonal matrix used for normalizationand Dii =

∑Kt

j=1(St−1Ct−1CtT StT )ij . As we mentioned

in Section 3.1, Ct and Ct−1 are the community indicatormatrices with respect to snapshot t and t-1. St andSt−1 represent the relationship between communities atsnapshot t and t-1, respectively. As we mentioned pre-viously, St can uncover the underlying relationships be-tween communities detected at snapshot t, and Ct−1CtT

demonstrates the number of common members betweenthe two snapshots’ communities. Thus, P can reflectnot only the common member relationship but alsothe underlying relationships between two snapshots’communities.

A natural definition of community progression net(from ct−1i at time t-1 to ctj at time t) is a flow startingfrom ct−1i , and transmits its strength to ctj . There are twoapplications that are worth discussing: First, we analyzehow the community strength from the current snapshottransmits to the next snapshot. Second, we analyze howthe current community strength succeeds from the previ-ous snapshot. For the former one, the strength transmitscommunity i at the current snapshot to the communityj at the next snapshot, which is defined as aitpij . Asmentioned before, ait is the strength of community iat time t and pij is the relationship among communityi and j, aitpij can reflect the influence community jobtained from community i. The network reflecting thistransmission relationship is named Strength TransmissionNet. Correspondingly, for the latter one, the strength thatthe current community j inherits from community i isdefined as pijajt, which is named Strength Reception Net.Notice that to measure the Strength Reception Net, weneed to normalize each column of P .

Examples for Strength Transmission Net and StrengthReception Net are depicted in Fig. 2(a) and 2(b), re-spectively. In each network, the values shown insidethe geometric shapes are the strength corresponding tothe communities. For example, from Fig. 2(a) we cansee that the circle community from the 1st snapshottransmits its current strength (0.46) to the succeedingcircle community with 0.44 and rectangle communitywith 0.02. Take another example, from Fig. 2(b), wecan see the diamond community at the 2nd snapshotinherits 0.35 and 0.03 strength from diamond communityand rectangle community at the 1st snapshot. In suchcases, we can find out that the members from diamondcommunity at the 2nd snapshot mainly inherits fromdiamond community at the 1st snapshot.

The community strength progression net can provideus with important information about the communityevolution. For example, in the field of biology, if onegene community’s strength mainly transmits to multiplesubsequent gene communities, it is very likely that thisgene community has splitted into these communities.Take another example in the social network, if several


7

friend communities’ strengths have transmitted to onesubsequent community, we would know that these pre-vious communities merge into a larger community.

0.46

0.34

0.2 0.19

0.26

0.44

0.01

0.02

0.01

0.07

Snapshotst nd1 Snapshot2

(a) Strength Transmission Net

0.42

0.2

0.380.35

0.15

0.41

0.01

0.01

0.04

0.03

Snapshotst nd1 Snapshot2

(b) Strength Reception Net

Fig. 2: Strength Progression Nets of the Toy Example.

4.2 Top-K strongest/weakest communitiesBy applying Algorithm 1, we obtain the communitystrength for each detected community at each snap-shot. Based on this output, we can compute an overallstrength for each community, which is useful to identifyinteresting communities that are the strongest/weakestthroughout the entire observation period. There aremainly two methods to aggregate the temporal commu-nity strength scores: unweighted and weighted. In theunweighted case, we can regard each temporal score tobe of equal importance and take the sum, i.e.

∑Tt=1 azt.

However, in some cases, the community strength is moreimportant at some particular snapshots, e.g. the earlystage of cancer. In such a case, we should give differentweights to different snapshots and the aggregation func-tion can be defined as

∑Tt=1 h

tazt, where ht is the weightfor the specific snapshot t. In addition, when choosingthe top strongest or weakest communities, we may alsowant to consider the size of the communities. When thetarget networks are very sparse, the penalty from theexternal connections may be very small, thus the penaltyfrom the external interaction would be very limited. Insuch a case, the community strength value would bebiased to the large-size communities which will containmore internal connections. To mitigate this effect, theaggregated function for community z can be redefinedas:

∑Tt=1 azt

|Cz| or∑T

t=1 htazt

|Cz| so that the community strengthis normalized by its size.

