Overlapping community detection in signed networks

Transcription

1 Ovelapping community detection in signed netwoks Yi Chen 1 Xiao-long Wang 1, 2 Bo Yuan 1 1 (Depatment of Compute Science and Technology, Shenzhen Gaduate School, Habin Institute of Technology, Shenzhen , China) 2 (School of Compute Science and Technology, Habin Institute of Technology, Habin , China) Signed netwoks that take both positive and negative links into consideation have gained consideable attention dung the past few yeas. Most of the existing algothms fo community detection in signed netwoks aim at poviding a had-patitioning of the netwoks, that is, a given node should belong to a community o not. Howeve, most actual signed netwoks ae made of ovelapping communities. In this pape, we popose a signed stochastic block (SSB) model fo ovelapping community detection in signed netwoks. Compaed with the pevious models, the most pominent advantages of ou methodology ae the soft-patitioning solutions and softmembeship distbutions of signed netwoks. Extensive expements on a numbe of eal wold and synthetic signed netwoks demonstate that the SSB model can (i) identify seveal types of stuctue egulaties, i.e., assotative stuctue and disassotative stuctue; (ii) detect ovelapping modules in signed netwoks; (iii) outpefom the state-of-the-at models at shedding light on the community detection in signed netwoks. I. INTRODUCTION Complex netwok [1] as a multi-elational epesentation method can model many eal wold complex systems, including infomation systems [2, 3], social systems [4, 5], ecological systems [6], and many othes [1, 7, 8]. Netwok stuctue detection is one of the po issues of complex netwok analysis. Ref. [9] classifies the netwok stuctues into assotative stuctue and disassotative stuctue. Assotative stuctue, i.e. the community stuctue, is a kind of stuctue in which nodes of the same goup connect moe densely than othe goups; disassotative stuctue, i.e. the multipatite stuctue, is a kind of stuctue in which most of edges ae acoss goups. A lage numbe of effective techniques have been developed fo detecting both types of netwok stuctues dung the last few yeas. And a detailed suvey about community detection in netwoks was pesented by Fotunato [10]. Netwoks that conside only positive links ae called positive netwoks and netwoks that take both positive and negative links into consideation ae called signed netwoks. The signed netwoks have gained consideable attention fom diffeent scientific disciplines, i.e. biology [11], compute [12], and social sciences [13]. In the language of social science, positive links in signed netwoks denote fendship, ageement, tust o moe than wheeas negative links denote hostility, disageement, distust o less than. The definition of community stuctue in signed netwoks is a little diffeent fom that in positive netwoks. It is a genealization of an old idea fom social balance theoy [14] that nodes of the same community ae mostly connected by positive links and nodes of diffeent communities ae mainly connected by negative links. It is noted that seveal studies have been done on the community detection in signed netwoks. In Ref. [15], Yang et al. poposed an agent-based appoach that pefoms a andom walk stating fom one node and caes on though a numbe of positive steps fo extacting community stuctues. Gomez et al. pesented a genealization of modulaty method to detect signed netwok stuctues [16]. Based on the extension of Potts model, an algothm able to find community stuctues in netwoks with negative links was povided in Ref. [17]. By genealizing the Newman mixtue model fom netwoks containing only positive links to netwoks with both positive and negative links, Shen et al. povided a statistical pobability model fo detecting the disassotative stuctue of signed netwoks [18]. The afoementioned methods have povided a hadpatitioning of the signed netwoks, that is, a given node should belong to a community o not. Howeve, most actual netwoks ae made of ovelapping communities. A node may belong to seveal communities simultaneously. Fo example, a peson in social netwoks may belong to both family and hobby goups. In signed netwoks, thee exist two kinds of ovelapping nodes: i) positive ovelapping nodes, nodes of diffeent communities ae connected by positive links, i.e. the node F shown in Fig. 1; ii) negative ovelapping nodes, nodes of the same community ae connected by negative links, i.e. the node E shown in Fig. 1. In a netwok of fends and advesaes, node F is fends of two communities while node E is advesaes of two communities. Seveal appoaches fo ovelapping community detection have been povided since this issue was aised by Palla et al. [19]. Clique Pecolation Method (CPM) [19] is a popula method to addess this poblem. It is based on the concept that edges of a community ae likely