Community Evolution of Social Network: Feature, Algorithm and Model
CCommunity Evolution of Social Network:Feature, Algorithm and Model
Yi Wang, Bin Wu, and Nan Du
Beijing Key Laboratory of Intelligent Telecommunications Software and Multimedia,Beijing University of Posts and Telecommunications, Beijing [email protected]{wubin,dunan}@bupt.edu.cn
Abstract.
Researchers have devoted themselves to exploring static fea-tures of social networks and further discovered many representative char-acteristics, such as power law in the degree distribution and assortativevalue used to differentiate social networks from nonsocial ones. However,people are not satisfied with these achievements and more and more at-tention has been paid on how to uncover those dynamic characteristicsof social networks, especially how to track community evolution effec-tively. With these interests, in the paper we firstly display some basicbut dynamic features of social networks. Then on its basis, we proposea novel core-based algorithm of tracking community evolution, Comm-Tracker, which depends on core nodes to establish the evolving relation-ships among communities at different snapshots. With the algorithm, wediscover two unique phenomena in social networks and further proposetwo representative coefficients: GROWTH and METABOLISM by whichwe are also able to distinguish social networks from nonsocial ones fromthe dynamic aspect. At last, we have developed a social network modelwhich has the capabilities of exhibiting two necessary features above.
Social network analysis has been a hot topic in the field of data mining. In theco-authorship network, a node is an author and a edge indicates a publishingcollaboration between them. Researchers are interested in these special networksfrom which they discover power law in the degree distribution, that is, only asmall proportion of nodes have high degrees while the rest has low degree. Socialnetworks also present positive assortative values while in nonsocial networks,such as Internet, biology network, the values are always negative, indicatingthat in social networks, higher degree nodes trend to connect with higher degreenodes while in nonsocial ones, it is largely possible that higher degree ones arelinked with lower degree ones. Moreover, researchers reveal community struc-tures where the vertices within communities have higher density of edges while
This work is supported by the National Science Foundation of China under grantnumber 60402011, and the National Science and Technology Support Program ofChina under Grant No.2006BAH03B05. a r X i v : . [ phy s i c s . s o c - ph ] A p r ertices between communities have lower density of edges. In the co-authorshipnetwork, a community reflects a group of scholars with similar interest. Appar-ently, from these static characteristics, people have gained much understandingof social networks. However, we are not satisfied with these achievements, butwill furthermore explore those dynamic features of social networks. For example,how can we track community evolution effectively? Does other dynamic featuresexist to distinguish social networks from nonsocial ones? How can we establisha more reasonable model of social network?With the interest to dynamic features of social networks, we firstly performexperiments in which after a long time duration has been divided into severalsnapshots, we find that about 80 percent of nodes appear in one or two snap-shots. The experiment indicates that most of nodes is so unstable that we cannot rely on them too much. We also discover that node with higher degree willappear in more snapshots. On its basis, we propose a core-based algorithm calledCommTracker to track community evolution effectively. With it, we not only findout a community evolution trace but also discover split or mergence points inthe trace. By the algorithm, we find two unique phenomena of social networks.One is that a larger community leads to a longer life and the other is that a com-munity with a longer life trend to have lower member stability. Correspondingly,we propose two representative coefficients: GROWTH and METABOLISM, bywhich we are able to tell social networks from nonsocial ones. At last, we proposea more reasonable model which focuses on node change. The model successfullydisplays two important phenomena discovered above.We validate our conclusions in 11 datasets including 6 social networks: 3 co-authorship networks in cond-mat, math and nonlinear fields, a call network, anemail networks and a movie actor network as well as 5 nonsocial ones involving 3software networks (tomcat 4, tomcat 5, ant), an Internet network, a vocabularynetwork.The rest of the paper is organized as follows: Section 2 reviews the relatedwork. Section 3 gives definitions. Section 4 introduces some basic dynamic fea-tures of our dataset. Section 5 presents the core-based algorithm of trackingcommunity evolution. Section 6 introduces two unique phenomena