5 EXPERIMENTS

In this section, we report experimental studies basedon both synthetic and real-world datasets. First, weevaluate the proposed method by comparing the de-tected strongest/weakest communities and communitystrength ranking with the ground truth on synthetic andreal-world social datasets. Then we evaluate the resultsobtained by the proposed method on actor relationship,bibliography and biological datasets using case studies.We perform comprehensive analysis to justify the top-K

strongest communities returned by the proposed algo-rithm.

5.1 Synthetic DatasetWe start with a synthetic dataset, which is generatedaccording to the method mentioned in [7]. We generatedata for a total of 30 consecutive snapshots. At the 1st

snapshot, we generate 100 nodes, which are dividedinto five communities of 20 nodes each. From the 2nd

to the 30th snapshots, edges are added randomly witha higher probability pin for within-community edgesand a lower probability probability pout for between-community edges. In this study, we set the two-tupleparameter (pin, pout) for these five communities asC1 = (0.22, 0.05), C2 = (0.2, 0.07), C3 = (0.18, 0.09),C4 = (0.16, 0.11) and C5 = (0.14, 0.13).Baselines

Since no previous methods target at the same prob-lem, we compare the proposed algorithm with severalvariations of previous approaches and the proposedapproach.PACSwithout: The first baseline, which we call

PACSwithout, adopts all the steps in the proposedmethod except the usage of the smoothness constraint.Comparison with PACSwithout will demonstrate the im-portance of the smoothness assumption.

KNN: The second baseline is the K-nearest neighbors(KNN) approach. At each snapshot, after we use KNN todetect the communities, we also calculate the strengthsfor them. Then, the strength of each specific communitycan be calculated via considering its top-K most similarcommunities’ structure information. Specifically, in KNN,the formula for strength of community i at time t can berepresented as

∑Kj=1 hijajt. Note that, ajt is the strength

of community j at time t which is calculated via Eq. 5,and hij here is the similarity between two communitiescalculated using Jaccard coefficient and

∑Kt

j=1 hij = 1.KNN represents the perspective of involving only kclosest communities in strength computation while theproposed approach conducts a global computation ofcommunity strength.

CID: The third baseline is based on community inter-nal density, which we call CID for short. As defined in[19], CIDt

c =∑

i∈c∑

j∈c wtij

|c|(|c|−1)/2 measures the internal edgedensity of the cluster c. Since CID is also proposed tomeasure the community quality, comparing with CIDcan help us understand which community quality indexcan better measure the community strength.Performance Evaluation on Synthetic Dataset

Due to the way we generate the synthetic data, wehave known that community C1 has the largest gapbetween pin and pout, which makes it the strongestcommunity throughout all the snapshots. Thus, we candirectly compare the strongest community discoveredby each method with C1. The higher similarity thedetected strongest community to C1, the more accuratethe corresponding method is. In this experiment, we


8

use Jaccard coefficient which is defined as the size of theintersection members divided by the size of the unionof the members to measure the community similarity.On the other hand, since all these five communities inthe synthetic data are more or less well separated, theweakest community should be the one composed ofmembers coming uniformly from these five communi-ties. To measure the distribution of members, we use en-tropy measure −

∑5i=1 P (xi)logP (xi), where P (xi) is the

percentage of members from community i. The higherthe entropy is, the weaker the discovered community is.Results on the Synthetic Dataset

Table 2 shows both the Jaccard coefficient for thestrongest community and the entropy for the weakestcommunity obtained by the four algorithms. From thetable we can see that the proposed method clearly out-performs the baselines on both strong and weak commu-nity detection. PACS performs better than PACSwithout,which indicates that the smoothness assumption con-tributes to the performance improvement. Moreover,CID does not perform well, since it only considers theinternal connections of the communities.