to fom k-cliques so that non-ovelapping nodes would get tapped inside its oginal community but ovelapping nodes failed if they moved on a gaph. The poblem of this method is that many leaf nodes which cannot be eached by any k-cliques may be left out. Anothe popula method is the fuzzy c-means clusteng pesented by Dunn [20]. It uses u ik to descbe the fuzzy pobability of node i belongs to goup k. Zhang et al [21] povided an appoximation mapping of netwok nodes into a dimensional space by combining a modulaty function, spectal mapping and fuzzing clusteng. A fuzzy community that allows a node belong to multiple communities accoding to its membeship degees was pesented by Ref. [22]. In addition, statistical pobability models ecently have gained populaty in the netwok stuctue analysis. Fo detecting ovelapping community, a Bayesian non-negative matx factozation model povided by Psoakis et al. [23] selects a community k fo an edge and geneates two nodes of the edge. Ren et al. [24] pesented a simple and easily undestandable pobability model that can also be used fo detecting ovelapping community. A genealized stochastic block model was poposed by Ref. [25] fo explong complex stuctual featues of netwoks, including the ovelapping stuctue and multipatite stuctue. The common dawback of above mentioned ovelapping detection methods lies in that they ae only capable of detecting ovelapping community stuctues in positive netwoks and fail to

2 wok in netwoks with negative links. In this pape, we addess the poblem of detecting ovelapping community stuctues in undiected signed netwoks by poposing a signed stochastic block (SSB) model. To give a clea illustation, an illustative signed netwok is shown in Fig. 1. Ou model will povide a coect ovelapping patition of the signed netwok, i.e. the community (A, B, C, D, E, F) and community (E, F, G, H, I). The SSB model is a vaant of the stochastic block model with diffeent pobabilities fo geneating positive and negative links. The most pominent advantages of this methodology ae (i) softpatitioning solutions in signed netwoks, such as nodes E and F belong to two communities simultaneously; (ii) soft-membeship distbutions, which quantify how stongly a node belongs to a community. FIG. 1. An undiected and unweighted signed netwok with two ovelapping communities (A, B, C, D, E, F) and (E, F, G, H, I). The solid lines denote positive links and the dotted lines denote negative links. In the following section we fist pesent the theoetical foundations of SSB model. Then an expectation-maximization algothm (EM) [26] is pesented fo infeng the paametes of SSB model. In addition, we also discuss two exteme netwoks with only positive o negative links and show that SSB in only positive and negative netwoks can detect assotative stuctue and disassotative stuctue, espectively. At last, extensive expements on a numbe of eal wold and synthetic signed netwoks demonstate that the SSB model can (i) identify seveal types of stuctue egulaties, i.e., assotative stuctue and disassotative stuctue; (ii) detect ovelapping modules in signed netwoks; (iii) outpefom the state-of-the-at models at shedding light on the community detection in signed netwoks. II. THE MODEL Befoe illustating detail pocedues, we define some basic notations about the undiected signed netwoks. Geneally, an adjacency matx A with n dimensions is used to epesent a netwok. The value A 0 denotes the weight of edge between node i and node j. We also use two matces A + and A - to epesent the positive and negative netwoks, espectively. And A A if A 0, othewise. Fo ovelapping community detection in a signed netwok, we suppose that n nodes of the netwok fall into K communities within which the positive links ae dense and between which negative links ae also dense. In this pape, fo geneating diffeent types of links, i.e. positive links and negative links, we popose a signed stochastic block model which is a vaant of standad stochastic block model. Stochastic block model as a geneative model has a long tadition of study in the social sciences and compute science. In a standad stochastic block model with K blocks, a geneated edge has K K possibilities fo selecting blocks because each node of the edge has K possibilities fo choosing communities. (Hee, K is assumed to be a pedefined value and we will give a detail discussion late). The matx element s denotes the pobability of a andom edge selecting community and community s, nomalized by the constaint s 1. Fo a signed netwok, s which denotes the pobability of geneating an edge within a community can be used fo modeling a positive link; s( s) which denotes the pobability of geneating an edge between diffeent communities and s can be used fo modeling a negative link. Afte the geneated edge choosing the communities denoted by ' and s ' espectively, the