A Real-Time Detecting Algorithm for Tracking Community Structure of Dynamic Networks

Transcription

1 A Real-Tme Detetng Algorthm for Trakng Communty Struture of Dynam Networks Jaxng Shang*, Lanhen Lu*, Feng Xe, Zhen Chen, Jaa Mao, Xueln Fang, Cheng Wu* Department of Automaton, Tsnghua Unversty, Beng,, Chna *Natonal CIMS Engneerng Center, Tsnghua Unversty, Beng,, Chna Insttute of Command Automaton, PLA Unversty of Sene and Tehnology, Nang, 7, Chna Insttute of Computer Sene, Natonal Unversty of Defense Tehnology, Changsha, 7, Chna {xef, {lulanhen, ABSTRACT In ths paper a smple but effent real-tme detetng algorthm s proposed for trakng ommunty struture of dynam networks. Communty struture s ntutvely haraterzed as dvsons of network nodes nto subgroups, wthn whh nodes are densely onneted whle between whh they are sparsely onneted. To evaluate the qualty of ommunty struture of a network, a metr alled modularty s proposed and many algorthms are developed on optmzng t. However, most of the modularty based algorthms deal wth stat networks and annot be performed frequently, due to ther hgh omputng omplexty. In order to trak the ommunty struture of dynam networks n a fnegraned way, we propose a modularty based algorthm that s nremental and has very low omputng omplexty. In our algorthm we adopt a two-step approah. Frstly we apply the Blondel et al s algorthm for detetng stat ommuntes to obtan an ntal ommunty struture. Then, apply our nremental updatng strateges to trak the dynam ommuntes. The performane of our algorthm s measured n terms of the modularty. We test the algorthm on trakng ommunty struture of Enron Emal and three other real world datasets. The expermental results show that our algorthm an keep trak of ommunty struture n tme and outperform the well known algorthm n terms of modularty. Categores and Subet Desrptors H.. [Database Applatons]: Data mnng. General Terms Algorthms, Measurement, Expermentaton. Keywords Real-Tme Detetng Algorthm, Modularty, Communty Struture, Data Mnng.. INTRODUCTION Many systems an be represented as networks or graphs [,, ], n whh nodes represent the ndvduals whle edges represent the relatonshp or nteratons between nodes. Examples nlude soal networks [], taton networks [,, 7], bologal networks [,, ], moble phone networks [,, ], terrorsm networks [,, ], et. One of the most mportant problems n network analyss s the dentfaton of ommunty struture, the dvson of network nodes nto subgroups, wthn whh nodes are densely onneted whle between whh they are sparsely onneted []. The dentfaton of ommuntes often reveals The th SNA-KDD Workshop (SNA-KDD ), August,, Beng, Chna. deeper propertes of networks and has many mportant applatons, suh as detetng rmnal organzatons, fndng ommon hobbes shared by people, et. To ombat the problem of fndng ommunty struture, many algorthms have been proposed reently. Grvan and Newman [] proposed a dvsve algorthm that uses edge betweenness as a metr to dentfy the boundares of ommuntes. In ths algorthm, Newman proposed modularty (Q) as a metr to evaluate the qualty of a ommunty struture and suessfully appled t to a varety of networks. However, as noted n [], the algorthm s tme-onsumng, whh restrts ts applaton to only small networks. After Newman proposed hs modularty-based algorthm, a number of faster algorthms have been proposed [,,,, ]. Sne fndng a global optmzaton Q s a NP-hard problem [, ], most of these algorthms take advantage of heurst strateges. To the best of our knowledge, the fastest modularty optmzaton based algorthm s proposed by Blondel et al [] (Blondel, Gullaume, Lambotte and Lefebvre, n the rest of the paper, we wll all ths algorthm the algorthm ). Expermental results show that ths algorthm outperforms all other state-of-art modularty based algorthms n terms of omputaton tme (It reveals ommuntes of a web network of mllon nodes and more than one bllon lnks wthn mnutes). Moreover, the qualty of the ommunty struture deteted s qute good, as measured by the modularty. The modularty optmzaton based algorthms mentoned above have been appled suessfully n many real world networks and some of them run qute fast [, ]. However, most of them ust fous on the representaton of graphs as stat networks and only gve some snapshots of networks. When the networks hange, these algorthms have to be re-performed to get the ommunty struture, makng t mpossble for them to trak the ommunty struture on a fne-graned level. Real world networks are usually dynam and some of them hange frequently. For some networks ther ommunty struture needs to be traked tmely. For example, n the terrorsm networks the rmnals usually do not ommunate wth eah other untl they are to ommt the rme. We have to be able to trak the ommunty struture so as to reveal the rmnal organzatons and avod terrorst ndent before t s too late. In order to keep trak of the ommunty struture of a network on a fne-graned level, our algorthm should have very low omputng omplexty n updatng the ommunty struture when network hanges. To ombat ths problem we model the hange of networks as sequental nrease of edges (ths s why we all our algorthm nremental). We lassfy nreased edges nto four types: ) nner ommunty edge, ) ross ommunty edge, ) half-new edge, and v) new edge. In our two-step approah, we

