Evolutionary Computation for Community Detection in Networks: a Review

Transcription

1 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X Evolutonary Computaton for Communty Detecton n Networks: a Revew Clara Pzzut Abstract In today s world, the nterconnectons among objects n many domans are often modeled as networks, wth nodes representng the objects and edges the exstng relatonshps among them. A key feature of complex networks s the tendency of enttes to group together to form communtes. The detecton of communtes has been recevng a great deal of nterest by researchers. In fact, the knowledge of how objects organze allows a better understandng of a network, and gves a deeper nsght of nterestng characterstcs, that could not be caught f consderng the network as a whole. In the last decade, evolutonary computaton technques have gven a sgnfcant contrbuton n ths context. The am of ths revew s to present the approaches based on evolutonary computaton to uncover communty structure. Especally, the representaton schemes wth the genetc operators apt for them are descrbed, and the most popular ftness functons employed by the methods are dscussed. The survey covers the most recent proposals optmzng ether a sngle objectve or multple objectves for dfferent types of network models, such as sgned, dynamc, multdmensonal. Index Terms Complex Networks, communty detecton, evolutonary computaton, sngle objectve optmzaton, multobjectve optmzaton. 1 INTRODUCTION Network scence, n recent years, has been attractng many researchers from dfferent domans. In fact, complex networks are an effectve formalsm n representng the relatonshps among objects composng many real world systems. Networks are modeled as graphs, where nodes denote the objects of a system, and edges represent the nteractons among these objects. Communty structure,.e. the dvson of a network nto groups of nodes havng dense ntra-connectons, and sparse nter-connectons [43], s an mportant characterstc of networks, ntensvely studed n the last years. The organzaton n communtes, n fact, takes place n both socety and complex systems, such as communcaton and transport, bology, nternet, World Wde Web [81]. The problem of uncoverng communty structure can be formalzed as an optmzaton problem where an approprate crteron functon, that at best catches the ntutve concept of communty, must be defned and optmzed. In the past years, a lot of approaches, employng dfferent types of heurstcs and a wde varety of crtera to optmze, have been proposed. Detaled surveys descrbng these methods can be found n [42], [92], [41], [24], [84], [109], [79], [91], [56], [1], [7], [90]. Clara Pzzut s wth the Natonal Research Councl of Italy (CNR), Insttute for Hgh Performance Computng and Networkng (ICAR), Va Petro Bucc, 4-11C, Rende (CS), Italy, e-mal: clara.pzzut@cnr.t. Manuscrpt receved ; revsed, Evolutonary Computaton s a powerful search and optmzaton technque nspred by the process of natural evoluton [37], [59], successfully appled for the soluton of many dffcult real world problems. Evolutonary methods are flexble methods that can be used, n prncple, to solve any type of problem, provded that the problem can be formulated as an optmzaton task. These methods consst of populaton ntalzaton, followed by varaton and selecton operators to mprove the value of a crteron, able to escape from local mnma, whle explorng the search space durng the optmzaton process. In the last decade, we have wtnessed an mpressve growth of new methods based on evolutonary computaton for the communty detecton problem. Ths ncreasng popularty s due to the capablty of evolutonary computaton n provdng a smple, but effcacous, methodology for solvng a complex problem, by requrng the defnton of few basc concepts: a sutable representaton for the problem, the functon to optmze, and how ndvduals of the populaton evolve. Compared to classcal metaheurstc methods, they present a number of advantages: the number of communtes s automatcally determned durng the search process, doman-specfc knowledge can be ncorporated nsde the method, such as based ntalzaton, or specfc varaton operators nstead of random, allowng a more effectve exploraton of the state space of possble solutons, beng populaton-based models, they are naturally parallel and effcent mplementatons can be realzed to deal wth large sze networks. The objectve of ths revew s to gve a comprehensve descrpton of the state-of-the art methods proposed so far that approach the problem of communty detecton wth computatonal models nspred by evoluton n nature. In partcular, the revew wll focus on methods based on Genetc Algorthms (GAs) [45], and evolutonary strateges n general [13], coverng also other nature nspred approaches, such as partcle swarm and ant colony optmzaton [65], [31], frefly and bat methods [111], [112] for fndng communtes, eventually overlappng, n dfferent types of networks, ncludng undrected, drected, weghted, sgned, mult-dmensonal, tme evolvng. The paper s organzed as follows. The next secton ntroduces defntons and concepts related to the problem. Secton 3 defnes the problem of communty detecton as

