An Overlap Communities Detection Algorithm in Asymmetric Dynamic

Transcription

1 An Overlap Communtes Detecton Algorthm n Asymmetrc Dynamc Socal Network 1 Wang Hong-yan, 2 Jang Gan-qng, 3 Lu Yong-shan, 4 Lu Wen-yuan, 5 Chen Z-jun 1 School of Informaton Scence and Engneerng, School of Economcs & Management, Yanshan Unversty, Chna, why925@163.com 2 School of Informaton Scence and Engneerng, Yanshan Unversty, Chna, snowear@163.com 3 School of Informaton Scence and Engneerng, Yanshan Unversty, Chna, ysulys@sohu.com 4 School of Informaton Scence and Engneerng, Yanshan Unversty, Chna, wylu@vp.163.com 5 School of Informaton Scence and Engneerng, Yanshan Unversty, Chna, zjchen@ysu.edu.cn Abstract Communty structure n socal network has features of lnk asymmetry and communty overlappng. The overlap communty detecton s mportant for obtanng global topologcal characterstcs of the network, desgnng data dstrbuton mechansm and controllng marketng costs. Based on the asymmetrc dynamc networks and the characterstcs of the communty, we propose a unversal algorthm for communty mnng: AFCAL. Not only can the algorthm detects the herarchcal structure and the overlappng areas of communtes, but also trace ther evoluton n dynamc moble networks effectvely. Smulaton results show that AFCAL algorthm compared wth exstng methods, exhbt better performance on the overlappng communty detecton accuracy and robustness. 1. Introducton Keywords:Asymmetrc, Overlap Communty, Dynamc Socal Network Socal network analyss (SNA) has drawn ncreasng attenton of researchers n varous felds, and also becomes the hot spot of varous busnesses to explore socal network marketng [1]. Frst, we detect the communty structure n socal network, and then select those customers who are the most lkely to receve the product and the most effectve secondary to spread the commodty, last, let these customers sell out merchandse by takng advantage of vral marketng. If the customers selected approprate, t wll maxmze marketng effectveness wth the least cost. In ths process, the detecton and analyss of communty structure are the keys to solve the problem. Accordng to symmetry or not, the communty detecton algorthm can be categorzed n two algorthms based on dgraph and undrected graph, and accordng to the dfference of applcaton, t can be dvded nto two algorthms used n statc network and dynamc network [2]. Palla and Derenv et al. proposed CFnder algorthm, based on the theory of k-clque n graph theory, whch can effectvely detect the overlap group structure n a statc network [3]. Steve Gregory proposed COPRA algorthm whch mproved Raghavan sgn genetc algorthm, COPRA algorthm s able to detect multple group structure overlap exstng n a statc network, the algorthm s stll able to mantan good effcency when t s used n large-scale and dense nodes network [4]. Nguyen put forward AFOCS algorthm whch can be used to detect overlap communty, and can be appled n a dynamc network [5]. Above algorthms abstract the network for an undrected graph, and does not consder the asymmetrc lnk features. Newman et al. mproved algorthm based on modularty, so that t can be appled n the drecton network to reflect the network herarchy, but the algorthm does not recognze more small communty structure of the network, and cannot detect the overlap area [6]. Blondel proposed a fast detect herarchcal communty structure aggregaton algorthm, whch s to some extent solves the problem of the resoluton of the Newman modular algorthm, s sutable for large-scale networks, but the algorthm easy to fall nto Local optmum, and cannot detect overlappng area [7]. Farkas I et al. proposed CPMd Advances n nformaton Scences and Servce Scences(AISS) Volume5, Number7, Aprl 2013 do: /aiss.vol5.ssue