2 frstly apply the algorthm to generate an ntal ommunty struture of networks. Then we apply dfferent strateges to update the ommunty struture aordng to the edges type. Whh strategy to apply follows a bas prnple: the strategy should be able to nrease the modularty of the ommunty struture, f not, t should make the lost n modularty as low as possble. The runnng tme for updatng the ommunty struture when nreasng an edge vares from O () to OS ( ), where S s the sze of the ommunty to be updated. From the expermental results we found that the nremental algorthm runs at the omputng omplexty of O() n most of the tme, whh guarantees the hgh effeny of our algorthm. The qualty of our algorthm s evaluated by the modularty. We test the performane on four real-world datasets, nludng emal ommunaton network, paper taton network, web votng network, et. The performane s ompared wth that of the [] and [] algorthm. Expermental results show that our algorthm outperforms both and n omputng tme and outperforms n terms of modularty. The rest of the paper s organzed as follows: Seton gves some related work n dynam networks. Seton ntrodues the modularty and algorthm. Seton presents our nremental algorthm for trakng ommunty struture of dynam networks; Seton gves expermental results to valdate the effetveness and the hgh omputatonal effeny of our algorthm. Seton draws the onluson.. RELATED WORK Reently sgnfant attenton s attrated to the lterature of ommunty deteton n dynam networks. In [7] the authors from Yahoo! studed the evoluton of ommunty struture wthn two large onlne soal networks: Flker and Yahoo!. Sne they have aess to the entre lfetme of the two networks, they are able to study ther dynam propertes n depth. By researhng the evoluton of edge densty, ommunty dameter, delay n reproty, degree and node dstrbuton n the two dynam networks, the authors reveal an nterestng unexpeted growng rule: rapd growth, delne, and then slow but steady growth. C. Tantpathananandh and T. Berger-Wolf [] model the dentfaton of ommuntes n dynam networks as a ombnatoral optmzaton problem (a olorng problem) based on some bas assumptons about the behavor of ndvduals. The optmzaton problem s shown to be NP-hard, so the authors present approxmaton algorthms usng dynam programmng and greedy heursts to solve t. In [7] Tantpathananandh and Berger-Wolf extend ther prevous model to arbtrary dynam networks and approxmately solve the optmzaton problem usng semdefnte programmng relaxaton and a roundng heurst. Greene et. [] use an event model to desrbe the evoluton of dynam networks where the lfe-yle of eah ommunty s haraterzed by a seres of sgnfant events. These events nlude brth, death, mergng, splttng, expanson and ontraton of dynam ommuntes. In ths paper a dynam network s represented as a set of tme step graphs. Frstly step ommuntes are dentfed at ndvdual tme steps. The step ommuntes an be found usng arbtrary ommunty deteton algorthms. Then the authors present a smple algorthm to math the ommuntes found at onseutve tme steps. The algorthm s evaluated on both synthet dynam networks and a real-world moble operator all network.. PRELIMINARIES The modularty s proposed by Newman [] and used to evaluate the qualty of network s ommunty struture. Let G ( V, E) be an undreted weghted graph [], where V s the set of nodes and E s the set of edges between nodes. Then we an use an adaeny matrx A to desrbe the graph: A w f nodes and are onneted, () otherwse where w s the weght of edge onnetng nodes and. In [] Newman gves the defnton of modularty as follows: kk Q A,, V () where s the ommunty that node belongs to, m A s the sum of weghts of edges n the graph,, V k A s the weghted degree of node, the funton V uv, s f u v and otherwse. Another expresson of Q s: () C Q e a where evw A, v, w s the fraton of, V weghts of edges that onnetng nodes n ommunty v and nodes n ommunty w, av k, v s the fraton of V weghts of edges that are attahed to nodes n ommunty v. The expresson of Q onssts of two parts. Intutvely, t measures the fraton of edges wthn ommuntes mnus the expeted fraton of edges n a network wth the same ommunty dvsons but random onnetons between nodes. Then the task of ommunty deteton turns to the task of fndng a dvson orrespondng to hgh value of Q. Unfortunately, fndng the exat dvson of a network orrespondng to the hghest modularty s a problem that s omputatonally hard []. So approxmaton algorthms are neessary when dealng wth large networks. To the best of our knowledge, the fastest approxmaton algorthm for optmzng modularty on large networks s proposed by Blondel et al [] (the algorthm), whh apples a two-phase teraton approah to generate herarhal ommunty struture. The dea of ths effent algorthm s motvated by the fat that the gan n modularty Q obtaned by movng an solated node to a ommunty C an easly be omputed by n k, n k n k Q k kn, m where n A,,, V () s the sum of the weghts of the edges nsde, k, s the sum of the V