next step of stochastic block model is to select nodes i and j fom the coesponding communities. We use to denote the pobability of community selecting node i. Note that the constaint fo is 1 athe than 1 because the SSB model assumes evey node paticipates in evey community instead of assigning a specific community to evey node. Similaly, denotes the pobability of community s selecting node j. With these teminologies and paametes, the geneative pocess of an edge A in signed netwoks is given as follows: 1) Check A belongs to A + o A -. If A belongs to A +, follow steps 2) 3) 4), othewise follow steps 5) 6) 7); 2) Select a community ' fo a positive edge with pobability ; 3) Select node i fom the community with pobability ; 4) Select node j fom the community with pobability j ; 5) Select two diffeent communities ' and s' s fo a negative edge with pobability s( s) ; 6) Select node i fom the community with pobability ; 7) Select node j fom the community s with pobability. Summazing in a mathematical expession, the pobability of geneating an edge A can be wtten as i A ( s s( s) P ( A, ) ( j) ) (1) A Whee ( j) denotes the pobability of a positive link within communities and ( s s( s) \ A ) denotes the pobability of a negative link between two diffeent communities. Then, the likelihood of the whole signed netwok can be modeled as ( P ( A, ) j) ( ) (2) s s( s) Ou next step is the model infeence and paametes estimation. The likelihood maximization fo paametes estimation of Eq. (2) is intactable because the selected communities of an edge ' and s ' ae all hidden vaables. The EM [26] algothm which is a geneal appoach to maximize the likelihood unde hidden vaables is intoduced to estimate the paametes of ou model. It computes the posteo pobability of ' and s ' unde the obseved data and estimated paametes in E step, and e-estimates paametes with these pobabilities in M step. Geneally, the likelihood maximization of Eq. (2) is solved by the log-likelihood maximization shown in Eq. (3) L ln A, ) A ln( j) ln( s( s ) s) (3)

3 In the E step, the algothm calculates the posteo pobability of hidden vaables unde the obseved quantities and model paametes. The Eq. (3) which contains logathm of sums ove the hidden vaables ' and s ' is difficult to be given a diect optimization. Using the Jensen s inequality, we tansfom the Eq. (3) to the expected log-likelihood L s( s) A,, ) ln A, s A,, ) ln A q Q s s( s) s,, ) A (ln ln ln ), s,, ) A (ln ln ln ) Whee to simplify the expected log-likelihood we have intoduced q A,, ) and Qs ( s), s A,, ), which denotes the pobability of a positive edge fom a community and a negative edge fom diffeent communities and s given the obseved data and model paametes, espectively. The value of q and Qs( s) can be calculated by Q q A,, ), A, ) A, ) s( s) j j, s A,, ), s, A, ) A, ) s s s( s) In the M step, the algothm e-estimates the model paametes with the obseved data and the existing posteo pobability of hidden vaables. Specifically, it optimizes the model paametes and given the hidden value of q and Qs( s). Combining with the afoementioned constaints that s 1 and 1, the Lagange fom of L is s( s) Whee, ~ L L (1 i (1 i ) j ) s s( s) (4) (5) (6) (7) ae the Lagange multiplies. By letting the devative of L ~ to be 0, the maximum of the Eq. (3) occus at the places whee q q Qs s( s) Qs s( s) (8) q Qs s( s) q Qs j js( s) q Qs s( s) In summay, the EM algothm computes the expectation of the pobability distbution of hidden vaables in the E step and calculates the model paametes fo maximizing the above expect in the M step. By iteating E and M steps, the log-likelihood of Eq. (3) will incease until convegence. Note that the EM algothm is sensitive to the stating values. To obtain a global maximum, it is necessay to pefom seveal times with divese initial paametes and take the highest log-likelihood ove all esults. Once the model paametes ae estimated in Eq. (8), the pobability of node i belonging to community, which is denoted as i, can be calculated by s s i (9) s s The pobability i povides a soft membeship fo nodes, that is, a node can belong to seveal communities simultaneously. If one wants a had-patitioning of a netwok, we can simply assign each node i to the most possible community accoding to ag max{ 1, 2,..., K}. Suppose a netwok has l positive edges and l negative edges, the time complexity of q and Qs( s) in the E step ae ( l K) and ( l K( K 1)). Theefoe, the oveall time cost of E step is ( l K l K( K 1)). The M step contains the calculation of, s( s) and, the time-complexity of which ae ( l K l K( K 1)), ( l K l K( K 1)) and ( l l K). Theefoe, the oveall time cost of M step is 2 ( l K l K ). The EM algothm conveges with T iteations and uns E and M steps in each iteation. In oveall, the 2 time-complexity of the SSB algothm is ( T( l K l K )). III. THE EXTREME CASES Paticulaly, in an exteme netwok with all positive edges, that is, A A, ou model can be simplified as In the E step, ( P ( A, ) j) (10) q j j (11) In the M step, q q (12) q j q This is simila to the SPAEM model povided by Ref. [24], fo detecting ovelapping community in positive undiected netwoks. This model assumes all links of the netwok ae geneated within K communities, which can be used to detect the assotative stuctue. And in anothe exteme netwok with all negative edges, that is, A A, ou model can be simplified as In the E step, ( P ( A, ) ) (13) s s( s)