discovered inthe social networks. Section 7 shows our model and Section 8 concludes. A lot of work has been dedicated to exploring the characteristics of social net-works. Barabasi and Albert show an uneven distribution of degree through BAmodels[1]. Newman has successfully discovered distinct characteristics betweensocial networks and nonsocial ones[20]. Various methods have been utilized todetect community structures. Among them, there are Newman’s betweennessalgorithm [13][14], Nan Du’s clique-based algorithm[12] and CPM[11] that fo-cuses on finding overlapping communities. Clustering is another technique togroup similar nodes into large communities, including L. Donetti and M.Miguel’smethod[5] which exploits spectral properties of the graph as well as Laplacianatrix and J. Hopcroft’s “natural community” approach[10]. Some social net-work models have been proposed [21][22][23].With respect to core node detection, Roger Guimera and Luis A.Nunes Ama-ral propose a methodology that classifies nodes into universal roles according totheir pattern of intra- and inter-module connections [4]. B. Wu offers a methodto detect core nodes with a threshold [3]. Shaojie Qiao and Qihong Liu dedicatethemselves to mining core members of a crime community[19].As to dynamic graph mining, Tanya Y.Berger-Wolf and Jared Saia studycommunity evolution based on node overlapping [6]; John Hopcroft and OmarKhan propose a method which utilizes “nature community” to track evolution[9].However, both methods have to set some parameters, which is too difficult tobe adaptive to various situations. In contrast, Keogh et al. suggests the notionof parameter free data mining[15]. Jimeng Sun’s GraphScope is a parameter-free mining method of large time-evolving graphs[8], using information theoreticprinciples. Our method in the paper shares the same spirit.As forerunners, A.L.Barabasi and H.Jeong study static characteristic vari-ations on the network of scientific collaboration[18]. Gergely Palla and A.-L.Barabasi provide a method which effectively utilizes edge overlapping to buildevolving relationship[7]. With the approach, they discover valuable phenomenaof social community evolution.
The table below lists almost all the symbols used in the paper.Sym. Definition C ( t ) i Community of index i in snapshot tN ( t ) i Node of index i in snapshot tW ( N ( t ) i ) Weight of a node of index i in snapshot tCen ( N ( t ) i ) Central degree of node N ( t ) i Core ( C ( t ) i ) Core node set of C ( t ) i N ode ( C ( t ) i ) Node set of C ( t ) i Edge ( C ( t ) i ) Edge set of C ( t ) i | N ode ( C ) | community C size C ( t ) i → C ( t +1) j C ( t ) i is a predecessor of C ( t +1) j or C ( t +1) j is a successor of C ( t ) i C ( t − k ) i ⇒ C ( t ) j C ( t − k ) i is an ancestor of C ( t ) j Evol ( C ( t ) i ) Evolution trace of C ( t ) i | Evol ( C ( t ) i ) | Span of evolution trace of C ( t ) i Definition 1. (COMMUNITY EVOLUTION TRACE).An evolution trace
Evol ( C ( t ) x ) is a time-series of C ( t + n ) as follows: Evol ( C ( t ) x ) := { C ( t ) x , C ( t +1) x , C ( t +1) y , C ( t +2) x . . . , C ( t + n ) x } ( n ≥ here each community C ( t + i ) x , i ∈ [1 , n ] satisfies the condition that there exists atleast one community C ( t + i − x , and then C ( t + i − x → C ( t + i ) x . Note that more thanone community is allowed to appear in the same snapshot t+i, like C ( t +1) x , C ( t +1) y both locating in the snapshot t + 1 . | Evol ( C ( t ) x ) | is n + 1 Definition 2. (ANCESTOR OF A COMMUNITY).The definition of a community’s ancestor is as follows: C ( t − k ) i ⇒ C ( t ) j if thereis an evolving chain C ( t − k ) i → C ( t − k +1) x , . . . , → C ( t ) j ( k ≥ Definition 3. (COMMUNITY AGE).The age of a community is time span between its birth snapshot and itscurrent snapshot. Here in the
Evol ( C ( t ) x ) defined in the Definition 1, the ageof C ( t ) x = 1 and C ( t +2) x = 3. Definition 4. (MEMBER STABILITY OF A COMMUNITY).The member stability of a community C ( t ) is as following: M S ( C ( t ) ) = N ode ( C ( t ) ) ∩ ( N ode ( C ( t +1)1 ) ∪ N ode ( C ( t +1)2 ) . . . ∪ N ode ( C ( t +1) n )) N ode ( C ( t ) ) ∪ ( N ode ( C ( t +1)1 ) ∪ N ode ( C ( t +1)2 ) . . . ∪ N ode ( C ( t +1) n )) where C ( t ) → C ( t +1) i ( i ∈ [1 , n ]) Definition 5. (MEMBER STABILITY OF A COMMUNITY EVOLUTIONTRACE).The member stability of a community evolution trace is the average stabilityvalue of all community having successors within the trace. Its definition is asfollowing: (cid:80)
M S ( C ( t ) ) /n , where C ( t ) is the community having successors and n is the corresponding number. In this section, we are interested in the following three aspects: (1) how the scaleof social networks evolves; (2) how the members of social networks evolve; (3)which nodes trend to live long lives.Note that the paper concentrates on social networks, but nonsocial networksare taken into account in that we must compare distinct characteristics betweenthem.