3 weghts of the edges ndent to nodes n, k s the sum of the weghts of the edges ndent to node, k n, s the sum of the weghts of the edges from to nodes n and m s the sum of the weghts of all the edges n the network. There s another smlar expresson used to evaluate the hange of modularty when node s removed from ts ommunty. Then we an evaluate the hange of modularty when a node hanges ts ommunty by removng t from ts urrent ommunty and then by movng t nto the new ommunty. Ths algorthm apples a two-phase approah. Intally, eah node s plaed n a separate ommunty. In the frst phase, for eah node we hek f movng the node from ts urrent ommunty to any ommunty to whh a neghbor belongs would yeld an nrease n modularty. If so, the node s moved to the neghborng ommunty that gves the hghest gan n modularty. The proess s ontnued untl all nodes are proessed. Then n the seond phase we onvert eah updated ommunty to a sngle node (alled ommunty-node) n a new network, wth edges between ommunty-nodes where there are edges between nodes n the ommuntes of the orgnal network. The weghts of the new edges are obtaned by summng over all prevous weghts. For eah new node, there s an edge alled self-loop, whose weght s equal to the sum of the weghts of prevous edges nsde the ommunty. Then, return to the frst phase and run the next round of teraton. The algorthm wll termnate when the modularty gan between two rounds of teraton s lower than a gven threshold. The fnal results orrespond to a herarhal ommunty struture. The algorthm runs qukly, takng ust mnutes to run on a network of mllon nodes and bllon edges. Moreover, the deteted ommunty strutures tend to have hgh modularty than other algorthms mentoned n [].. INCREMENTAL ALGORITHM The algorthm performs qute well on stat networks. However, when the network hanges, the algorthm has to be reperformed to trak the ommunty struture. Ths s obvously mpossble f the networks hange frequently or when they beome very large. Tantpathananandh, Greene, et. propose algorthms to fnd ommuntes n dynam networks. However, there s a ommon shortomng n these algorthms: they are unable to trak the ommuntes on a fne-graned level. Some of them [, 7] an handle only small networks, due to the hgh omputng omplexty n solvng optmzaton problems.. Type of Edges In order to takle these problems, we propose an nremental algorthm wth low omputng omplexty. We vew the evoluton of networks as the observaton of nteratons between ndvduals. For example, Bob and Ale are loser together beause we observe that they nterat more often wth eah other, a new member oned n the ommunty beause we observe that he or she nterats wth at least one of the members n the ommunty. Ths s qute a dfferent perspetve from that of state-of-art studes. Then we an model the hange of networks as sequental observaton of nteratons, whh an be represented as sequental nrease of edges (ths s why we all our algorthm nremental). For eah nreased edge l (, w, ), where s the soure node, s the target node and w s the weght, t ould be assgned to one of the four types:. Inner ommunty edge: the two nodes ndent to the edge already exst and belong to the same ommunty.. Cross ommunty edge: the two nodes ndent to the edge already exst and belong to dfferent ommuntes.. Half-new edge: one of the nodes ndent to the edge s new.. New edge: both of the nodes ndent to the edge are new. Fgure shows the four types of nreased edges and the hange t brngs to the network. The orgnal network graph s dvded nto two ommuntes {, } and {,, } aordng to the algorthm, as shown on the left sde. After the four types of edges are added, the ommunty struture s affeted n dfferent ways, as shown on the rght sde. () () () () l(,,) l(,,) l(,,) l(,7,7) Fgure. The types of edges. Algorthm Then we take dfferent operatons to update the ommunty struture aordng to the edges type. These operatons nlude:. keepng the ommunty struture unhanged;. ombnng two ommuntes nto one;. assgnng nodes to an exstng ommunty;. reatng a new ommunty wth new nodes. We see from these operatons that operaton redues the number of ommuntes whle operaton nreases t. Operaton atually does nothng on the ommunty struture. Whh operaton to be taken follows a bas prnple: the operaton should be able to nrease the modularty of the ommunty struture, f not, t should make the lost n modularty as low as possble. In Eq. (), by wrtng e n and a, We gve another expresson of Eq. () as follows: Q n C () It wll be used to dede the operaton to be taken. Now we llustrate n detal how to hoose operatons to update the ommunty struture when an nreased edge l (,, w ) omes.. If the nreased edge l (,, w ) s an nner ommunty edge, as llustrated n Fgure, we see that t nreases the nner onnetons of the ommunty and keeps the nter-ommunty 7 7