4 Q s ( s) s s s( s) (14) In the M step, Qs s( s) Qs s (15) Qs js Qs s( s) This model assumes all links of the netwok ae geneated between K communities, which can be used to identify the disassotative stuctue. IV. EXPERIMENTAL RESULTS In this section, we demonstate the effectiveness of SSB model on detecting ovelapping community stuctues in signed netwoks. We fist validate ou model on two exteme netwoks with only positive and negative links. Then we conduct a geat deal of expements on both eal wold and synthetic signed netwoks. At last we also discuss the model selection issue, i.e., how to detemine the optimal numbe of communities. A. Community detection in positive netwoks The tested positive netwok is the Zachay club netwok which chaactezes the acquaintance elationship between 34 membes [27]. The netwok is splited into two goups because a dispute aose between the club s administato and its kaate teache. It has been eseached by seveal models fo ovelapping community detection. Setting ou model with the community numbe K = 2, Fig. 2 illustates the clea ovelapping community stuctues. As shown in Fig. 2, on one hand, the SSB model can exactly obtain the oginal assotative community stuctue; on the othe hand, it can detect seveal ovelapping nodes, such as nodes 3, 9, 14, 20, 31, 32. To validate the effectiveness of identifying ovelapping nodes, we make compasons with seveal popula models GSB [25], NMM [9], SPAEM [24]. Table I shows the belonging coefficient of above 6 ovelapping nodes in diffeent models. We can see that GSB, SPAEM and ou models poduce the same esults. Because the SPAEM model is a special case of the GSB model in a netwok with assotative stuctue and it is an exteme case of ou model in a netwok with only positive edges. And the GSB model outpefoming the NMM model on the Zachay club netwok has been explained in Ref. [25]. In conclusion, ou model can poduce the same esults as the best popula GSB and SPAEM models fo ovelapping community detection in netwoks with assotative stuctues. And the GSB and SPAEM models can only detect ovelapping community stuctues in positive netwoks while ou model can apply to netwoks with both positive and negative links. TABLE I. Mixed membeship of 6 ovelapping nodes in fou algothms. NodeID GSB NMM SPEAM SSB 3 (0.51, 0.49) (1.00, 0.00) (0.51, 0.49) (0.51, 0.49) 9 (0.30, 0.70) (0.04, 0.96) (0.30, 0.70) (0.30, 0.70) 14 (0.76, 0.24) (1.00, 0.00) (0.76, 0.24) (0.76, 0.24) 20 (0.67, 0.33) (0.87, 0.13) (0.67, 0.33) (0.67, 0.33) 31 (0.29, 0.71) (0.08, 0.92) (0.29, 0.71) (0.29, 0.71) 32 (0.17, 0.83) (0.00, 1.00) (0.17, 0.83) (0.17, 0.83) B. Community detection in negative netwoks Ou model in netwoks with only negative links can detect disassotative stuctue. The tested dataset is a netwok of 112 common adjectives and nouns in the novel David Coppefield by Chales Dickens [28]. And its edges denote that a pai of wods appeang adjacent to each othe in the text. To adapt the input fo ou model, we change the oginal edges with negative signs. By setting the community numbe K = 2, Fig. 3 illustates the clea netwok stuctue esults. As shown in Fig. 3, ou model indeed detects the bipatite stuctue which is a kind of disassotative stuctues. In addition, we also make compasons with thee afoementioned popula methods fo evaluating the accuacy of a had-patitioning of the netwok. 100 of the 112 nodes ae coectly classified by GSB, NMM and ou models while only 60 of the 112 nodes ae classified by the SPAEM model. Because the SPAEM model assumes that netwoks ae composed of assoative stuctues. Theefoe, ou model can poduce the same esults as the best popula GSB and NMM models in netwok with disassotative stuctues. Fig. 2. The positive netwok of the Zachay club with 34 nodes. The eal communities of this netwok ae denoted by two diffeent shapes, squae and cicle. The shades of nodes indicate a soft patitioning distbution obtained by the SSB model. Those nodes in the ellipse ae ovelapping nodes identified by ou model. Fig. 3 The negative netwok of 112 common adjectives and nouns in the novel David Coppefield by Chales Dickens[27]. The adjectives and nouns node goups ae denoted by cicle and squae shapes, espectively. The shades of nodes indicate the soft patitioning distbution obtained by SSB model.