As Fig.1 shows, in each co-authorship network (cond-mat, math, non linear),the node number of networks at different snapshot gradually increase. The phe-nomenon is also observed in the network of movie actor. However, in the callnetwork, such an increase trend is not very apparent and in the email network,we can see a fluctuant rise, but it falls in the latest snapshots. In our analysis,co-authorship datasets and movie dataset reflect worldwide cooperating situa-tions, which is relatively complete. In contrast, the call network only considersthe situation of one province and the email network is from the Enron company.Both of them might reflect the partial change. In all, we can get the conclusionthat social network scale inflates when it evolves.
Although the size of a network increases in the evolution, its members is alwayschanging, that is, some members will leave the network and some will enter it.We make a statistics which indicates that during the whole evolution process,about 80% nodes appear in less than two snapshots (See Fig.2). Therefore, weconcludes that members of social networks change dramatically and only a smallproportion exists in the networks stably. .4 Discovery of long life members.
We are also interested in which nodes will get high appearance times in the net-work. Here, node degree is taken into account as a critical factor, which indicatesthe importance of some node in the network to some extent. We respectively cal-culate the correlation coefficient between node degree and appearance frequencyin six social networks: cond-mat is 0.12; math is 0.13; non linear is 0.22; call is0.28; email is 0.44; movie actor is 0.14; In conclusion, nodes with higher degreewill exist in the network with a larger possibility.According the conclusions in this section, we understand that a large propor-tion of nodes is so unstable that we can not rely on them too much but focus onthose small stable nodes, especially when we want to track community evolution.
Fig. 1.
Network scale (node number) evolution. Snapshot id (X axis) and net-work scale (Y axis)
As discussed above, community structures are mined by many algorithms inevery network snapshots. We are interested in how these communities evolves.For example, there exists a community in snapshot t , and what about its statein the next snapshot t + 1? Does it split into smaller ones or merge into a largerone with another community? ig. 2. Node appearance distribution. Node appearance times (X axis) and per-centage (Y axis)Our algorithm, CommTracker, heavily relies on core nodes instead of theoverlapping level of nodes or edges between two communities. From the exper-iments above, we have realized that most of nodes lacks stability. Therefore,taking advantage of not all nodes that include those high fluctuating ones butthese representative and reliable core nodes, will be more accurate and effectiveto track community evolution. A good example is the co-authorship communitywhere core nodes represent famous professors and ordinary ones are other stu-dents. The research interest of professors is usually that of a whole community.Moreover, it is harder for professors to change their research interest than forthose ordinary students.In this section, the algorithm of core node detection is firstly introduced andthen we present our core-based algorithm of tracking community evolution.
As discussed above, core nodes are of greatest importance in our evolution algo-rithm, so its preparation work, selecting core nodes from a community, is a keystep. The structure of a community is too dynamic and unpredictable to set anempirical threshold to distinguish core nodes from ordinary ones. Unlike [3], thefollowing method concentrates on not only effectiveness but also parameter free.A node can be weighed in terms of many aspects, such as degree, betweenness,page rank and so on. Generally, the higher a node’s weight is, the more importantt is in a community. Here, we give a node N i a weight value W ( N i ) accordingto its degree.In our algorithm, both the community topology and the node weight areconsidered as critical factors to distinguish core nodes from ordinary ones. InAlgorithm 1, we present the whole algorithm. Fig. 3.