TABLE 2: Performance Comparison with Baselines

Method Jaccard coefficient Entropy(Strong Community) (Weak Community)

PACS 0.95 1.38PACSwithout 0.90 1.33

KNN 0.85 1.1CID 0.85 1.33

5.2 Social Evolution Dataset

The social evolution dataset was collected by MIT hu-man dynamics lab [20], which recorded the daily livingof 80 students in a dormitory with mobile phones. Weconstruct the student interaction networks from the rawdata. In these networks, nodes are students and an edgeexists between them if the corresponding students haveinteractions in one of the following three ways: call,message and music sharing. The weight of each edge isthe number of interactions. It is easy to see that, the morefrequently two students interact with each other, thehigher weight is assigned on the edge connecting them.In addition, there are five snapshots which correspondto the time before 10/19/2008 - 10/19/2008 (T1), 10/20/2008- 12/13/2008 (T2), 12/14/2008 - 3/5/2009 (T3), 3/6/2009 -4/17/2009 (T4) and 4/18/2009 - 5/22/2009 (T5). We willdiscuss why the time intervals are divided in this waylater.

Note that the number of communities at time t canbe determined by the modularity function [11]. To deter-mine the best community number K at time t, we trieddifferent candidates for K in a specific range and usedthe one which leads to the highest modularity functionvalue. From these student interaction networks, we hopeto rank friend communities based on their strengths. In

other words, we try to find out strong friend clans fromtheir regular social interactions.

The evaluation of temporal community strength isdifficult due to the lack of ground truth. Fortunately,besides the student interaction information, this datasetalso provides a series of temporal surveys about thecloseness degree between students, which can be usedto validate our discoveries. The surveys were mainlymade on five dates: 10/19/2008, 12/13/2008, 3/5/2009,4/17/2009 and 5/22/2009. (This is the reason why we cutthe snapshots of student interaction networks in the waymentioned above). In each survey, every student needsto indicate his/her current relationship level with theothers in the six kinds of surveyed relationships, whichare sorted and weighted by us in view of the closenessdegree in Table 3.

TABLE 3: Weights for Various Closeness Categories

Original Redefined WeightClose Friend Friend 3

Socialize Twice Per Week Acquaintance 2Political Discuss

Facebook All Tagged Photos Not familiar 1Blank Do not know 0

Performance Evaluation on Social Evolution DatasetWe first calculate the gap between the average inside-

community closeness degree and the average outside-community closeness degree for each community in thecommunity pool C, and then we sort these communitiesbased on their gaps as a ranking list L1. Note thatthe higher a community is ranked, the stronger it is.It is obvious that a strong friend clan should simulta-neously obtain close relationships among the membersand obtain a relatively weaker relationship with non-members. The dense internal connections and sparseexternal connections situation ensures a low probabilityof current members leaving and new members joining,which makes it a strong friend clan. Also, we calculatethe community strength for each community in thecommunity pool C using the proposed method or one ofthe baselines, and then also rank them as a ranking listL2. Then we can directly compare the predicted resultL2 with L1. Because L1 denotes the ground truth ofthe communities’ strengths ranking, the estimated resultcan be validated through the comparison of two rankinglists.

The similarity between ranked list L1 and L2 canbe measured globally or locally. The global approachmeasures how close the overall similarity is betweenL1 and L2. To measure this, we use rank correlationmeasure proposed in [21], which is also commonly re-ferred to as Kendall’s tau (τ) coefficient. The Kendall’stau coefficient ranges in [-1,1]. Using this measure, twoidentical rankings will receive value 1, and the oppositerankings (i.e., one ranking is the reverse of the other)has value -1. Besides measuring the proposed methodfrom the global perspective, we are also interested in the


9

method’s local precision at the two extremes (strongestand weakest) in a ranking list. To measure the accuracyof detected communities in the top/bottom x elements,we use a cover rate function of |L

Tx1 ∩L

Tx2 |+|L

Bx1 ∩L

Bx2 |

|LTx1 |+|L

Bx1 |

,

where LTx denotes the elements of the top x ratio of listL and LBx denotes the elements of the bottom x ratioof list L. This function is used to measure the commonelements ratio of two ranking lists in terms of their 2x(top x and bottom x percent) percent elements. In theexperiment, we vary the ratio x as 10%, 15%, 20%, 25%and 30%.