2 algorthm on the bass of CFnder, whch can be appled to the weghted network, and s able to detect communty overlap regon, but the algorthm requres a pre-specfed value of k, dfferent values of whch have a great mpact on the number of communtes [8]. The above three algorthms can be appled n drected graph, but they are not applcable to dynamc network. DongshengDuan et al. proposed an algorthm sutable for dynamcal weghted networks, but the algorthm cannot detect the overlappng regons of the network, and t requres the same number of nodes to be changed n every change of the network, whch s often unrealstc [9]. Yang T B et al. use a Bayesan approach to detect communtes and ther evolutons n dynamc socal network, but they dd not take lnk asymmetry nto account [10]. Due to a varety of factors, the lnk connectng A to B s asymmetrc n socal networks, for example, A beleves B s a close frend and B regards A as a general frend. Ths asymmetry wll affect the network topology, thus affect the busness achevement of marketng programs. Some of the nodes n the socal network may be shared by a number of communtes. For example, a person may pass nto the part of several groups. As showed n fgure 1, an abstract network model, some nodes may be n an overlappng regon of two or more communtes. These nodes are relay nodes for the transmsson of nformaton between the communtes, and play an mportant role n the network. Therefore, to detect the overlap regon between the communtes has crucal sgnfcance for nformaton transfer and routng algorthm desgn. Fgure1. The overlapped communty structures In summary, the socal network communty structure should have asymmetrc, dynamc and overlap features, so, ths paper presents AFCAL(Adaptve Fndng Communty n Asymmetrc Lnk) algorthm whch can be appled to detect the overlappng regon n an asymmetrc dynamc network. 2. Descrpton of the problem 2.1. Related defntons and symbols n Descrpton Takng nto account the dynamcs and lnk drectonal characterstcs of the network, we abstract the network to a drectonal weghted graph, and express t usng a two-dmensonal matrx R. Defnton 1 (Drected weghted graph). G(V,E,W) s a drected weghted graph, V s the set of all nodes, E={<u,v> u,v V} represents the set of all edges, <u,v> s an ordered pars of nodes, e uv E represents a drected edge from node u to node v, W={w uv R u,v V;R s real number} s the weght set of drected edges, w uv s the weght of e uv. Defnton 2 (Neghbor nodes). If there s at least one lnk e uv from u to v, then defne u s a neghbor node of v, Ne(v) represents the neghbor nodes collecton of v. Defnton 3 (The really close neghbor nodes). For any node u,v V, f exst drect lnk between u and v, then defne u s a real nearest neghbor node of v, TNe(u) represents the true neghbor nodes collecton of v. Let C={C 1,C 2,,C k } represents the communty structure of the network, Com(C ) represents the label of communty C, whch marked n each communty s dfferent. For any C C, ts nternal nodes 972

3 s a subset of V, and the graph formed by them and ther adjacent lnks represents the communty structure of G. Defnton 4 (The nteractvty). For any communty C C, f u V, let w u n denotes the sum of lnk weghts from node u to the other nodes n communty C, w u out denotes the sum of lnk weghts from the node n communty C to node u, f w u n and w u out are not equal to 0, then we call there have node nteractvty between node u and communty C. Defnton 5 (Communty outler). For any communty C C, f u V and w u n =0 or w u out =0, then call u s an outler relatve to communty C. Defnton 6 (Communty boundary). For any C, C j, C, f u C and u s both nteractve wth C and C j,then u s called a boundary node of communty C, and the set F ={u u C, u s a boundary node of C } s called the boundary of communty C. Defnton 7 (Brdge node). If C=C C j, u C, and u s the real close neghbor to at least one node wthn C and C j, then call u s a brdge node between C and C j Dynamc Network Model Let G =(V, E, W ) s the correspondng abstract dagram model of the network at the tme t. C =(C 1,C 2,,C k ) s the network communty structure at t, G 0 =(V 0,E 0,W 0 ) s the orgnal nput map. ΔV,ΔE ΔW respectvely represent the change values of the node numbers, lnk numbers, and lnk weghts at Δt perod, Δt=t -t -1. ΔG =(ΔV,ΔE,ΔW ) denotes the network changes durng ths tme, G G G 1 Г=(G 0,G 1,G 2, ).. Dynamc network Г can be expressed as a sequence of tme-varyng graphs, 2.3. Objectve Functon We use Q-functon whch was proposed by Newman et al. n 2004 to be the objectve functon [11]. The Q functon s defned as the dfference of the actual numbers of lnk wthn the communty and the expected numbers of lnk whch s obtaned by random deployment, and t can quanttatvely measure the pros and cons of communty structure. The Q value s between 0-1, a hgher number ndcates the network communty structure s more obvous and the better dvson of the communty structure, otherwse, the worse. The process of detectng the communty structure s the process to maxmze the Q value. However, fndng the optmal Q value s an NP-Hard problem [12]. Therefore, the algorthm based on modular s an approxmaton algorthm n most cases. Based on Blondel algorthm, ths paper presents an overlappng communty detectng algorthm whch s used n the dynamc and lnk-asymmetrc network. Lnk weghts between nodes are correspondng to the elements of the drected and weghted graph R, then, the Q functon can be wrtten as [13]: 1 w out n w j Q ( w j ) ( C, C j ) (1) W j W Where w j s the weght of the lnk from node to node j; node to other nodes; w w s the sum of weghts from out j j n wj wj s the sum of weghts from other nodes to node ; W wj j s the total sum of the weghts of the network; σ(c,c j ) s the decson functon, If C =C j, then σ(c,c j )=1,else σ(c,c j )=0. The Q functon of the equaton (1) can be smplfed as: Q ( epp apbq ) p 1 (2) 973