Core detection illustration.The basic idea behind the algorithm is similar to a vote strategy. For eachnode N i , it is entitled to evaluate the centrality of those nodes linked with it.Assuming that W ( N i ) is higher than the weight of a linked node, W ( N j ), then N i is considered as more important node than N j , so N i ’s centrality value shouldbe incremented by a specified value while N j ’s value is reduced by a specifiedvalue. Here, | W ( N i ) − W ( N j ) | is employed to represent the centrality differencebetween two nodes. Through the “vote” of all round nodes, if N i ’s centrality isnonnegative, it is regarded as a core node. Otherwise, it is just an ordinary node.As Fig.3 shows, W ( N ) = 6. The running result is that Cen ( N ) = 23, Cen ( N ) = 12 whereas Cen ( N ) = Cen ( N ) = − Cen ( N ) = Cen ( N ) = Cen ( N ) = − Cen ( N ) = Cen ( N ) = − Cen ( N ) = −
7. Therefore, the coreset are { N , N } .In general, Algorithm 1 is effective to detect core nodes in a small networkscope, like community, where node distances are no more than 3 hops and eachnode has large probability to connect to all other ones. Tanya Y.Berger-Wolf and Jared Saia propose a method based on the overlappinglevel of nodes that C ( t +1) is a successor of C ( t ) if nodeoverlap ( C ( t ) , C ( t +1) ) ≥ s [6]. However, to set a proper s is challenging for users. When members of acommunity change dramatically and s is given a higher value, C ( t +1) will beconsidered to disappear because of too low overlapping level between them, butin fact C ( t +1) still exists. Otherwise, if s is set a bit low, doing so will giveirrelevant communities more opportunities to become the successors of C ( t ) ,leading to “successors explosion” and masking those real successors. lgorithm 1 CoreDetection( C ) if W ( N ) = W ( N ) = . . . = W ( N n ) then
2: return C3: end if Cen ( N i ) = 0 , i ∈ [1 , n ]5: for every edge e ∈ Edge ( C ) do N i , N j are nodes connected with e if W ( N i ) < W ( N j ) then Cen ( N i ) = Cen ( N i ) − | W ( N i ) − W ( N j ) | Cen ( N j ) = Cen ( N j ) + | W ( N i ) − W ( N j ) | end if if W ( N i ) ≥ W ( N j ) then Cen ( N i ) = Cen ( N i ) + | W ( N i ) − W ( N j ) | Cen ( N j ) = Cen ( N j ) − | W ( N i ) − W ( N j ) | end if end for
16: coreset = {} for every node N i ∈ Node ( C ) do if Cen ( N i ) ≥ then
19: input N i into coreset;20: end if end for
22: return coreset
Gergely Palla and A.-L. Barabasi provide an approach utilizing the over-lapping of edge between two communities[7], but it fails to deal with split andmergence amongst communities. As there are one C ( t ) and two C ( t +1) i , C ( t +1) j ,in snapshot t and t + 1 respectively, if the edge overlapping level between C ( t ) and C ( t +1) i is higher than that between C ( t ) and C ( t +1) j , C ( t +1) i becomes thesuccessor of C ( t ) while C ( t +1) j is considered as a new born community. Actually, C ( t ) may split into two parts. The similar problem also exists in the process ofcommunity mergence.The disadvantage of the method above is to treat all nodes in an unprejudicedway and it is not accorded with the reality where different nodes have differentinfluences. Our method has deeply paid attention to such a difference so that itputs emphasis on core nodes.The basic thought of our algorithm can be described as: C ( t ) i → C ( t +1) j if and only if (1) at least one core node of C ( t ) i appears in C ( t +1) j , that is, Core ( C ( t ) i ) ∩ N ode ( C ( t +1) j ) (cid:54) = ∅ (2) at least one core node of C ( t +1) j must appear in some ancestor community of C ( t ) i , that is, there existsone C ( t − m ) k , C ( t − m ) k ⇒ C ( t ) i , N ode ( C ( t − m ) k ) ∩ Core ( C ( t +1) j ) (cid:54) = ∅ . see Fig.4For the first condition, it is reasonable to consider C ( t ) i ’s core nodes appearin some succeeding community C ( t +1) j , due to the representative quality of core ig. 4. Community Evolution illustration: core nodes are colored red and ordi-nary ones grey. As we seen, (1) in snapshot t + 1, C ( t +1) contains two core node N , N of C ( t ) . (2) Node N has also been in C ( t − m ) , an ancestor of C ( t ) . There-fore, C ( t +1) becomes the succeeding community of C ( t ) . In practice, if C ( t ) hasno ancestor, then communities satisfying the first condition will become C ( t ) ’ssuccessors automatically. Algorithm 2