The experimental results of global and local rankingmeasurement comparing with baselines mentioned inSection 5.1 are shown in Fig. 3 and Fig. 4, respectively.It can be observed from these figures that the pro-posed method outperforms other baselines consistentlyin both the global measure and local measure. To bemore specific, for the global measure, we got the similarperformance results as those on synthetic dataset. For thelocal measure, two patterns are found: for the PACS andPACSwithout, the cover rates are high at the beginning,and begin to decrease as x increases; for the KNN andCID, they start with low cover rates and the cover ratesincrease as the x increases. The reasons behind thesepatterns are as follows. First, the farther communitiesranked from two extremes (i.e. strongest and weakest),the differences between them are more trivial. In otherwords, the communities which are closer to the middleare hard to rank. This is the reason why the coverrates of PACS and PACSwithout drop when x increases.Second, since the cover rate measures the two rankinglists’ common elements ratio in terms of their 2x percentelements, even two ranked lists are totally opposite, theywill reach 100% cover rate when x equals to 50%. Also,different with PACS and PACSwithout whose beginningcover rate is very high, the beginning cover rate of CIDand KNN is very low, thus the possible growing spaceof cover rate is larger.

Fig. 3: Global Performance on the Social EvolutionDataset

To evaluate how the temporal smoothness parameterα affects the performance, we increase α from 0 to 1with a step of 0.1 and report the Kendall’s tau valueof the detected community ranked list in Fig. 5. As α

0.1 0.15 0.2 0.25 0.30.4

0.5

0.6

0.7

0.8

0.9

1

x

Cover

Rate

PACS

PACSwithout

CID

KNN

Fig. 4: Local Performance on the Social Evolution Dataset

increases, we emphasize more on the current network.We get a hill-shape curve, which demonstrates thatboth historical and current networks contribute to thetrue community strength estimation. In other words,the temporal smoothness assumption adopted in ourframework is helpful. On the other hand, we validatethe influence from different number of communities inFig. 6. For simplicity, we set the Kt value identical foreach timestamp. In the test, different Kt values are testedand the Kendall’s tau coefficients are recorded. It can beobserved that the different number of communities willnot cast much impact on the efficacy of our framework.

Finally, we report the average inside-community close-ness degree (as mentioned in Table 3) and the averageoutside-community closeness degree at each snapshotfor the top three strongest and weakest communitiesin Table 4. We can observe that, for the top threestrongest communities, the average inside-communitycloseness degrees are obviously larger than the averageoutside-community closeness degrees. On the contrary,the average inside-community closeness degrees of weakcommunities are basically less than the average outside-community closeness degrees. This demonstrates thatthe temporal community strength detected by the pro-posed method can reflect the true community relation-ships.

0 0.2 0.4 0.6 0.8 1

0.58

0.59

0.6

0.61

0.62

0.63

α

Ke

nd

all’s

τ

Fig. 5: Parameter Sensitivity


10

6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

The Number of Communities

Ke

nd

all’ τ

Fig. 6: The Performance on Different Number of Com-munities

T1 T2 T3 T4 T5In/Out In/Out In/Out In/Out In/Out

Top three strongest communities1st 2.1/1.2 2.1/1.2 2.1/1.3 2.0/1.3 2.1/1.32st 1.8/1.2 1.8/1.2 2.0/1.3 1.9/1.3 2.1/1.33st 2.0/1.2 2.0/1.2 2.0/1.3 1.9/1.3 2.1/1.4

Top three weakest communities1st 0.8/1.0 0.7/1.0 0.7/0.9 0.7/0.9 0.6/0.92st 0.1/1.1 0.1/1.0 0.7/1.1 0.9/1.2 0.7/1.23st 1.5/1.6 2.0/1.5 1.3/1.3 1.1/.2 0.1/1.0

TABLE 4: Inside/Outside Closeness Degrees of the TopThree Strongest/Weakest Communities

5.3 Twitter DatasetThe Twitter Dataset is crawled from Twitter.com2. Wetracked the topic “English Premier League” and relatedfootball teams for over a week and then divide the datainto 5 intervals, each spanning 60 hours of interactions.There are in total 1582 users involved in the networks.The weight on the edge stands for the number of inter-actions between two users in the specific timestamp. Tovalidate our results, we create a word frequency vectorfor each user and calculate the cosine similarity betweena given pair of users. Thus we create a text basedsimilarity matrix as ground truth. Similar with SocialEvolution Dataset, we implement our proposed methodon user interaction network, and make comparison withthe ground truth information.

