Predicting the Evolution of Communities in Social Networks Using Structural and Temporal Features

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1 Predctng the Evoluton of Communtes n Socal Networks Usng Structural and Temporal Features Mara Evangela G. Pavlopoulou, Grgoros Tzortzs, Dmtros Vogatzs and George Palouras Department of Informatcs and Telecommuncatons Natonal and Kapodstran Unversty of Athens, Athens, Greece Emal: mary18pav@gmal.com Insttute of Informatcs and Telecommuncatons NCSR Demokrtos, Athens, Greece Emal: {gtzortz, palourg}@t.demokrtos.gr Insttute of Informatcs and Telecommuncatons NCSR Demokrtos, Athens, Greece & The Amercan College of Greece, Deree, Athens, Greece Emal: dmtrv@t.demokrtos.gr Abstract Durng the last years, there s ncreasng nterest n analyzng socal networks and modelng ther dynamcs at dfferent scales. Ths work focuses on predctng the future form of communtes, whch represent the mesoscale structure of networks, whle the communtes arse as a result of user nteracton. We employ several structural and temporal features to represent communtes, along wth ther past form, that are used to formulate a supervsed learnng task to predct whether a communty wll contnue as currently s, shrnk, grow or completely dsappear. To test our methodology, we created a reallfe socal network dataset consstng of an excerpt of posts from the Mathematcs Stack Exchange Q&A ste. In the experments, specal care s taken n handlng the class mbalance n the dataset and n nvestgatng how the past evolutons of a communty affect predctons. I. INTRODUCTION Socal networks evolve over tme as a result of the actvty of ther users. New users jon the network, old ones cease to be actve or depart, whle edges representng user nteracton can be created, destroyed or exhbt a complex ntermttent behavour, gvng rse to a dynamc network. Predctng the future form of a socal network presents an nterestng challenge wth numerous applcatons, such as n marketng to locate approprate groups of users on whch to target advertsements, crmnology to dentfy growng clques of delnquent ndvduals that requre mmedate attenton and journalsm to uncover developng stores. One of the frsredcton problems to be nvestgated n the context of socal networks was edge predcton. Edge predcton refers to predctng whether an nteracton (edge) wll occur between two users of the network [1] [3]. A related problem s that of edge sgn predcton, where the goal s to nfer whether an nteracton between two users has a postve or negatve context [4] [6]. Communtes represent the mesoscale structure of the socal network and are mplctly formed as users wth the same nterests closely nteract. As the nterests of users change over tme so do the communtes, whch may /17/$31.00 c 2017 IEEE reduce or ncrease n sze, or, even completely dsappear from the network. Communty evoluton predcton concerns the predcton of the future form of a communty gven ts present and past form and has been a hot research topc lately [7] [10]. In ths work we focus on four popular evolutonary phenomena of communtes; growth, shrnkage, contnuaton and dssoluton [7]. We present a framework for predctng these types of evoluton that covers all the necessary steps nvolved, ncludng the preprocesng of the data, the detecton and trackng of the communtes, the extracton of features to represent the communtes and fnally the tranng of a predctve model that dscrmnates the four evolutonary events. Partcular focus s placed on employng an extensve set of structural and temporal features that capture varous characterstcs of the communtes n order to get accurate predctons. To test the proposed framework, experments are performed on a reallfe socal network dataset obtaned from the Mathematcs Stack Exchange Q&A ste. Results confrm the effcacy of our framework and the mportance of usng a mxture of structural and temporal features. The rest of paper s organzed as follows. In Secton II we provde a revew of related on work on methods for communty evoluton predcton. In Secton III we present our framework for communty evoluton predcton placng specal focus on the extracton of approprate features to represent communtes. Next, n Secton IV we present experments usng a real-lfe socal network. Fnally, n Secton V concludes ths work and offers drectons for future work. II. RELATED WORK Varous approaches have been presented n the lterature to predct the evoluton of communtes. Brodka et al. [7] tred dfferent classfers to predct sx evolutonary events of communtes (defned as grow, shrnk, contnue, merge, splt and dssolve). Classfers were traned usng as features the sze of the communtes and ther evolutonary events over the last three tmeframes. An extended verson of ths work s presented n [11], where a larger set of features and past tmeframes are