4 onnetons unhanged, whh ondes wth the bas prnple of the modularty, so operaton wll be taken, n ths ase, the ommunty struture wll keep unhanged.. If the nreased edge l (,, w ) s a ross ommunty edge, whh means nodes and belong to two dfferent ommuntes, suppose they are and respetvely. Then two anddate operatons wll be taken: operaton or operaton. The former keeps the ommunty struture unhanged, whle the latter ombnes ommuntes and nto one. Our purpose s to nrease the modularty or make the loss of t as low as possble. So we ompare the modularty gan brought by these two operatons. If the ommunty struture of the network keeps unhanged, the new modularty value wll be:, Q n w C w ( ) ( ) w w n n w w If the two ommuntes and are ombned nto one, the new modularty value an be formulated as:, Q n w C w ( ) w n n w w Suppose the modularty of the ommunty struture before any operatons are taken s Q, then the modularty gan brought by the two operatons wll be Q Q Q and Q Q Q respetvely. By omparng them, we get: Q Q Q Q ( )( w w ) w w w When w ( w ) ( w )( w ), namely, Q Q, we take operaton, else operaton.. If the nreased edge l (,, w ) s a half-new edge, whh means that one of the nodes ndent to l (assume ) already exsts n the network whle the other (node ) s new (there are no edges between and any of the other nodes before l s nreased). Two anddate operatons wll be taken: operaton or operaton. The former assgns to the ommunty that belongs to (assume ), whle the latter reates a new ommunty and assgn to t. Smlarly, we ompare the modularty gan brought by the two operatons. If s assgned to ommunty, the new modularty value wll be: () (7) () Q n w C w ( ) w n w w If a new ommunty s reated for, the new modularty value wll be: Q n w C w ( ) w w n w w () () Comparng the modularty gan brought by the two operatons, we get: w( m ) w Q Q () ( w ) Sne s dentally true, Eq. () wll always be smaller than zero, whh means operaton wll brng more gan or less loss n modularty than operaton, so operaton wll be taken to update the ommunty struture.. If the nreased edge l (,, w ) s a new edge, whh means that both of the nodes ndent to l are new (there are no edges among, and any of the other nodes of the network before l s nreased). Two anddate operatons wll be taken: operaton or operaton. The former assgns and to an exstng ommunty (assume k ), whle the latter reates a new ommunty and put both and nto t. Agan we ompare the modularty gan brought by the two operatons. If we assgn, to an exstng ommunty k, the new modularty value wll be: k Q n w C w k ( ) k w n w w () If we reate a new ommunty and put both and nto t, the new modularty value wll be: k Q n w C w k ( ) k w n w w w () Comparng the modularty gan brought by the two operatons, we get: w Q ( w) k Q () where Q Q wll be dentally true, so operaton wll be taken.

5 The relatonshp between dfferent types of edges and the orrespondng operatons adopted on them are summarzed n Table, n whh row tems represent the operatons whle olumn tems represent the dfferent types of edges. In ths table, ICE s short for Inner ommunty edge, CCE for Cross ommunty edge, HNE for Half-new edge and NE for New edge. Y means the operaton wll be taken on the orrespondng type of edge, P means the operaton s possble to be taken on the edge and N means the operaton wll not be taken on t. Table. Relatonshp between dfferent types of edges and operatons ICE CCE HNE NE Opt Y P N N Opt N P N N Opt N N Y N Opt N N N Y From ths table we see that f the nreased edge s a ross ommunty edge, operaton and operaton are possble to be taken. The algorthm s desrbed n Algorthm. Algorthm : The nremental algorthm. Deeper Inspeton of the Algorthm In ths part we wll take a deep look at the nremental algorthm. The nremental algorthm s desgned for trakng ommunty struture of dynam networks of whh the hange s nremental and frequent. Suppose an extreme example: the ntal network ontans no nodes or edges ( GV (, E), V, E ) and the ommunty struture s nrementally generated wth our algorthm. The fnal network s shown n Fgure (), whh onssts of nodes and 7 edges. We frstly onsder the sequene of nreased edges as {(,, ), (,, ), (,, ), (,, ), (,, ), (,, ), (,, )}. When the frst edge (,, ) s nreased, from the defnton of the types of edges, we see t s a new edge, so operaton wll be taken, n ths ase, a new ommunty s reated and nodes and are assgned to t. Then edge (,, ) omes, sne node already exsts n the network now, operaton wll be taken and vertex s assgned to, the ommunty that vertex belongs to. The next nreased Intalze: Run the GBL algorthm to generate an ntal ommunty struture for := to Number of nreased edges do swth Typeof(edge ) ase: ICE Keep ommunty struture unhanged. ase: CCE f (deltaq(opt ) > deltaq(opt )) then Combne ommuntes wth opt. else Keep ommunty struture unhanged. end ase: HNE Update ommunty struture wth opt. ase: NE Update ommunty struture wth opt. end edge s (,, ), whh s an nner ommunty edge, operaton wll be taken and eah node keeps ts ommunty afflaton unhanged. Then edge (,, ) s nreased and a new ommunty s reated. Ths proess keeps on untl all the edges are nreased. Fnally we get the ommunty struture of the network, whh onssts of two ommuntes. The nodes of the two ommuntes are olored, as shown n Fgure (). Let us onsder another rumstane and make the sequene of nreased edges as {(,, ), (,, ), (,, ), (,, ), (, ), (,, ), (,, )}. After the exeuton of the nremental algorthm, we wll get a ommunty struture onsstng of only one ommunty, as shown n Fgure (). () - - () - - () Fgure. The evoluton of a dynam network. From