5 C. Ovelapping community detection in signed netwoks The fist signed netwok is the illustative example shown in Fig. 1. The netwok contains two ovelapping communities and has two ovelapping nodes E and F. Fitting ou model with goups K = 2, the obtained communities and the detailed nodes belonging coefficient ae shown in Fig. 4. Seen in Fig. 4(a), the SSB model indeed identifies two coect ovelapping community stuctues (A, B, C, D, E, F) and (E, F, G, H, I), which ae a soft-patitioning of the signed netwok. Fig. 4(b) povides the soft membeship distbutions, in which nodes A, B, C, D, G, H, I ae assigned 100% to the belonging communities and nodes E, F ae ovelapping nodes belonging to two communities simultaneously. connects nodes 5 and 13 which ae in othe diffeent communities by negative links. It is tue that these ovelapping community stuctues have been ignoed by most cuent methods fo detecting community stuctues in signed netwoks. FIG. 4. Ovelapping community detection in the illustative signed netwok shown in Fig. 1. (a) The ovelapping community stuctues ae detected by SSB model. (b) The belonging coefficients of all nodes ae obtained by SSB model. The nodes in the squaes ae ovelapping nodes identified by ou model. The second signed netwok is a elation netwok of 10 paties of the Slovene Paliamentay in 1994 [29]. Seen as in Fig. 5(a), the weights of links in the netwok wee estimated by 72 of 90 membes of Slovene National Paliament by questionnaies. The questionnaies wee designed to estimate the distance of 10 paties on the scale fom -3 to 3 and the final weight wee the aveaged value and multiplied by 100. Using the SSB algothm, Fig. 5(b) and (c) pesent the ovelapping community stuctues and the belonging coefficient of each node, espectively. As seen in Fig. 5(b), the SSB model can detect the coect community stuctues in which nodes of the same community ae mostly connected by positive links and nodes of diffeent communities ae mainly connected by negative links. As seen in Fig. 5(c), all nodes but 10 ae assigned 100% to the belonging communities. Node 10 which is a case of negative ovelapping is identified as an ovelapping node because it connects nodes 5 and 7 by negative links. The ovelapping community stuctues have been neglected by most cuent methods fo detecting community stuctues in signed netwoks. The thid signed netwok is the Gahuku-Gama Subtbes netwok about the cultues of highland New Guinea [30]. Seen as in Fig. 6(a), it descbes the political alliance and enmities among the 16 Gahuku-Gama subtbes and the positive and negative links of netwok coespond to political aangements espectively. Using the SSB algothm, Fig. 6(b) and (c) illustate the ovelapping community stuctues and the belonging coefficient of all nodes, espectively. As seen in Fig. 6(b) and (c), the SSB model can detect the exact community stuctues. All nodes but 7 ae assigned 100% to the belonging communities and node 7 is identified as being two communities simultaneously because it FIG. 5. Ovelapping community detection in the Slovene Paliamentay signed netwok. (a) The initial adjacency matx of 10 Slovene Paliamentay paty. (b) The ovelapping community stuctues ae detected by the SSB model. The solid lines denote the positive links and the dotted lines denote the negative links. The eal communities of this netwok ae denoted by two diffeent shapes, squae and cicle. The shades of nodes indicate the soft patitioning distbutions obtained by ou model. (c) The belonging coefficients of all nodes ae obtained by the SSB model. The nodes labeled in (b) and (c) ae ovelapping nodes identified by ou model. D. Community detection in synthetic signed netwoks It is common to test on synthetic netwok to validate the pefomance of algothms fo community detection. Hee we also have applied the SSB algothm to the synthetic signed netwoks. We have adopted the method poposed in Ref. [15] fo geneating synthetic signed netwoks. The synthetic signed netwok is denoted as SG(c, n, k, p in, p +, p - ), whee c is the numbe of communities, n is the numbe of nodes in each