Community Evolution( C ( t ) i )
1: Evol( C ( t ) i ) = { C ( t ) i } Core ( C ( t ) i ) = CoreDetection( C ( t ) i )3: for every community C ( t +1) j in snapshot t + 1 do Core ( C ( t +1) j ) = CoreDetection( C ( t +1) j )5: if Core ( C ( t ) i ) ∩ Node ( C ( t +1) j ) (cid:54) = ∅ and Node ( C ( t − m ) k ) ∩ Core ( C ( t +1) j ) (cid:54) = ∅ and C ( t − m ) k ⇒ C ( t ) i then
6: establish the relationship C ( t ) i → C ( t +1) j
7: Evol( C ( t +1) j ) = Community Evolution( C ( t +1) j )8: Evol( C ( t ) i ) = Evol( C ( t ) i ) ∪ Evol( C ( t +1) j )9: end if end for
11: return
Evol ( C ( t ) i ) nodes. As to the second condition, if some community C ( t +1) j wants to becomethe succeeding one of a specified community C ( t ) i , it must suffice that its corenodes appear in some ancestor of C ( t ) i , because of the stable quality of corenodes, that is , core nodes do not appear suddenly without any evidence in thepast snapshots.We describe the whole algorithm in Algorithm 2.From the perspective of successors and predecessors, we provide a very straight-forward way to identify community split , community mergence , communitybirth and community death . Note that they are four phenomena that occurs ina single evolution trace. – Community Split: a community has more than one successor. – Community Mergence: a community owns more than one predecessor. – Community Birth: a community has no predecessor.
Community Death: a community has no successor.Fig.5 shows a typical example of community evolution.
Fig. 5.
Community evolution illustration. Red square points are core nodes.
In [7], Palla has performed two experiments only on cond-mat co-authorshipand call networks: one is to find out the correlation between community sizeand age; the other is to uncover the correlation between evolution trace spanand member stability. In his paper, he obtains conclusions that communities oflarger size lead to longer lives and that if an evolution trace span is longer, itsmember stability is lower. We are interested in the two situations in other socialnetworks and nonsocial ones. The results are shown in Fig.6 (a) and (b).Firstly, depending on CommTracker, we can discover similar phenomena withthose proposed in the Palla’s paper, proving that our method is effective andcorrect. Secondly, it is obvious that 6 social networks display two common be-haviors we discuss above. On the contrary, nonsocial networks fails to own suchbehaviors. In nonsocial networks, it seems that the size of a community can notreflect its age and that a community with higher stability will live for a longerlife.We calculate the correlation coefficients between community size and age(GROWTH) as well as between evolution trace span and member stability(METABOLISM) (See Table 1). Apparently, in the 1st experiment, social net-works’ values are positive while those of nonsocial ones are nearly all nega-tive. In the 2nd experiment, the values of social networks are negative whereasthose of nonsocial ones are all positive. Two experiments reveal that we candifferentiate social networks from nonsocial ones according to GROWTH andMETABOLISM.One important reason contributing to such distinctions is that in social net-works, a community represents a group of persons with close connection and innonsocial ones a community is just a cluster of objects. As we know, in socialnetworks, if a community want to obtain a long life, it must undertake suitablemember changes, that is, when some old core members retire, new ones takeover responsibility in time so that the development of the community is wellsupported. Otherwise, if a community refuses to absorb new members, when the ig. 6. (a) The correlation between community size (X axis) and community age(Y axis). (b) The correlation between evolution trace span (X axis) and memberstability (Y axis).
Table 1.