We conducted the experiments to display the globaland local performance, shown in Fig. 7 and Fig. 8 respec-tively. The stronger community should have discussionson relatively consistent topics, and thus more similarword patterns over time. In the results, our proposedmethod outpaces the two baseline in Twitter dataset.Also we can observe the similar local performance pat-terns with that in Social Evolution Dataset from Fig. 8.

5.4 IMDB DatasetIn this part, we consider the co-starring network con-structed from a subset of the Internet Movie Database

2. http://www.twitter.com/

Fig. 7: Global Performance on the Twitter Dataset

0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Co

ver

Rate

PACS

PACSwithout

CID

KNN

Fig. 8: Local Performance on the Twitter Dataset

(IMDB) dataset3. In the network, nodes are actors andan edge exists between them if the corresponding actorshave participated in an American made comedy movie(excluding TV movies and TV series) together in agiven period of time. There are totally four snapshotsfrom 1991 to 2002, and the network of each snapshotdemonstrates co-starring relations over a period of threeyears 1991-1993 (T1), 1994-1996 (T2), 1997-1999 (T3) and2000-2002 (T4). To remove noise and outliers, only actorswho have participated in at least 10 movies of any genresfrom 1990 to 2010, and at least one American madecomedy movie are selected at each snapshot. There aretotally 700 actors satisfying the above requirements, andwe build snapshot networks with these 700 actors asnodes.

Top three strongest actor communities detected byPACS and the major movies of each community areshown in Table 5. Without loss of generality, we demon-strate the case study on the first community and showthat the method is effective in discovering strong com-munities. Among all the 18 actors that are includedin this community, 12 are indicated as voice actors inWikipedia, and the rest of them have also contributedtheir voices to some cartoon characters at least threetimes. Moreover, in their nine key movies, seven are

3. www.imdb.com/interfaces


11

comedy cartoons. As we know, the particularity of dub-bing makes the cooperation between voice actors morefrequent than with the actors who are not voice actors,which demonstrates that this actor community detectedby the proposed method is indeed a strong community.

5.5 DBLP DatasetWe evaluate the proposed method on a subset of DBLPDataset used in [22]. To be more specific, we focuson work published on conferences or journals during1991-2000 with 144,179 papers in total. We track thestrength of the author communities within five timeintervals: 1991-1992(T1), 1993-1994 (T2), 1995-1996 (T3),1997-1998 (T4) and 1999-2000 (T5). Only authors whohad at least one publication (in a selected set of 43conferences/journals) at each timestamp are considered.There are in total 1059 authors who are represented asnodes in the networks. Each node pair is connected ifthe corresponding authors have joint publications, andthe weight connecting the nodes denotes the times ofcollaboration at this timestamp.

The strongest author community detected by PACSand its related collaboration venues within five times-tamps are shown in Table 6. This author communityconsists of 14 authors, and due to space limit, we onlyprovide their abbreviated names. We show the numberof collaborations among the authors for each confer-ence/journal they co-published in. As for the top fivefrequent venues in which they co-published in, on aver-age they have around four co-authored papers in everytwo years. Furthermore, their publications on these jour-nals/conferences are relatively consistent throughout theobservation period. The high frequency and stability ofcollaboration has made this author community a strongcommunity. Therefore it demonstrates that the strongauthor community detected by the proposed method isreasonable.