community, k is the degee of each node, p in is the pobability of each node connecting othe nodes in the same community, p + denotes the pobability of positive links appeang between communities and p - denotes the pobability of negative links appeang within communities. Note that we only focus on a had-patitioning of the netwok, that is, a given node belongs to a community o not, although the SSB model can also povide the belonging coefficient infomation fo all nodes in synthetic signed netwoks. Because the synthetic signed netwoks poduce ovelapping nodes with paametes p + and p - which act on all nodes, that is, all nodes ae eithe non-ovelapping o ovelapping nodes. To measue the pefomance of SSB model on the hadpatitioning, we adopt the Nomalized Mutual Infomation (NMI) [31]. The NMI is a widely used method fo evaluating the community detection and is defined as P nmi ' ' 2MI( G, G ) P nmi ( G, G ) ' (16) H( G) H( G )

6 Whee G ( G1, G2,..., GK ) is a defined community stuctue, ' ' ' ' G ( G1, G2,..., G K ) is a community stuctue given by an ' algothm, H (G) and H ( G ) ae the entopies of community stuctues G and G ', MI ( G, G ) is the mutual infomation between two stuctues. The highe value of P nmi, the bette the detection pefomance and P nmi beging 1means the pefect detection. FIG. 7. Identifying the community stuctue fom a andom signed netwok SG(20, 30, 16, 0.8, 0, 0). The netwok contains 20 communities, each community includes 30 nodes, each node has 16 edges, 80 pecent of edges ae positive links appeang within communities and 20 pecent of them ae negative links appeang between communities. (a) The initial adjacency matx. (b) The SSB output. FIG. 8. Identifying the community stuctue fom a andom signed netwok SG(4, 30, 16, 0.1, 0, 0). The netwok has 10 pecent of positive links and 90 pecent of negative links. (a) The initial adjacency matx. (b) The SSB output. FIG. 6. Ovelapping community detection in the Gahuku-Gama Subtbes signed netwok. (a) The initial adjacency matx of 16 Gahuku-Gama Subtbes. (b) The ovelapping community stuctues ae detected by SSB model. The solid lines denote the positive links and the dotted lines denote the negative links. The eal communities of this netwok ae denoted by thee diffeent shapes, squae and cicle, tangle. The shades of nodes indicate the soft patitioning distbution obtained by ou model. (c) The belonging coefficients of all nodes ae obtained by SSB model. The nodes labeled in (b) and (c) ae ovelapping nodes identified by ou model. Fig. 7(a) illustates a synthetic signed netwok SG(20, 30, 16, 0.8, 0, 0). The netwok contains 20 communities and each community includes 30 nodes, each node has 16 edges, 80 pecent of edges ae positive links appeang within communities and 20 pecent of them ae negative links appeang between communities. It is a pue netwok in which all positive links ae geneated within communities and all negative links ae geneated between communities. Seen as in Fig. 7(b), the exact community stuctues ae detected by the SSB model with P nmi being 1. Fig. 8(a) shows a synthetic netwok SG(4, 30, 16, 0.1, 0, 0). Simila to the netwok mentioned in Fig. 7(a), it is also a pue netwok without any ovelapping nodes. The main diffeence between the two netwoks lies in the p in paamete. Each node in Fig. 8(a) netwok has only 10 pecent of 16 edges appeang within communities, that is, the netwok stuctue is vey vague. Shown as in Fig. 8(b), the SSB model can still detect the exact community stuctues with P nmi being 1 even though the positive edges within fou community stuctues connect spasely. FIG. 9. Identifying the community stuctue fom a andom signed netwok SG(4, 30, 16, 0.8, 0.2, 0.2). The netwok has 20 pecent of positive links appeang between communities and 20 pecent of negative links appeang within communities. (a) The initial adjacency matx. (b) The SSB output. FIG. 10. Identifying the community stuctue fom a andom signed netwok SG(4, (32, 64, 96, 128), 24, 0.7, 0.2, 0.2). The node sizes of 4 communities ae 32, 64, 96, 128. (a) The initial adjacency matx. (b) The SSB output.