ths example t s observed that dfferent orders of nreased edges result n ompletely dfferent ommunty strutures, the frst one of whh s onsstent wth the result of the algorthm. Ths s manly beause the ommunty struture s bult from null and there s no heurst nformaton about the ntal partton of the network. If the network s large and there s a good ntal partton, the above stuaton s hardly to happen, espeally when the edges are randomly nreased, whh we suppose to be the ase of most real world networks. An mportant reason we hoose the algorthm to generate the ntal ommunty struture s beause ts qualty s qute hgh, ompared to other algorthms.. Complexty Analyss Sne our nremental algorthm orents from the algorthm, we wll frst revew the omputatonal omplexty of t. The algorthm s teratve and we all eah round of teraton as a pass, whh s further dvded nto two phases. Assume that we start wth a weghted network of N nodes and M edges. In the frst phase, eah node s onsdered to be an solate ommunty. So there are as many ommuntes as number of nodes n ths ntal partton. For eah node we ompute the gan of modularty by movng from ts urrent ommunty to the neghbor ommuntes. Then s moved to the ommunty for whh the gan of modularty s the maxmum and postve, otherwse wll stay n ts orgnal ommunty. From Eq. () we

6 see the tme omplexty for movng node to ts neghbor ommunty s O (). For node, all of ts neghbor nodes wll be onsdered to get the maxmum gran of modularty. So the omputng tme for movng node s Od ( ), where d s the degree of node. In the frst phase eah node has to be handled, so the overall tme omplexty of the frst phase s N Od ( ) ( ) OM. After the frst phase many solate ommuntes wll dsappear whle some of them wll beome larger, makng the al number of ommuntes muh smaller. The seond phase of the algorthm onssts n buldng a new network whose nodes be the ommuntes found durng the frst phase. The man work durng ths phase s summng up the weghts of edges between nodes of ommuntes. In pratal mplementaton, ths work has been done durng the frst phase, whh means the al omputng tme of the frst pass s OM ( ). Assume that after the pass we get a network of N nodes and M edges, then the tme omplexty of the seond pass wll be OM ( ). If the algorthm stops after the kth pass, the al tme omplexty k of ths algorthm wll be OM ( M). The expermental results show that the number of passes s usually smaller than and the algorthm onverges very qukly after the frst pass, due to the quk derement of number of edges and nodes after a pass. So the omputng tme s manly spent on the frst pass, makng the al tme omplexty of the algorthm the same magntude as OM ( ). In the nremental algorthm, dfferent types of nreased edges orrespond to dfferent operatons, therefore wth dfferent omputng tme. For an nreased edge l (,, w ), f operaton s taken, sne no new node s nreased and all nodes keep ther ommunty afflaton unhanged, all that needs to do s updatng n and wth O () tme. Smlarly, the tme omplexty of operaton and operaton s O (). For operaton, the two ommuntes and are ombned nto one. So eah vertex n one of the ommuntes (assume ) has to hange ts ommunty afflaton to the other. Suppose the sze of the new ommunty s S, then the tme omplexty wll be OS ( ). Aordng to the statstal results of our experments, the perentage of ourrenes of operaton s usually very low (often less than %) ompared to that of other operatons, so the nremental algorthm wll run at a hgh speed n most of the tme.. PERFORMANCE EVALUATIONS In ths seton we wll evaluate the performane of our nremental algorthm by the modularty on four real world datasets, nludng ommunaton networks, taton networks, and soal networks. In our experments we model the hange of network as sequental nrement of edges, whle not onsder the derease of edges. Ths s beause for real word networks, we usually know from the observaton of nteratons of two ndvduals that a relatonshp s bult, whle we ould not tell that the relatonshp s ended by the same observaton. The expermental results are ompared wth the well known and algorthms. We also ompare the omputng tme of our algorthm wth that of and GBL when dealng wth dynam networks. It s shown that our algorthm outperforms both and n omputng tme and outperforms n terms of modularty. k. Evaluatons Under the Enron Dataset The Enron dataset [] ollets emal ommunaton douments from senor exeutves n the Enron orpus. Ths dataset was orgnally prepared by the CALO Proet (A Cogntve Assstant that Learns and Organzes) and ontans a al of about.m messages. Then a number of folks and organzatons worked hard on t to orret the problems (e.g. nvald emal addresses, et.) of the dataset, makng t more omfortable to researhers. By now ths dataset has been largely used by researhers who are nterested n ommunty dsovery, dynam soal networks, emal tools mprovement, et. The latest verson of the dataset (about M) was publshed n August,. The elder verson (Marh, ) s no longer aessble. The dataset onssts of, vertexes and 7 edges. We get the network data from the web page of Stanford SNAP Graph Lbrary, who has preproessed the ommunaton data and generates a well organzed