GROWTH and METABOLISM. (1) cond-mat (2) math (3) nonlinear(4) call (5) email (6) movie actor (7) Internet (8) vocabulary (9) ant (10) tomcat4(11) tomcat5 (12) randomld core members exit from the community, it is possible that new core ones havenot been cultivated, leading to quick disintegration. In contrast, the members ofnonsocial networks are objects, not persons. For example, in software network, acommunity is a class cluster with similar functions. If a class cluster is designedwell, it must experience little change and be used for a long time.
Nowadays, many social network models have been established. However, whenwe get some snapshots generated from these social networks, most of them failto display the characteristic behaviors we have proposed above. In our view, amain defect is that a node will permanently exist in the network once it is addedinto the network. However, from the experiments shown in Section 4, a lot ofnodes enter into the network and then quickly exist from it. Hence, how to reviseexisting models to make them more reasonable is a problem to be solved.
Our model is based on the one proposed in Emily’s model[21], which takes intoaccount social network aspects completely, such as meeting rate between pairs ofindividuals, decay of friendships, etc. Moreover, Emily’s model indeed presentsmany static features of social networks. Therefore, we decide to adopt it as ourmodel basis. Our model can be simulated directly using the following algorithm.Let n p = N ( N −
1) where N is the network initial scale. Let n e = (cid:80) z i where z i is the degree of the i th vertex. And let n m = (cid:80) z i ( z i − n p r pairs of vertices uniformly at random from the network tomeet. If a pair meet who do not have a pre-existing connection, and if neitherof them already has the maximum z ∗ connections then a new connection isestablished between them.2. We choose n m r vertices at random, with probability proportional to z i ( z i − z ∗ of connections.3. We choose n e γ vertices with probability proportional to z i . For each ver-tex chosen we choose one of its neighbors uniformly at random and delete theconnection to that neighbor.4. We choose one vertex, if its degree z i > z , the average degree, we deleteit with the probability α ; otherwise, we delete it with the probability β . Theprocess doesn’t stop until k d vertices have been deleted.5. We add k a new vertices. For each new one, it establishes a link with avertex v randomly and then it also connects to the vertex with highest degreefrom the neighbor vertices of v .Note that the first 3 steps have already existed in the Emily’s algorithm whilethe last 2 steps are added by ourselves. The 4th step is responsible for deletingome existing vertices according to their degrees. The last step focuses on addingnew vertices. In this step, we eliminate the limit of maximum connection inorder to allow some vertices to get high degree. In reality, a community consistsof vertices with distinct degrees while in the Emily’s social network model, acommunity trends to be a clique due to the limit of maximum connection.As pointed out in [21], the network is initialized by starting with no edges,and running the first two steps without the other three ones until all or mostvertices have degree z ∗ (we set the limitation as 85%). Then all five steps areused for the remainder of the simulation. Six experiments have been performed with different parameters α , β , k a and k d shown in Fig.7. In all stimulation, z ∗ = 5, N = 250, r = 0 . r = 2, γ = 0 . Fig. 7.
Model stimulation. (a) α = 0 . β = 0 . k a = k d = 3 (b) α = 0 . β =0 . k a = k d = 3 (c) α = 0 . β = 0 . k a = k d = 3 (d) α = 0 . β = 0 . k a = k d = 3(e) α = 0 . β = 0 . k a = k d = 6 (f) α = 0 . β = 0 . k a = 5, k d = 3 Conclusions.
In the paper, we firstly perform some basic experiments to explore those dynamiccharacteristics of social networks and it is discovered that a large percentage ofnodes are so instable that we can not rest on them too much and that nodes withhigher degree will appear more frequently during the evolution of a social net-work. Under the experimental results, we propose a novel core-based algorithmto track community evolution, which has the following features:(1) it is effective;(1) it is parameter-free; (2) it is suitable to discover split and mergence points.With the algorithm, we uncover two representative dynamic features of socialnetworks and define two coefficients: GROWTH and METABOLISM by whichwe also achieve the goal of telling social networks from nonsocial ones. In theend, we propose a revised social network model which can display two typicalcharacteristics. The experiments are based on 6 social networks (co-authorshipnetwork, call network, movie actor network and email network)and 5 nonsocialnetworks (Internet, vocabulary network and software network).
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