2 used. Sequental and non-sequental classfers were evaluated n [8] to nfer four types of communty evolutonary phenomena: contnuaton, shrnkage, growth and dssoluton. Features related to the structure, content and context of communtes over the past one, two or three tmeframes were consdered, to test how past evolutons of communtes affecredctons. Ilhan and Oguducu [12] ntroduced a tme seres ARIMA model to estmate how communty features values wll change n future tmeframes and predct sx types of evolutonary events (survve, shrnk, contnue, merge, splt and dssolve) usng those feature estmates for tranng a classfer. Patl et al. [13] addressed a smlar problem to the above, that of o predctng the stablty of communtes,.e., whether a communty wll dsappear or thrve n the future. Takaffol et al. [14] consdered fve evolutonary events, namely survve, merge, splt, sze and coheson. These events are treated as beng non mutually exclusve and, thus, may occur together at the same tme for a partcular communty. Hence, they learn separate models to predct each of them, usng structural and temporal nformaton about the communtes. The sze and coheson evolutons are meanngful for a survvng communty only, therefore a two-stage technque s employed to predct them. Frst, the survval of a communty s decded and, f t s found to survve, the predcton for these two evolutons follows usng the correspondng models. Karam et al. [9] dstngushed between two types of growth for a communty, dffuson and non-dffuson growth, and analyzed the processes whch govern them. Dffuson growth occurs when a communty attracts new members through tes to exstng members, whereas n non-dffuson growth ndvduals wth no pror tes become members. They generated models whch explot a communtys structure and past growth behavour to predct ts future rate of growth and longevty of growth. In [15], structural nformaton extracted from the early stages of a communty were utlzed to nfer the lfespan of a communty usng lnear regresson. Results ndcated that there s a correlaton between the lfespan of a communty and ts structural propertes. Fnally, n [10] a method was proposed that examnes the structural characterstcs of a socal network to extract an approprate subset of features for representng communtes. Usng ths feature subset that s talored to the ndvdual network, the accuracy of predctons n communty evoluton tasks s mproved and, also, a speedup wth regards to run tme s acheved. III. PREDICTING COMMUNITY EVOLUTION Communtes n dynamc socal networks evolve, snce ther members and the nteractons between the members change as tme passes. We consder four popular and mutually exclusve evolutonary events and present a framework for predctng them that covers all the necessary steps nvolved. In partcular, we wsh to buld a predctve model that dscrmnates whether a communty n the future wll grow or shrnk n sze, contnue to exst wth almost no change of ts current form, or dssolve and, thus, completely dsappear from the network. Our framework conssts of the followng steps: 1) Segment the socal network data nto tmeframes. 2) Detect the communtes n each tmeframe. 3) Track communtes across tme to dentfy ther evoluton and correspondng evolutonary events. 4) Compute structural and temporal communty features. 5) Tran a classfer to predct communty evoluton. Below we analyze n detal each of these steps. A. Segmentaton nto Tmeframes Data acqured from socal networks are tmestamped and come n the form of data streams. To handle the contnuous tme dmenson of the data stream, we dscretze t nto a predefned number of tme-ordered tmeframes F t, t = 1,..., T. Data s assgned to tmeframes accordng to ts tmestamp, such that each tmeframe contans the same number of elements (e.g., user posts from a socal network). Consecutve tmeframes are allowed to overlap, wth an overlap O [0, 1], n order to have a smoother transton between them and, thus, better montor the evoluton of communtes, as suggested n [16]. The amount of overlap desgnates the percentage of the prevous tmeframe that s also part of the next tmeframe, as shown n the examples of Fgure 1. Fg. 1. Examples of overlap between two tmeframes. B. Communty Detecton Havng segmented a socal network dataset nto tmeframes, the next step s to ndependently detect the communtes n each tmeframe. We model the socal network as a sequence of undrected graphs {G 