text fle. The frst several lnes of the text fle gve some smple ntroduton about the dataset, after that are M lnes (edges), eah of whh s a par of nteger numbers dentfyng the two vertexes of the edge. The network s dreted and unweghted, so the weght of eah edge wll be assgned. Sne our algorthm deals wth undreted network, edges l (,,) and l(,,) wll be treated as a sngle edge. So the number of edges s redued to,... Modularty We frstly evaluate the qualty of ommunty struture by ts modularty. The network data (edges) s randomly dvded nto two parts wth equal sze: the orgnal data and the nremental data. For the orgnal data, we perform the algorthm to generate an ntal partton of the ommunty struture. The algorthm s also performed as a omparson. For the nremental data, we add the edges to the network sequentally to update the ommunty struture. Due to the hgh omputng omplexty of the and algorthm and the large amount of nremental data, t s not possble to fnsh the experment wthn fnte tme ( hours) f they are performed after eah edge s nreased. So we further dvde the nremental data nto ten equal subsets, whh means that eah subset ontans % of the data. To update the ommunty struture, the nremental algorthm s performed after eah edge s nreased whle the and algorthm s performed on the aggregated data only after all the edges of a subset are nreased. After eah subset s nreased, we alulate the modularty of the ommunty struture. Fgure shows how the modularty hanges along wth the nrease of subsets. We see that the modularty value of the algorthm (denoted wth trangles) vares obvously among dfferent exeutons, from the lowest. to the hghest.. It s beause the algorthm ntrodues a random mehansm to mprove ts performane. Sne the nremental algorthm take advantage of the algorthm to generate the ntal partton, they have the same modularty value n the begnnng. After the subsets are sequentally added, the modularty of the nremental algorthm gradually dereases, from. to., about % derement. Even so, the smallest modularty value s muh better than that of the algorthm, about.. Also, we should observe ths derement s produed after half of the al data s nreased. In applatons we an perform the algorthm perodally to

7 Fgure. The hange of modularty over the nrement of subsets generate stat ommunty struture at dfferent tme steps and then apply our nremental algorthm to trak the dynam ommunty struture. It s noteworthy that the modularty dereases very slowly. For example, f we take the frst subset as the overall nremental data, the modularty wll derease from. to., about only.% derement. If the ntal partton of network s good enough, t s beleved that the result wll be even better... Tme Complexty Compared to ts promsng behavor n modularty, the bggest advantage of our nremental algorthm s ts hgh tme effeny, whh makes t possble to trak the ommunty struture of networks n a fne-graned way. We ompare the overall omputng tme spent on trakng ommunty struture by our algorthm wth that of and. Eah of the ten subsets of nremental data nludes about, edges. One an edge s nreased, the nremental algorthm s performed to update the ommunty struture. If all the, edges of a subset are nreased, the and algorthm wll be performed as omparsons. The result s shown n Fgure. The urves denote the relatonshp between the overall omputng tme and the number of nreased subsets. We see that our nremental algorthm only takes. seonds to trak the ommunty struture of the network whose amount of edges vares from about, to,, whle the and algorthms needs and 7 seonds respetvely. Even so, they only get ten snapshots of the ommunty struture, stll unable to keep trak of Tme(S) Modularty Number of nreased subsets Number of nreased subsets Fgure. Computng tme over the nrement of subsets. t on a fne-graned level.. Other Real World Datasets We also test our nremental algorthm on three other real world datasets: the Arxv Hgh Energy Physs Theory paper taton network (t-hepth), the Wkpeda who-votes-on-whom network (wk-vote) and the ForSun orp IM ommunaton network (FSIM). Table gves a smple summarzaton of these three datasets. Table. Introduton to the three real world datasets Name Nodes Edges Type t-hepth 7,77,7 unweghted, dreted wk-vote 7,, unweghted, dreted FSIM,7, weghted, undreted The Arxv HEP-TH taton network omes from the e-prnt arxv and overs all the tatons wthn a dataset of 7,77 papers wth,7 edges. If a paper tes paper, there s a dreted edge n the network graph ponts from to. If a paper tes, or s ted by a paper outsde the dataset, no edge wll be reated. The data of wk-vote s from soal ommunty Wkpeda, a free enylopeda wrtten ollaboratvely by volunteers around the world. Some of the Wkpeda ontrbutors are admnstrators, who are users wth aess to addtonal tehnal features. If a user wants to beome an admnstrator, he has to be ssued a request for admnshp and then voted by the ommunty. The Stanford SNAP graph lbrary extrated all the admnstrator eletons and vote hstory data (tll January ), resultng n a network graph, where vertexes represent Wkpeda users and a dreted edge from vertex to represents that user voted on user. We get the two