5.6 Biological DatasetThe biological dataset used in the experiment is fromStevenson et al. [23]. They collected the gene expressiondata from two groups of rats: the rats (eight replicates)exposed to cigarette smoke (i.e. exposure group) andthe rats (eight replicates) exposed to room air only (i.e.the control group). Various intervals up to eight monthsare used to identify the molecular changes induced bycigarette smoke inhalation that may drive the biologicaland pathological consequences leading to diseases, suchas asthma and lung cancer. The dataset includes elevensnapshots (1, 3, 5, 14, 21, 28, 42, 56, 84, 112 and 182 days),and there is a gene expression matrix created for eachsnapshot. In this study, we focus on 3672 genes whosep-values (via t-test) are smaller than 0.05. To constructthe gene co-expression networks, we first calculate thePearson correlation coefficient between the gene pairs basedon each snapshot’s gene expression matrix and thenmaintain the edges whose correlation coefficients are

larger than a cutoff threshold (which is set to 0.8). Notethat this is a commonly used way to construct gene co-expression network as used in many previous work [24],[25].

A widely used method to analyze gene clusters isto divide them into functional categories for biologi-cal interpretation. This is usually accomplished usingGene Ontology (GO) categories [26]. The GO providesbiologists a list of gene annotations which are usedas inferences for understanding the genes communities’biological functions instead of investigating each geneindividually. When GO is used on the strong genecommunities detected from the exposure group of rats,we can find the strongest and most significant genefunctions that influence this group of rats throughout theentire observation period. Similarly, we can also obtainthe significant gene annotations influencing the controlgroup.

We compare the gene annotations between these twogroups and filter the common gene annotations. Thenthe unique gene annotations influencing the exposuregroup can be obtained, which can tell us the mostsignificantly affected annotations in the chronic responseto cigarette smoke. Table 7 shows all the significant geneannotations in the strongest gene community detectedby the proposed approach (for the sake of simplicity,we name this community C∗), under the p-value cut-off threshold 2.0E-06. Furthermore, from all these anno-tations, we select the gene annotations which are onlydetected in C∗ but not detected by the top-10 strongestcommunities from control group, which are shown inTable 7 with a star mark (*). Among all these uniqueannotations, majority of them have been proven byprevious studies to be really driven by the cigarettesmoke. As shown in [23], carbon fixation (i.e., GO ID19752), metabolism (i.e., GO ID 43436, 6082, 42180, 44281,8152) and inflammation (i.e., GO ID 6954, 2526) are somespecial functions distinguishing between the exposuregroup and control group. In addition, the strength pro-gression of this community is shown in Fig. 9. From thisplot, we can see that C∗ becomes much stronger after the4-th snapshot (14 days). This is validated by the previousresult provided in [23], which demonstrated that carbonfixation, metabolism and inflammation show differencesafter two weeks. This interesting result shows that theproposed approach is not only effective on detecting thetop strongest communities globally, but also effective ontracking the progression of community strengths locally.Besides the proven gene annotations, we believe that therest of the unique annotations (i.e. those not proven byprevious work) may provide important hints for learningthe effect of smoking cigarettes.

6 RELATED WORK

There have been extensive research studies on commu-nity detection in networks. [27] came up with an efficientalgorithm to conduct overlapping community detection


12

TABLE 5: Members and Corresponding Major Movies in the Top Three Strongest Actor Communities during 1991-2002

Actors Key workDebi Derryberry, Christopher McDonald An American Tail: Fievel Goes West (1991)

Erik von Detten, Bob Bergen, Phil Proctor, Sherry Lynn Aladdin (1992), Toy Story (1995)Jim Varney, Mickie McGowan, Tom Hanks, Jack Angel House Arrest (1996), A Smile Like Yours (1997)

Wallace Shawn, R. Lee Ermey, Harry Shearer, John Mahoney A Bug’s Life (1998), Toy Story 2 (1999)Earl Boen, Ben Stein, Charlton Heston, Wayne Knight The Iron Giant (1999), Recess: School’s Out (2001)Patrick Richwood, Kathleen Marshall, Garry Marshall Frankie and Johnny (1991)Larry Miller, Hope Alexander-Willis, Hector Elizondo A League of Their Own (1992), Exit to Eden (1994)Marvin Braverman, Rosie O’Donnell, Shannon Wilcox Dear God (1996), The Other Sister (1999)Sean O’Bryan, Donal Logue, Greg Lewis, Jane Morris Runaway Bride (1999), The Princess Diaries (2001)