1, G 2,..., G T }, where G t = (V t, E t ) denotes the graph of tmeframe F t wth vertex set V t, n(f t ) = V t, and edge set E t, m(f t ) = E t. Each user of the socal network n a partcular tmeframe s represented by a node n the tmeframe graph, and there s an edge between two nodes f an nteracton between the correspondng users occurs n ths tmeframe (e.g., one responds to the other s post). A communty corresponds to a densely connected subset of users (.e., a subgraph) of the tmeframe graph that s loosely connected to the rest of the graph. Any graph clusterng algorthm can be employed to uncover the communtes n socal networks. One popular choce s the Louvan algorthm [17], whch optmzes the modularty measure and scales well to large networks. We use the set C t = { 1, 2,..., Kt } to denote the communtes detected at tmeframe F t. Each communty k C t s represented by a graph G k t = (Vt k, Et k ), whch s a subgraph of G t wth vertex set Vt k V t, n( k ) = Vt k, and edge set Et k = {(u, v) E t : u, v Vt k }, m( k ) = Et k. C. Communty Trackng A communty n a tmeframe may be matched to a communty n a followng (not necessarly consecutve) tmeframe, n the sense that the latter s the evoluton of the frst. The nstances of the same communty at dfferent tmeframes form what s called a dynamc communty. Formally, a dynamc communty s defned as a sequence of matched communtes M = { k1 1,..., C kp,..., km m }, where 1 t 1 < t 2 <... < 41

3 t m T, 1 k j K tj j = 1,..., m. For a communty C kp M, ts past nstances, C kj t j, j < p, are referred to as the ancestors of the communty. Gven the communtes n each tmeframe, communty trackng algorthms that fnd matchng communtes based on a smlarty measure, such as GED [7], can be employed to locate the dynamc communtes arsng n the dataset. These algorthms also assgn a label to each communty of the dynamc communty sequence that descrbes the type of evoluton whch took place as the communty evolved (e.g., contnue, grow, shrnk, dssolve 1 ). These labels serve as the groundtruth for tranng a model to perform communty evoluton predcton. Note that some socal networks provde annotatons whch allow to readly track communtes, wthout needng the nterventon of trackng algorthms. Such a case s descrbed n our experments wth the Mathematcs Stack Exchange socal network. D. Communty Feature Engneerng To attan an nformatve representaton of communtes that captures a varety of ther propertes, we employ a comprehensve set of structural and temporal features, amng to predct communty evoluton accurately. Structural features capture dfferent aspects of the communty graph, whle temporal features capture characterstcs of the evoluton of the communty by extractng nformaton from ts ancestors. On the followng, we analytcally present those features. 1) Structural Features: Relatve Sze [8] s the normalzed value of communty s C k t sze n tmeframe F t : RS(C k t ) = n(c k t ) /n(f t ) (1) Relatve Edges Number s the normalzed value of edges belongng to communty C k t n tmeframe F t : RE(C k t ) = m(c k t ) /m(f t ) (2) Densty [8] s the rato of the actual edges of communty k to the maxmum number of edges the communty could have: D(C k t ) = m(c k t ) n(c k t )(n(c k t ) 1)/2 Coheson [8] s the product between the densty and the nverse fracton of edges (out of all possble edges) pontng outsde of communty C k t : Ch(C k t ) = D(C k t ) n(ck t )(n(f t ) n(c k t )) m out (C k t ) (3), (4) where m out (C k t ) = {(u, v) E t : u V k t, v / V k t }. Rato Assocaton [18] s the average nternal degree of a communty s members: RA(C k t ) = 2m(C k t ) / n(c k t ) (5) 1 A communty dssolves when t s the last communty of the dynamc communty sequence, hence t was not matched to a communty of a subsequent tmeframe. Rato Cut [18] s the average external degree of a communty s members: RC(C k t ) = m out (C k t ) / n(c k t ) (6) Normalzed Cut [18] measures the edge volume that ponts outsde of the communty: NC(C k t ) = m out (C k t ) / (2m(C k t ) + m out (C k t )) (7) Average Path Length of communty k s the average path length on the communty s graph G k t, as defned n graph theory. Dameter of communty k s the dameter of the communty s graph G k t. Clusterng Coeffcent [14] of communty k shows how often, on average, the neghbours of a node of the communty are also connected to each other, based on the communty graph G k t. Centralty measures capture how central (.e., centre of mportance) each node (.e., user) of a communty k s. We use three centralty measures as features, namely closeness, betweenness and egenvector centralty [19], [20]. Closeness centralty shows how close a node s to other nodes n the communty graph G k t, n terms of the edges that must be traversed to reach the other nodes. Betweenness centralty measures the number of shortesaths a node les on. Hence, t shows the mportance of the node n controllng the communcaton between other nodes of the communty. Fnally, egenvector centralty reflects the dea that a node s more central, f t s connected to central nodes. As centralty measures are defned on a per node bass, we take the average over all nodes to calculate the centralty of the entre communty. 