4 where, epp apbq s the rato of the sum of lnk weghts from nodes wthn communty p to nodes wthn communty q to the total amount of all lnk weghts n the network, Obvously, e pp s the proporton of the sum of lnk weghts wthn communty p to the total amount of all lnk weghts; ap p j ( V p) w j s the proporton of the sum of lnk weghts from nodes wthn communty p to W the nodes wthn other communtes to the total amount of all lnk weghts n the network; bq ( V q) j q w j s the proporton of the sum of lnk weghts from nodes out of communty q to W the nodes wthn communty q to the total amount of all lnk weghts n the network. 3. Algorthm Descrpton In ths paper, AFCAL algorthm s a two-phase algorthm. In the frst phase, by the orgnal nput graph G 0 =(V 0,E 0,W 0 ), FCAL s able to detect the ntal communty structure C 0 ={C 01,C 02,,C 0k } at the tme t 0 ; In the second phase, the dynamc network s abstracted nto graphc sequence Г =( G 0,G 1,G 2, ), whch s changeable wth tme. The AFCAL algorthm can determne the communty structure C +1 at the tme t +1 based on the communty structure C at the tme t. Specfc algorthms are descrbed below FCAL Algorthm Communty Structure Intalzaton In the ntal case, each node n the network s seen as a communty structure, and s gven a dfferent communty structure label Com(C ). For node, calculate the module functon ncrements ΔQ when node s joned nto the communty of ts neghbor node j. Q epq eqp apbq aqbp (3) Let p and q are two communtes n the network, Q 1 s the network module degree before p and q merger, Q 2 s the network module degree after p and q merger, Q(p) s the module of communty p, equaton (2) shows: Q( p) epp apbp (4) If ΔQ s postve and the maxmum, node wll be joned nto the communty correspondng to neghbor node j; otherwse, the node wll stay n the orgnal communty. Repeat the operaton untl all the nodes n the network are assgned to correspondng communtes, whch lead to consttutng the ntal coarse-graned communty structure C 0. We sequentally combne each of the communty n C 0 wth adjacent communty, and calculate the ncremental ΔQ of the global Q value, f ΔQ s greater than zero and the maxmum, then merge these two communtes, else retan the orgnal communty structure. Repeat the operaton untl there are no communtes to be merged, the ntal communtes structure C 0 ={C 01,C 01, C 0m } s determned Detect communtes overlap regon In order to detect the overlappng area between communtes, we set an overlappng threshold value λ. Let F s the boundary of C 0, u s a pont on F, C 0j s an nteractve communty wth u. We calculate ΔQ 1 whch s the change after deletng u from C 0 and ΔQ 2 whch s the change after jonng u nto C 0j. Let ΔLQ=ΔQ 1 +ΔQ 2, f the absolute value ΔLQ <λ, then u s an overlappng area node of C 0j. 974