7 The two synthetic signed netwoks mentioned above ae all pue netwoks. Fig. 9(a) illustates a netwok SG(4, 30, 16, 0.8, 0.2, 0.2) with ovelapping nodes. In this netwok, 80 pecent of 16 edges ae positive edges, 20 pecent of which locate between communities and 20 pecent of 16 edges ae negative edges, 20 pecent of which locate within community. The abnomal 20 pecent of positive and negative edges bng noises fo the signed netwok. Fig. 9(b) shows the community stuctue detected by ou model. We can see that ou model povides a clea community stuctue with P nmi being 1 although the andom signed netwok SG(4, 30, 16, 0.8, 0.2, 0.2) contains significant noises. In addition, we have also applied the SSB model to a netwok SG(4, (32, 64, 96, 128), 24, 0.7, 0.2, 0.2) which contains fou communities with diffeent sizes and noises, as shown in Fig. 10(a). Compaed with thee synthetic netwoks mentioned above, the node sizes of each community in this netwok ae diffeent. Coespondingly, the density of each community is also divese, which bngs difficult fo community detection. Fig. 10(b) shows the SSB model output matx. We can see that the SSB model succeeds in finding all pedefined communities with P nmi being 1. To validate the obustness of SSB model on synthetic signed netwoks fo community detection, we have applied it to netwoks SG (4, 30, 16, p in, p +, p - ). Fo compasons, we have also conducted the same expements using FEC algothm [15], Taag algothm [17] and SNMM algothm [18]. Fig. 11 illustates the expemental esults of fou algothms on the netwoks SG(4, 30, 16, p in, 0, 0), with p in changing gadually fom 0.05 to 1. The value being 0.0 is kicked off because thee ae no gound-tuth communities when the netwok stuctues connect without any positive links. Each point in the cuves is calculated by taking the aveage ove 30 synthetic andom netwoks. Seen as in Fig. 11, The P nmi fo all methods follow a descending tend as the p in deceases fom 1 to SNMM, Taag and SSB models obviously outpefom the FEC model because in the ange of 0.15 p in 0. 6, the P nmi of thee models is still 1 while that of FEC model is less than 1. Moeove, we can see that The P nmi of SSB model is slightly bette than SNMM and Taag models in the ange of 0.05 p in In conclusion, the SSB model outpefoms the othe thee models when p in changes gadually fom 0.05 to 1. fom 0 to 0.5. It is simila to the paamete p in that the anges of p + and p - fom 0.5 to 1 ae kicked off because thee ae no goundtuth communities. Seen as in Fig. 12, the P nmi dawn in each gaph is calculated by taking the aveage ove 30 synthetic andom netwoks. Fist, the Taag and SSB models outpefom the FEC model because the P nmi of FEC model dops to 0 when p + is 0.5 while both Taag and SSB models get non-zeo values. Second, the Taag and SSB models ae supeo to the SNMM model, because in the ange of 0 p 0. 35, the P nmi of Taag and SSB models is always 1 while that of SNMM model is less than 1. Thid, in the ange of 0 p 0. 45, Taag and SSB models possess a simila esult. The main diffeence between two models lies in the situation p 0. 5 and the SSB model outpefoms the Taag model if 0.2 p Although SSB model fails to get the best value in the ange of 0 p 0. 2, the P nmi is found to be still acceptable, that is, not less than 60 pecent. In conclusion, the SSB model outpefoms the othe thee models when p + and p - change gadually fom 0.5 to 1. FIG. 12. Community detection pefomance of fou algothms on the netwok SG(4, 30, 16, 0.8, p +, p -). The P nmi dawn in each gaph is calculated by taking the aveage ove 30 synthetic andom netwoks. The highe value of P nmi the bette the detection pefomance and P nmi being 1means the pefect detection. In summay, expements on exteme netwoks with only positive and negative links demonstate that the SSB model can poduce the same esults as the state-of-the-at models on identifying the assotative stuctue and disassotative stuctue, espectively; expements on eal wold signed netwoks demonstate the SSB model can detect ovelapping community modules which ae neglected by most cuent popula models; expements on seveal synthetic signed netwoks demonstate the SSB model outpefoms the state-of-the-at models at shedding light on the community detection in signed netwoks. E. Model selection issue FIG. 11. Community detection pefomance of fou algothms on the netwoks SG(4, 30, 16, p in, 0, 0). Each point in the cuves is calculated by taking the aveage ove 30 synthetic andom netwoks. The highe value of P nmi the bette the detection pefomance and P nmi beging 1means the pefect detection. Fig. 12 shows the expemental esults of fou models on the netwok SG(4, 30, 16, 0.8, p +, p - ), with p +, p - changing gadually The pevious expements ae conducted with a pedefined community numbe. Howeve, many eal wold netwoks ae pesented without any po knowledge about the community numbe. Theefoe, it is necessay fo ou model to povide a cteon fo detemining the community numbe. The detemination of community numbe in signed netwoks elates to the eo cteon function pesented in [32]. Let N be the total weight of negative links within communities and let P be the total weight of positive links between communities. A geneal cteon function is defined as P ( C) N (1 ) P (17)