2) Temporal Features: To present these features, we wll use as reference a dynamc communty M = { k1 1,..., C kp,..., km m }, as defned n Secton III-C, and assume that we want to compute the temporal features of communty C kp usng ts n most recent ancestors n tme. We shall splt temporal features nto three groups. Structural features and evolutonary events of ancestors: The structural features (descrbed above) of the n ancestor communtes, as well as the evolutonary events assgned to them through trackng, form our frst group of temporal features. The temporal features belongng to the second group are defned between pars of communtes and depct how a communty has evolved compared to ts prevous nstance n tme. We calculate these features between the followng pars of communtes from M when we want to represent communty C kp usng n ancestors: (C kp n n, C kp n+1 n+1 ),..., (C kp 1 1, C kp ). These features are: Jaccard Coeffcent s the fracton of members that are common n both nstances of the communty: k V JC( k, C k 1 t ) = V k 1 V k t V k 1 (8) 42

4 Jon Nodes Rato [14] s the percentage of new members jonng the communty compared to ts prevous nstance: JNR( k, C k 1 ) = Vt k \ V k 1 / V k t (9) Left Nodes Rato [14] s the percentage of members leavng the communty compared to ts prevous nstance: LNR( k, C k 1 ) = V k 1 \ Vt k / V k 1 (10) Actveness [12] measures the new edges per node that a communty contans compared to ts prevous nstance: Act( k, C k 1 ) = Et k \ E k 1 / V k t (11) The temporal features belongng to the thrd group are defned for ndvdual communtes nstead of pars. When representng communty C kp usng n ancestors, we calculate these features for C kp and ts n ancestors. These features are: Lfespan [14] of communty kw w s the rato of the ancestors the communty has based on the correspondng dynamc communty, to the maxmum number of ancestors t could have. Obvously the maxmum number of ancestors equals t w. Agng [12] of communty kw w s the average age of the communty members. The age of a member s ncreased by 1 every tme t s found to be also a member of an ancestor communty of kw w n the correspondng dynamc communty. Agng s normalzed by dvdng wth the maxmum possble age of members, whch equals w. E. Learnng a Predctve Model for Communty Evoluton Communty evoluton predcton s formulated as a classfcaton problem, where the am s to tran a classfer that dstngushes between four types of evolutonary events (.e., classes). These events are: contnuaton, shrnkage, growth and dssoluton. The nstances for tranng the classfer are the communtes that have been extracted from the socal network tmeframes, whch are represented wth vectors comprsng of the structural and temporal features descrbed above, along wth ther correspondng class label obtaned through trackng. Any classfer that can handle nstances n vectoral form can be appled to learn the predctve model. In our experments we use Support Vector Machnes (SVMs) as the underlyng classfer. IV. EXPERIMENTAL EVALUATION To test the applcablty of our framework n practce and nvestgate the effcacy of the presented features n predctng communty evoluton, we perform experments over a dataset acqured from the Mathematcs Stack Exchange Q&A ste 2, a real-lfe socal network. Mathematcs Stack Exchange s a queston and answer ste for people studyng math, where users post questons, answer questons posted by other users and 2 comment on the users posts. All questons are tagged wth ther subject areas (.e., topcs), whle answers and comments nhert the topcs of the queston they correspond to. Our dataset contans tmestamped posts under varous topcs, publshed between 2009 and To conduct our experments we splt the dataset nto 10 tmeframes contanng an equal number of posts, set the overlap of tmeframes to O = 0.6 and set out to detect and track the communtes to obtan the nstances that wll be used to tran a support vector machne (SVM) classfer for predctng the evoluton of communtes. Each tmeframe s modeled wth an undrected