John Cusack, Kathleen Doyle, Peter McRobbie Shadows and Fog (1991)Woody Allen, Tony Sirico, Michael Rapaport Manhattan Murder Mystery (1993)

Paul Herman, David Ogden Stiers Bullets Over Broadway (1994), Mighty Aphrodite (1995)Jeff Mazzola, Brian McConnachie Everyone Says I Love You (1996)John Doumanian, Colicchio Victor Deconstructing Harry (1997), Celebrity (1998)

Natasha Lyonne, Jack Warden, Alan Alda Small Time Crooks (2000)Alan Alda, Steven Randazzo, Paul Giamatti The Curse of the Jade Scorpion (2001)

TABLE 6: Authors and Corresponding Major Publications in the Strongest Co-Author Communities during 1990-2000

Authors 1991-1992 1993-1994 1995-1996 1997-1998 1999-2000S. Suri, M. Sharir, Da. Dobkin D. Geometr (2) D. Geometr (8) D. Geometr (9) D. Geometr (4) D. Geometr (8)

L. Guibas, J. Snoeyink Comput. Geo. (7) Comput. Geo. (2) Comput. Geo. (3) Comput. Geo. (11) Comput. Geo. (1)J. Hershberger, P. Agarwal SICOMP (1) SICOMP (4) SICOMP (2) SICOMP (7) SICOMP (2)

M. Grigni, B. Chazelle, M. Berg SWAT (3) SWAT (2) SWAT (0) SWAT (2) SWAT (2)D. Halperin, M. Overmars J. Algorithms (3) J. Algorithms (4) J. Algorithms (4) J. Algorithms (0) J. Algorithms (1)B. Aronov, D. Kirkpatrick Others (17) Others (20) Others (11) Others (13) Others (5)

TABLE 7: Gene Annotations for C∗

GO-ID p-value Description10038 2.61E-10 response to metal ion10035 1.33E-09 response to inorganic substance19752* 2.21E-08 carboxylic acid metabolic process43436* 2.21E-08 oxoacid metabolic process6082* 2.47E-08 organic acid metabolic process42180* 2.96E-08 cellular ketone metabolic process9719 1.50E-07 response to endogenous stimulus

44281* 2.77E-07 small molecule metabolic process10033 3.31E-07 response to organic substance9725 3.38E-07 response to hormone stimulus

71396* 4.24E-07 cellular response to lipid2526* 5.01E-07 acute inflammatory response44283* 5.18E-07 small molecule biosynthetic process6954* 8.71E-07 inflammatory response8152* 1.69E-06 metabolic process6952* 1.92E-06 defense response

in large-scale social networks. In [28], a novel methodwas proposed for the community discovery in complexnetworks based on an extremal modular optimizationframework. [29] introduced the modularity concept insocial networks and leveraged eigenvectors of charac-teristic matrix for the detection task. Also, [30] discusseda benchmark method to test the detected communities.However, these methods focus on the static scenario andcannot be easily extended to dynamic networks.

With the availability of many online datasets, dynamicnetwork analysis has become a hotly discussed topictoday. In [1], an optimization framework based on lo-

2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

Snapshot

Str

en

gth

Fig. 9: Community Strength Progression of CommunityC∗

gistic regression was proposed to estimate the networkevolution. [2] discussed the biological dynamic networksand proposed the method to predict the state of proteincomplexes. [3] analyzed the dynamic molecular interac-tions, which is crucial in regulating the functioning ofcells and organisms. In [31], the authors investigated thesubgraph discovery in dynamic networks.

Besides, the community analysis of dynamic networkshas been extensively studied in various research areas.Most existing community research on dynamic networksfocuses on community discovery and community evo-lution pattern detection. Chi et al. [6] and Lin et al. [7]proposed community discovery algorithms for dynamicgraphs where the communities detected at each snapshot


13

are based on the optimization defined on both the cur-rent and historical networks. Similarly, Ahmed et al. [32]proposed a machine learning algorithm named TESLAfor recovering the underlying structure of time-varyingnetworks, which could be also used on temporal com-munity detection. Although these methods can outputsome stable and consistent communities at each snap-shot, they may not be able to provide any evolutionaryinformation of the communities, such as the histori-cal/successive structure information. Some studies [33],[34] focused on detecting the evolution of communities,which captures the changes (e.g. merging, splitting andsurviving) between successive communities. However,the information provided by these studies is limited toonly adjacent snapshots which cannot give us a wholepicture of the community evolution.