network graphs from the Stanford SNAP graph lbrary and onvert them nto weghted (eah edge s assgned wth weght ) undreted graphs. The t- HepTh and wk-vote are both well known datasets, dfferent from them, the FSIM dataset s provded by a ompany named ForSun orp. The network graph s modeled by the nstant message ommunaton data of about, volunteers. For prvay onsderaton ther personal nformaton wll not be publshed. The data s olleted from July to Otober, about months of data, gvng us a network graph of,7 vertexes and, edges. In the network eah vertexes represent a user, an undreted edge from vertex to represents that there are messages exhanged between the two users. The weght of an edge s evaluated by the frequeny of the ommunaton. The expermental results of the three datasets are show n Fgure, where we see our nremental algorthm outperforms that of s on both modularty and omputng tme n the frst two datasets. In Fgure -() we fnd an nterestng thng that the and algorthms almost get the same modularty value. Ths s manly aused by the speal network struture of the dataset. In fat, the network graph s not a onneted graph and has many solated star-lke ommuntes. An solated star-lke ommunty s ntutvely defned to ontan a entral node and several neghbors onnetng to the enter, and there are no onnetons between the nodes n the ommunty to the rest of the graph. The best performane of our algorthm s on the wk-vote

8 Modularty Modularty Modularty Fgure. Expermental results on three other real world datasets: (a) t-hepth; (b) wk-vote; ()FSIM dataset. When all the subsets are nreased to the network, the derements of modularty for the three datasets are.%,.7% and.% separately, whh are aeptable. As we mentoned n seton, dfferent types of edges are handled by dfferent operatons, whh orrespond to dfferent tme omplexty. Fgure gves the statstal nformaton about the perentage of operatons taken n updatng the ommunty struture wth our nremental algorthm. The perentage of operaton on the four datasets s.%,.%,.% and Perent of operatons(%) Number of nreased subsets Number of nreased subsets Number of nreased subsets Our Tme(S) (a) Tme(S) (b) Tme(S) - () Number of nreased subsets Number of nreased subsets Number of nreased subsets Enron t-hepth wk-vote FSIM Dataset Operaton Operaton Operaton Operaton Fgure. Dstrbuton of operatons taken n updatng the ommunty struture.% (three of whh are lower than %), whh guarantees the hgh tme effeny of our algorthm. We observe an nterestng thng that the operaton dstrbuton on the FSIM dataset s qute dfferent from the former three datasets. We guess ths may be aused by the speal sparsty and star-lke haratersts of the dataset. Ths may also explan why the modularty value of our algorthm s lower than that of on the FSIM dataset. In table we summarze the modularty value and omputng tme of our algorthm and that of and on the four datasets after all subsets are nreased. Table. Modularty and omputng tme Q/Tme(s) Our Enron./7././. t-hepth./././. wk-vote././7../. FSIM././..7/.. CONCLUSION In ths paper we propose an nremental algorthm to keep trak of ommunty struture of dynam networks whh s large and hange frequently. The ommunty struture s defned as dvsons of network vertexes nto sub groups, wthn whh nodes are densely onneted whle between whh they are sparsely onneted. In, Newman proposed a metr alled modularty to evaluate the qualty of a ommunty struture. Though fndng the optmal soluton s NP hard, a lot of work s done to fnd the suboptmal solutons, and the performane of some proposed algorthms s qute good. However, most of the modularty optmzaton based algorthms are stat and ust gve a snapshot of the networks. When dealng wth trakng ommunty struture whh hanges from tme to tme, they all suffer from hgh omputng omplexty. Many researhers propose algorthms to deal wth dynam networks, but they are stll unable to trak the ommunty struture on a fne-graned level. The greatest strength of our nremental algorthm s ts low omputng omplexty, as low as O (), makng t possble to keep trak of the ommunty struture of dynam networks n a fne-graned way. The effeny of our algorthm s evaluated by both modularty and ts omputng tme. We ompare our algorthm wth the ones proposed n [] and []. The expermental results show that our algorthm has reasonably good performane on both modularty and omputng tme. As we dsussed n seton, the performane of our nremental algorthm partly depends on the ntal ommunty struture of the network. However, fndng a good ntal ommunty struture s not an easy thng. In ths paper we use the modularty as the metr to evaluate the ommunty struture. But hgh modularty does not neessarly orrespond to good ommunty struture. In the future we wll try some other metrs to evaluate our nremental algorthm. In the algorthm we model the hange of networks as sequental nrement of edges, whle not onsder the derement of edges. Ths wll also be onsdered n our future work. 7. ACKNOWLEDGMENTS We thank Anyan Chen, Le Xu, Handong Mao, Chengka Guo, Gang Wang, Le Guo and Xlong Jn for ther help n fnshng ths work. Speal thanks to Deke Guo and Yngwen Chen, they provde us wth muh preous adve.