graph where every dfferent user who posted a queston, answer or comment n ths tmeframe s a node of the graph and an edge s added between two users f one posts an answer or comment to respond to the other s post n the tmeframe. To detect the communtes n each tmeframe, we take advantage of the topcs assocated wth posts n Mathematcs Stack Exchange and do not employ a communty detecton algorthm. Specfcally, we consder that users belong n the same communty f they make posts (questons, answers or comments) about the same topc. Hence, each communty s assocated wth a partcular topc. Note that communtes that contan less than four members are consdered as artfact communtes and are gnored n our experments. Trackng the communtes across tme to obtan the dynamc communtes and ther evolutonary events s also done by utlzng the topcs. For each detected communty k n tmeframe F t, we obtan the topc assocated wth the communty and look for a matchng communty wth the same topc n a subsequent tmeframe F t, t > t. Tmeframes are processed sequentally, stoppng at the frst tmeframe a match s found (.e. a communty wth the same topc). If the topc s not found n any of the followng tmeframes (.e., no matchng communty s found), then we set the evolutonary event of k as dssoluton. Otherwse, a match s found to a communty k, t > t (.e. k s the most recent ancestor of k ) and the evolutonary event s set as: If n( k ) n( k ) > th, we set the evolutonary event of communty k as shrnkage. Else f n( k ) n(ck t ) > th, we set the evolutonary event of communty k as growth. Else, we set the evolutonary event of communty k as contnuaton. We call th > 0 the event threshold. Ths threshold determnes how much dfferent the szes of two matchng communtes should be n order to decde that a sgnfcant dfference exsts and thus label the evoluton as growth or shrnkage. Note that by defnng dfferent values for the event threshold, we obtan a dfferent ground-truth for the dataset. Fgure 2 llustrates the number of the dfferent evolutonary events n our dataset as we vary the threshold value. Notce the mbalance that exsts n the dataset n terms of the dfferent types of evolutonary events for all th values. Havng obtaned the communtes along wth ther labels, as descrbed above, we compute ther structural and temporal features (Secton III-D) to form the dataset for tranng and testng an SVM classfer. The SVM mplementaton avalable 43

5 Fg. 2. Number of communty evolutonary events for dfferent values of event threshold th. n Weka 3 wth RBF kernel s used for our experments. To tran and test the classfer, we employ a varant of the popular k-fold cross valdaton technque that s approprate for tmestamped data, called tme seres cross valdaton. In ths varant, folds correspond to tmeframes and each fold s once used as the test set and performance s averaged over all folds. When the -th fold s used as the test set, only the frst 1 folds are used n the tranng set, thus tme seres cross valdaton ensures that the nstances of the tranng serecede n tme those of the test set, respectng the natural orderng of the data. In all experments we apply the SMOTE oversamplng technque and the spreadsubsample undersamplng technque of Weka to counter the mbalance that exsts n our dataset (Fgure 2). The experment results obtaned wthout the use of oversamplng and undersamplng technques were nferor to the ones presented below, and thus were omtted due to space lmtaton. Moreover, we measure classfcaton performance usng the popular F1 score and report the per-class performance, as well as the performance over all classes usng the macro-f1 score whch s sutable for evaluatng mbalanced datasets. We am at nvestgatng whether the addton of the temporal features on top of the structural features mproves predctons and studyng how the number n of ancestors consdered when computng the temporal features affects predctons. To examne ths we perform experments for n {0, 2, 4, 6} and use ) only the structural features and the evolutonary events of the ancestors as temporal features and ) the complete set of temporal features. When n = 0 no ancestors are consdered, hence only the structural features are used to represent the communtes. For each value of n we try th {2, 3, 4, 5, 6} and optmze the nternal parameters of the SVM classfer usng grd search and report the beserformance. Results are shown n Tables I and II and were obtaned for th = 6 for all tred values of n. It s evdent that temporal features help n mprovng performance. Also the use of 2 or 4 ancestors seems to be benefcal whle for 6 ancestors performance