5 3.2. AFCAL Algorthm Let G={ G 1, G 2,, G k } represents the topology changes n the network n the perod of t, where, G represents a fundamental change. The network changes wthn ths perod of tme are attrbuted to the combnaton of the followng sx basc changes: (1) Add a new node (V+v): A new node v and ts adjacent lnks are added to the network. (2) Remove an orgnal node (V-v): An orgnal node v and ts adjacent lnk have been removed from the network. (3) Form a new lnk (E+e): A lnk s formed between two nodes n the network. (4) Remove the orgnal lnk (E-e): A lnk n the network s removed. (5) Increase the lnk weght (W+ w): A lnk weght n the network s ncreased. (6) Reduce the lnk weghts (W- w): A lnk weght n the network s decreased. In order to accurately detect the changes n communty structure n a dynamc network, we propose AFCAL algorthm. It requres FCAL frst to detect the ntal communty structure, next to detect the changes n communty structure whch are generated by the fundamental changes, so as to accurately reflect the dynamc changes of the communty structure n the network. Formaton and dsappearance of the lnk can be seen as the specal cases of lnk weghts ncreased and reduced. Therefore, we only put forward the algorthm to treat (1), (2), (5), (6) four basc changes n AFCAL. The specfc algorthm s descrbed below: Processng the changes n the nodes If u and ts adjacent lnk are added to the network n the perod of t, t can potentally produce sgnfcant changes to the orgnal communty structure. As showed n fgure 2 (b), when v as a new node s added to the network, the orgnally solated two communty groups C 1 and C 2 are lnked, and v becomes the neghbor wth part of the nodes n C 2, whch causes the formaton of a new communty C 3. Smlarly, the dsappearance of the orgnal node may lead to a varety of changes n the communty structure. Sometmes, specal nodes dd not change the exstng communty structure too much. As showed n fgure 2 (a), u as a new node s just added to communty C 1, whch wll keep other communtes structure constant. Therefore, f the node changes are specal changes, we need only add ths node to the correspondng communty or delete from the orgnal communty, and other communty structure s mantaned unchanged. Otherwse, we need to run the FCAL algorthm n the local area, to determne local changes n the communtes. (a) u s added n C 1 (b) v gathers some nodes from C 2 to form C 3 Fgure2. A new node s added If u only nteracts wth the communty C 1, Lemma 1 shows that t strengthens the nternal nodes contact n C 1 and ncreases the Q values, then we wll jon u nto communty C 1 and mantan the structure of other communtes unchanged. Let v be the orgnal node dsappearng n the perod of t, f node v does not belong to any network, then remove v from the network, and keep the other communty structure unchanged. Lemma 1: If u s a new node, and only have nteractve relatonshp wth communty C, then f u s added to the communty C, the module Q(C) of C wll be ncreased. 975

6 Proof: Let Q(C) be the ntal module degree of C, the module degree of C wll be ncreased to Q(C) f u s added to C. By formula (2) and (3) we can obtan: Q = Q(C) - Q(C) = (Q(C) + sum(u)) - Q(C) = sum(u) > 0 Among, sum(u) s the sum of lnk weghts between node u and the nodes n communty C Dealng wth changes n the value of the lnk weghts Same to the node changes, the lnk weghts changes wll drectly affect the sze of the communty modularty, and wll produce a varety of changes to communtes. If e uv s a lnk n C, by Lemma 2 we know that t wll enhance the nternal lnks of C by ncreasng the weghts w uv, so as to make the communty structure more obvous. If e uv s a lnk connectng of two communtes C and C j whch do not overlap, then ther respectve communty structure wll be more obvous by reducng the weghts value w uv. Therefore, for the above two partcular changes n the weghts, we mantan the exstng communty structure constant. For other cases, such as the ncreased value of the weghts to a certan lnk between the two communtes, the decreased value of the weghts to a certan lnk wthn one communty, whch wll cause the changes n communty structure havng dversty and uncertanty, and we wll judge these by FCAL algorthm n the local area. Lemma 2: If w uv s ncreased where u and v are belong to communty C, then, the module degree Q(C) of C wll be ncreased. Proof: Let Q(C) be the ntal module degree of C, and t changes to Q(C) after ncreasng w uv. By formula (2) and (3), we can obtan: Q = Q(C) - Q(C) = (Q(C) + w uv ) - Q(C) = w uv >0 Among, w uv s the ncrements of w uv. 4. Experments and Evaluaton 4.1. Comparson the number of communtes and NMI value In order to analyze the performance of the algorthm, we apply LFR judge reference map proposed by Fortunato for smulaton, contrast the number of detectng communty and Normalzed Mutual Informaton(NMI) wth CPMd, and test the accuracy n detectng communty of the overlappng area under dfferent threshold [5] [14]. In ths artcle we set the parameters of the LFR judge benchmark chart as follows: The sze of the mnmum and maxmum communtes are C mn = 10 and C max = 50; Each node belong to two communtes at most, O m =2; The number of nodes s N=1000; Network weghts mxng rato parameter μ respectvely are 0.1 and 0.3, whch ndcates the degree of the asymmetry of the network; The communty overlap rato β (Overlappng Fracton) determnes the number of nodes n the overlappng area n the network, whch s set a value of 0 to 0.5. (a)n=1000,μ=0.1 (b)n=1000,μ=0.3 Fgure3. Communtes number found by AFCAL and CPMd 976