8 Whee 0 1. Geneally, 0. 5 denotes the two pats ae equally weighted. Doeian et al [32] established a theoem that Fo any signed netwok, thee will be a unique lowest value of the cteon function that occus fo patitions with a single numbe k. We can use the geneal cteon function fo detemining the community numbe. As tests, we etest two eal wold netwoks: Slovene Paliamentay paty netwok and Gahuku-Gama Subtbes netwok. As shown in Fig. 13, the goup numbes of two netwoks ae coectly identified by the geneal cteon function. Specially, the Gahuku-Gama Subtbes netwok obtains the minimum value at K 3 and K 4. By checking the belonging coefficient of K 4, we find that 16 nodes actually fall into 3 communities fo a had-patitioning of netwok. V. CONCLUSIONS In this pape, we have studied the exploation of ovelapping community detection in signed netwoks using a pobabilistic model SSB. The SSB model is a vaant of the stochastic block model with diffeent pobabilities fo geneating positive and negative links. The most pominent advantages of this methodology ae the soft-patitioning solutions and softmembeship distbutions of signed netwoks. Extensive expements on a numbe of eal wold and synthetic signed netwoks demonstate that the SSB model can (i) identify seveal types of stuctue egulaties, i.e., assotative stuctue and disassotative stuctue; (ii) detect ovelapping modules in signed netwoks; (iii) outpefom the state-of-the-at models at shedding light on the community detection in signed netwoks. In addition, the geneal cteon function is employed and tested to detemine the optimal numbe of communities. As futue wok, we will apply ou model to eal scalable signed netwoks and investigate the possible applications. Fig. 13. Model selection fo (a) the Slovene Paliamentay paty netwok and (b) Gahuku-Gama Subtbes netwok. ACKNOWLEDGMENTS The wok is suppoted by the National Natual Science Foundation of China( ) and the Stategic Emeging Industy Development Special Fund of Shenzhen (ZDSY ). [1] S. H. Stogatz, Natue 410, 268 (2001). [2] J. Chang, and D. M. Blei, The Annals of Applied Statistics 4, 124 (2010). [3] G. W. Flake et al., Compute 35, 66 (2002). [4] P. Doeian, and A. Mva, Social netwoks 31, 1 (2009). [5] A. M. Vedey et al., Social netwoks 34, 112 (2012). [6] D. A. Gea, L. T. Luong, and P. J. Hudson, Ecological Applications (2013). [7] J. Chen, B. J. Aonow, and A. G. Jegga, BMC bioinfomatics 10, 73 (2009). [8] M. Kitsak et al., Natue Physics 6, 888 (2010). [9] M. E. Newman, and E. A. Leicht, Poceedings of the National Academy of Sciences 104, 9564 (2007). [10] S. Fotunato, Physics Repots 486, 75 (2010). [11] M. J. Mason et al., BMC genomics 10, 327 (2009). [12] G. Facchetti, G. Iacono, and C. Altafini, Physical Review E 86, (2012). [13] H. Deng et al., Social netwoks 34, 253 (2012). [14] F. Haay, The Michigan Mathematical Jounal 2, 143 (1953). [15] B. Yang, W. K. Cheung, and J. Liu, Knowledge and Data Engineeng, IEEE Tansactions on 19, 1333 (2007). [16] S. Gómez, P. Jensen, and A. Aenas, Physical Review E 80, (2009). [17] V. Taag, and J. Buggeman, Physical Review E 80, (2009). [18] H.-W. Shen, in Community Stuctue of Complex Netwoks (Spnge, 2013), pp. 93. [19] G. Palla et al., Natue 435, 814 (2005). [20] J. C. Dunn, Jounal of Cybenetics 4 (1973). [21] S. Zhang, R.-S. Wang, and X.-S. Zhang, Physica A: Statistical Mechanics and its Applications 374, 483 (2007). [22] T. Nepusz et al., Physical Review E 77, (2008). [23] I. Psoakis et al., Physical Review E 83, (2011). [24] W. Ren et al., Physical Review E 79, (2009). [25] H.-W. Shen, X.-Q. Cheng, and J.-F. Guo, Physical Review E 84, (2011). [26] A. P. Dempste, N. M. Laid, and D. B. Rubin, Jounal of the Royal Statistical Society. Sees B (Methodological), 1 (1977). [27] W. W. Zachay, Jounal of anthopological eseach, 452 (1977). [28] M. E. Newman, Physical Review E 74, (2006). [29] A. Feligoj, and A. Kambege, Developments in Statistics and Methodology, 209 (1996). [30] K. E. Read, Southwesten Jounal of Anthopology, 1 (1954). [31] D. Leon et al., Jounal of Statistical Mechanics (2005). [32] P. Doeian, V. Batagelj, and A. Feligoj, Genealized blockmodeling (Cambdge Univesity Pess, 2005), Vol. 25.