degrades, ndcatng that gong too far back tme s not helpful. It s 3 TABLE I. RESULTS IN TERMS OF F1 SCORE WHEN ONLY THE STRUCTURAL FEATURES AND EVOLUTIONARY EVENTS OF ANCESTORS ARE USED AS TEMPORAL FEATURES. Ancestors Contnue Shrnk Grow Dssolve Overall TABLE II. RESULTS IN TERMS OF F1 SCORE WHEN ALL TEMPORAL FEATURES ARE INCLUDED. Ancestors Contnue Shrnk Grow Dssolve Overall noted that F1 scores are rather low n these experments and ths occurs because low th values were used, resultng n low qualty ground truth, as shown below. Havng shown that the temporal features mprove predctons we perform a more detaled experment to examne f the outrgherformance scores can be mproved. Specfcally, we experment wth a greater range of event threshold values, ncludng large ones, th {5, 10, 15, 20, 25, 30, 60}. When assgnng evolutonary events usng hgher values for event threshold, we become more strct whle decdng whether a communty has grown or shrunk, snce the dfference n sze between two matched communtes must be larger n order to assgn these labels. Hence, more communtes are labeled as contnung as the event threshold ncreases. Although a hgher threshold ncreases mbalance (Fgure 2), t may stll lead to better performance f the underlyng ground-truth s of hgher qualty. Results over all classes are reported n Table III, where also the macro recall and macro precson are shown. It s evdent thaerformance has consderably ncreased n these experments and for all number of ancestors tred the best results were obtaned for th = 30. Ths shows that a groundtruth of hgher qualty s constructed n ths case allowng 44

6 TABLE III. RESULTS WHEN ALL TEMPORAL FEATURES ARE INCLUDED AND AN EXTENDED SET OF EVENT THRESHOLD VALUES IS USED. Ancestors Macro F1 Macro Recall Macro Precson for a more accurate predcton of communty evoluton. In accordance wth the prevous experment, we observe that performance ncreases as we move from smaller to greater values for the number of ancestors, untl we use 6 ancestors, at whch ponerformance declnes. Overall, our experments have shown that structural features when combned wth temporal features mprove predcton accuracy and that usng some, but not too many, ancestors to compute temporal features s also benefcal. V. CONCLUSIONS Ths work amed aredctng the evoluton of communtes that are formed n socal networks as a result of user nteracton, usng a mxture of structural and temporal features. Four types of evoluton that commonly arse n socal networks were examned, namely the contnuaton, growth, shrnkage and dssoluton of communtes. We presented a framework that ncorporates all necessary steps for buldng a predctve model to nfer communty evoluton. These steps are: segmentaton nto tmeframes, detecton and trackng of communtes, calculaton of communtes features and classfer tranng. We performed experments usng real-lfe socal network data acqured from the Mathematcs Stack Exchange Q&A ste. Experments demonstrated tharedcton accuracy mproves when temporal features are used on top of the structural ones. Also, the extent of past evolutons of a communty consdered (.e., the number of ancestors) affects predctons and usng four ancestors gave the best results n our dataset. It seems that the past of a communty encapsulates nformaton about ts future evoluton and can help n mprovng predctons, f we do not go too far back n tme. Future work wll focus on the predcton of other types of communty evoluton, such as merges and splts where there s no one-to-one correspondence between communtes as they evolve. The ncorporaton of other types of features n order to mprove predctons, such as features derved from the text posted by socal network users (e.g., topcs of dscusson and sentment) and features related to the context of a partcular socal network (e.g., reputaton n the Mathematcs Stack Exchange ste and hashtags n Twtter), could be also examned. In addton, usng other classfers, apart from SVMs, for predctons and performng tests wth more datasets, as well as comparng our approach to exstng ones from the lterature, such as [14], s n our plans. Moreover, fndng the optmal tmeframes for splttng the data stream of a socal network poses an nterestng problem tself. Such optmal tmeframes would contrbute n a more accurate detecton of communtes and subsequently ther trackng and predcton. Fnally, usng frst-order logc to capture the knowledge governng communty evoluton and applyng Markov Logc Networks [21] to predct the evoluton of communtes, s another nterestng research drecton. REFERENCES [1] D. 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