7 Fgure 3 (a) and (b) show the comparson of the number of foundng communtes and the actual number of communtes n dfferent communtes overlap rato β by algorthm AFCAL and CPMd. In fgure 3 (a), the weghts mxng rato parameter μ = 0.1, whle n fgure 3 (b), μ = 0.3. As can be seen from the fgure, the number of communtes found by AFCAL has a very hgh degree of concdence wth the actual number of communtes, whch ndcates the algorthm can more accurately fnd the number of communtes n the actual network than CPMd. Fgure 4 (a) and (b) show the values of NMI n dfferent β by algorthm AFCAL and CPMd. NMI values can be used to measure the overlappng communty qualty, the NMI value s the closer to 1, the better s the qualty of the communty structure detected. As can be seen from the fgure, when μ = 0.3, wth ncreasng communtes overlap rato, the NMI value n CPMd drops by an average of 21%, and the NMI value n AFCAL drops by only 9%. In summary, AFCAL can accurately fnd the actual number of overlappng communtes, and the communty structure detected has hgher qualty. (a)n=1000,μ=0.1 (b)n=1000,μ=0.3 Fgure4. Normalzed Mutual Informaton score (a)nmi score (b)runnng tme Fgure5. The runnng tme and NMI score of AFCAL 977

8 4.2. AFCAL algorthm runnng tme and NMI value measurement In order to test the performance of AFCAL algorthm wth the network dynamc changes, we do dynamc smulaton for the network. Wheren the sze of the ntal network s 1000 nodes and the network dynamcs s dvded nto four stages, each stage have an average of 20 nodes and 200 lnks dynamc changes. As showed n fgure 5(a), the NMI value of communty structure at each stage s verfed, whch can be seen that the communty structure NMI value s around 0.8; Fgure 5 (b) verfes the AFCAL runnng tme at each stage, whch can be seen that the average runnng tme s about 1s. Therefore, AFCAL can effectvely detect the communty structure n dynamc networks, and ts speed s relatvely fast, whch mples that t can be able to reflect the dynamc changes n the network. 5. Concluson Ths paper presents AFCAL algorthm whch can detect communty structure n an asymmetrc lnk dynamc network. The algorthm can detect the overlap nodes between communtes, and can n real tme reflect the evolutonary process of the communty structure n a dynamc network. Expermental results show that AFCAL s wth hgher effcency and detecton accuracy. The communty structure research s the bass of socal network marketng, followng further study wll focus on the dstrbuton mechansm and user partcpaton ncentve mechansm based on ths data structure, whch s the drecton of our future research. 6. Acknowledgement The work s supported by the Natonal Scence Foundaton of Chna (No and No ). 7. References [1] Yjng Lu, Zhshu L, Mng Lu, Nanbo Lu, Yalan Ye, Socal Selfshness Aware Mult-copy Routng n Moble Socal Networks, JDCTA, vol. 6, no. 9, pp , [2] Mao-Guo Gong, Lng-Jun Zhang, Jng-Jng Ma, L-Cheng Jao, Communty Detecton n Dynamc Socal Networks Based on Multobjectve Immune Algorthm, Journal of Computer Scence and Technology, vol. 27, no. 3, pp , [3] Gergely Palla, Imre Dereny, Illes Farkas, Tamas Vcsek, Uncoverng the overlappng communty structure of complex networks n nature and socety, Nature, vol. 435, no. 7043, pp , [4] Steve Gregory, Fndng overlappng communtes n networks by label propagaton, New J. Physcs, vol. 10, no. 12, pp , [5] Nam P.Nguyen, Thang N.Dnh, Sndhura Tokala, My T. Tha, Overlappng Communtes n Dynamc Networks: Ther Detecton and Moble Applcatons, Proceedngs of the 17th annual nternatonal conference on Moble computng and networkng, pp , [6] Newman M.E.J, Lecht E.A, Communty structure n drected networks, Physcal Revew Letters, vol. 100, no. 11, pp , [7] Blondel V.D, Gullaume J.L, Lambotte R, Lefebvre E, Fast unfoldng of communty herarches n large networks, Journal of Statstcal Mechancs: Theory and Experment, vol. 10, no. 10, pp , [8] Gergely Palla, Illes J. Farkas, Peter Pollner, Imre Dereny, Tamas Vcsek, Drected network modules, New J. Phys, vol. 9, no. 6, pp. 186, [9] Dongsheng Duan, Yuhua L, Yanan Jn, Zhengdng Lu, Communty Mnng on Dynamc Weghted Drected Graphs, Proceedngs of the 1st ACM nternatonal workshop on Complex networks meet nformaton & knowledge management, pp ,

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