Gupta et al. [5] investigated and tackled the problemof identifying evolutionary community outliers given thediscovered communities from two snapshots of an evolv-ing dataset. The target problem is to detect communityoutliers that evolve against the trend, which is differ-ent from ours. Newman and Girvan [11] proposed theModularity function, which measures the quality of graphpartitioning. In particular, a graph is considered to havehigh modularity if it has dense connections between thenodes within the communities but sparse connectionsbetween nodes across different communities. Modularityfunction captures community strength in some sense,but it is only used as a global index which measuresthe quality of a particular clustering or the standard topartition the objects into communities [35] in a staticgraph.

7 CONCLUSIONS

In this paper, we introduced a new problem of analyzingthe progression of community strengths. Communitystrength is a temporal measure which represents theprobability that a particular community has a stablemembership at the current snapshot. To solve this prob-lem, we propose a framework that provides reliable andconsistent community strength scores which are not onlyinsensitive to short-term noise in the current networkbut also adaptive to long-term network evolution. Theresults of community strength analysis can be also usedto find the top-K strongest or weakest communitiesand track the change of strengths via constructing thecommunity strength progression net. Extensive experi-mental analysis demonstrated that the proposed methodis very effective on both synthetic and real dynamicdatasets. Case studies on three real datasets showed thatinteresting and meaningful communities can be revealedby community strength detection.

8 ACKNOWLEDGEMENT

This work was supported in part by the National ScienceFoundation under Grant NSF IIS-1218393, 1016929 and1319973. Any opinions, findings, and conclusions or

recommendations expressed in this paper are those ofthe authors and do not necessarily reflect the views ofthe National Science Foundation.

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Dr. Nan Du received his Ph.D degree fromComputer Science & Engineering departmentin State University of New York at Buffalo, NY,with supervision by Prof. Aidong Zhang. Prior tothat, he received the BS degree from Guang-dong University of Technology in 2006, and theMS degree from Southern China University ofTechnology in 2009. His research interests arein the areas of data mining, machine learning,and bioinformatics.

Xiaowei Jia is currently a Ph.D. candidate incomputer science from State University of NewYork at Buffalo. He is working in data miningand machine learning with a focus on socialnetwork analysis, recommender systems andhealth-care signals.

Dr. Jing Gao is currently an assistant professorin the Department of Computer Science at theUniversity at Buffalo (UB), State University ofNew York. She received her PhD from ComputerScience Department, University of Illinois at Ur-bana Champaign in 2011, and subsequentlyjoined UB in 2012. She is broadly interested indata and information analysis with a focus on in-formation integration, crowdsourcing, ensemblemethods, mining data streams, transfer learningand anomaly detection. More information about

her research can be found at: http://www.cse.buffalo.edu/ jing. She is amember of IEEE.

Vishrawas Gopalakrishnan is a Ph.D. candi-date in the department of Computer Science andEngineering at State University of New York atBuffalo. His research focus is on using graphtechniques and machine learning in the field oftext and web mining.

Dr. Aidong Zhang is SUNY Distinguished Pro-fessor and Chair in the Department of ComputerScience and Engineering at State Universityof New York at Buffalo. Her research interestsinclude data mining, bioinformatics, multimediaand database systems, and content-based im-age retrieval. She is an author of over 250research publications in these areas. She haschaired or served on over 100 program com-mittees of international conferences and work-shops, and currently serves several journal ed-

itorial boards. She has published two books Protein Interaction Net-works: Computational Analysis (Cambridge University Press, 2009)and Advanced Analysis of Gene Expression Microarray Data (WorldScientific Publishing Co., Inc. 2006). Dr. Zhang is a recipient of theNational Science Foundation CAREER award and State University ofNew York (SUNY) Chancellor’s Research Recognition award. Dr. Zhangis an IEEE Fellow.


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