9 . REFERENCES [] S. H. Strogatz. Explorng Complex Networks. Nature, -7,. [] R. Albert, A.-L. Barabás. Statstal mehans of omplex networks. Rev. Mod. Phys. 7, 7-7,. [] M. E. J. Newman. The Struture and Funton of Complex Networks. SIAM Revew, 7-,. [] Stephen P. Borgatt, Aay Mehra. Network Analyss n the Soal Senes. Sene (Feb ) Vol. No. pp. -,. [] Je Tang, Jng Zhang, and Lmn Yao. ArnetMner: Extraton and Mnng of Aadem Soal Networks. In Proeedngs of the th ACM SIGKDD nternatonal onferene on knowledge dsovery and data mnng (KDD ),. [] Je Tang, Jmeng Sun, and Ch Wang. Soal Influene Analyss n Large-Sale Networks. In Proeedngs of the th ACM SIGKDD nternatonal onferene on knowledge dsovery and data mnng (KDD ),. [7] Matthew L. Wallae, Yves Gngras, and Russel Duhon. A New Approah for Detetng Sentf Spealtes from Raw Cotaton Networks. Journal of the Ameran Soety for Informaton Sene and Tehnology. Vol, Issue, Pages -. Feb. [] P. C. Cross, J. O. Lloyd-Smth, and W. M. Getz. Dsentanglng Assoaton Patterns n Fsson-Fuson Soetes usng Afran Buffalo as an Example. Anmal Behavour, :-,. [] Mkal Rubnov, Olaf Sporns. Complex Network Messures of Bran Connetvty: Uses and Integpretatons. Journal of NeuroImage, Vol, Issue, Pages -, Sep. [] Pan Hu, Eko Yonek, Shu Yan Chan, and Jon Crowroft. Dstrbuted Communty Deteton n Delay Tolerant Networks. In Proeedngs of nd ACM/IEEE nternatonal workshop on Moblty n the evolvng nternet arhteture (MobArh 7), 7. [] Mro Musoles, Cela Masolo. A Communty Based Moblty Model for Ad Ho Network Researh. In Proeedngs of the nd nternatonal workshop on Mult-hop ad ho networks: from theory to realty (REALMAN ).. [] M. magdon-ismal, M. Goldberg, W. Wallae, and D. Sebeker. Loatng Hdden Groups n Communaton Networks Usng Hdden Markov Models. In Pro. ISI,. [] J. Baumes, M. Goldberg, M. Magdon-Ismal, and W. Wallae. Dsoverng Hdden Groups n Communaton Networks. In Pro. nd NSF/NIJ Symp. on Intellgene and Seurty Informaton,. [] B. Maln. Data and Colloaton Survellane Through Loaton Aess Patterns. In Pro. NAACSOS Conf.,. [] Aaron Clauset, M. E. J. Newman, and Crstopher Moore. Fndng Communty Struture n Very Large Networks. Phys. Rev. E 7, (). [] M. E. J. Newman, M. Grvan. Fndng and Evaluatng Communty Struture n Networks. Phys. Rev. E, (). [7] R. Kumar, J. Novak, and A. Tomkns. Struture and Evoluton of Onlne Soal Networks. In Proeedngs of the th ACM SIGKDD nternatonal onferene on Knowledge dsovery and data mnng (KDD ),. [] JanHua Ruan, Wexong Zhang. An Effent Spetral Algorthm for Network Communty Dsovery and Its Applatons to Bologal and Soal Networks. on 7th IEEE Internatonal Conferene on Data Mnng (ICDM), 7. [] M. E. J. Newman. Fast Algorthm for Detetng Communty Struture n Networks. Phys. Rev. E., (). [] V. D. Blondel, J-L. Gullaume, R. Lambotte, and E. Lefebvre. Fast Unfoldng of Communtes n Large Networks. Journal of Statstal Mehans: Theory and Experment,. [] Ken Wakta and Toshyuk Tsurum. Fndng Communty Struture n Mega-Sale Soal Networks. In Proeedngs of the th nternatonal onferene on World Wde Web (WWW 7), 7. [] U. Brandes, D. Dellng, M. Gaertler, et. Maxmzng Modularty s Hard. In Data Analyss, Statsts and Probablty. arxv: physs/,. [] C. Tantpathananandh, T. Berger-Wolf and D. Kempe. A framework for ommunty dentfaton n dynam soal networkks. In Proeedngs of the th AMC SIGKDD nternatonal onferene on knowledge dsovery and data mnng (KDD 7), 7. [] D. Greene, D. Doyle and P. Cunnngham. Trakng the Evoluton of Communtes n Dynam Soal Networks. In Internatonal Conferene on Advanes n Soal Networks Analyss and Mnng (ASONAM),. [] Nathan Eagle, Alex Pentland, and Davd Lazer. Inferrng Frendshp Network Struture by Usng Moble Phone Data. In Proeedngs of the Natonal Aademy of Sene of the unted States of Amera. Vol, No,. [] M. E. J. Newman. Analyss of Weghted Networks. Phys. Rev. E 7, (). [7] C. Tantpathananandh and T. Berger-Wolf. Fndng Communtes n Dynam Soal Networks. In IEEE th Internatonal Conferene on Data Mnng (ICDM),. [] B. Klmat and Y. Yang. The Enron Corpus: A New Dataset for Emal Classfaton Reaserh. In ECML,.