2 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X An example of undrected network s shown n Fgure 1. Ths toy network wll be used n the paper to llustrate genetc operators. Fg. 1. An example network wth 12 nodes, 20 edges, and three communtes. an optmzaton problem. Secton 4 descrbes the encodng schemes, whle Secton 5 descrbes the varaton operators. Secton 6 llustrates the most popular objectve functons, then Secton 7 explans how the problem has been faced wth multobjectve optmzaton. A comparson between sngle objectve and mutobjectve approaches s reported n Secton 8. Sectons 9, 10, and 11 descrbe partcular network models and overlappng approaches. Secton 12 reports the most recent proposals of other bo-nspred methods. Secton 13 concludes the paper by summarzng all the descrbed approaches n three tables reportng for each of them, the man characterstcs, and dscusses future desrable developments. 2 PRELIMINARIES A network N can be modeled as a graph G = (V, E, W ) where V s a set of n objects, called nodes or vertces, E s a set of m lnks, called edges, that connect two elements of V, and W : V V R s a functon whch assgns a weght to a couple (, j) of nodes and j, f there exsts an edge connectng and j, and 0 f an edge between and j does not exst [81], [4]. A graph G can be represented wth the adjacency matrx A, whose elements are denoted as A j. The values of A j determne the knd of graph. Thus, an undrected network s such that A j = A j. If A j > 1 the network s sad weghted, f A j { w, 0, w}, the network s sgned. A communty (also called cluster) [41] n a network s a group of vertces (.e. a sub-graph) havng a hgh densty of edges wthn them, and a lower densty of edges between groups. A communty structure (or clusterng) s defned as a dvson C = {C 1,..., C k } of the network n k subgraphs such that V = k =1 C. When C C j =, j, we have a parttonng of the nodes, otherwse we allow nodes to partcpate n more that one cluster, thus havng overlappng communtes. The degree k of a generc node, s k = A j. The degree k (C) of a node wth j respect to the communty C t belongs, can be splt as k (C) = k n (C) + k out (C) where k n (C) = A j s the j C number of edges connectng to the other nodes n C, and (C) = A j s the number of edges connectng to the k out j / C rest of the network. 3 COMMUNITY DETECTION AS AN OPTIMIZA- TION PROBLEM The detecton of communty structure n a network can be consdered as a problem of clusterng and, as such, t can be formally defned as an optmzaton problem. The problem can be faced n two dfferent ways: sngle objectve optmzaton and multple objectve optmzaton [35]. Let Ω = {C 1,..., C r } be the set of feasble clusterngs of a network. For sngle crteron optmzaton, the communty detecton problem can be formulated as the optmzaton problem (Ω, F) of fndng a dvson C for whch F(C ) = mn F(C), subject to C Ω (1) where F : Ω R s the sngle crteron functon that determnes the feasblty and qualty of the clusterng obtaned. For multple objectves, the problem can be formulated as a multobjectve clusterng problem (Ω, F 1, F 2,..., F h ) F(C ) = mn F (C), = 1,..., h subject to C Ω (2) where F = {F 1, F 2,..., F h } s a set of h competng sngle crteron functons F : Ω R that must be smultaneously optmzed. The am s to fnd a domnant soluton C such that, for each soluton C Ω and for each objectve F F F (C ) F (C) = 1,..., h (3) Often, however, a domnant soluton does not exst and the problem s how to fnd an effcent soluton,.e. one whch s as good as possble respect to each crteron. Pareto optmalty theory [33] allows to fnd these solutons. Gven C 1 and C 2 Ω, soluton C 1 s sad to domnate soluton C 2, denoted as C 1 C 2, f and only f : F (C 1 ) F (C 2 ) s.t. F (C 1 ) < F (C 2 ) (4) Multobjectve optmzaton ams to the generaton and selecton of nondomnated solutons, called Pareto-optmal, for whch an mprovement n one objectve requres a degradaton of another. The set of Pareto-optmal solutons Π s defned as Π = {C Ω : C Ω wth C C} The vector F maps the soluton space nto the objectve functon space. When the nondomnated solutons are plotted n the objectve space, they are called Pareto front. Thus, the Pareto front represents the better compromse solutons satsfyng all the objectves as best as possble. The many proposed methods can be dvded n two man categores, those optmzng only one ftness functon, and those optmzng two, or more, objectves. However, ndependently of the number of crtera, some general prncples are common for all the methods,.e. the choce of the representaton, and the type of crossover and mutaton operators. In the followng,

3 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X poston label Fg. 2. Labels-based representaton of the network dvson of the example of Fgure 1. poston neghbor (a) a descrpton of the representaton schemes proposed n the lterature s reported, along wth the genetc operators apt for each representaton and the most popular ftness functons adopted by approaches. It s worth pontng out that these basc schemes have been ntroduced by sngle objectve methods, and then exploted by the multobjectve ones. Thus, unless explctly stated, the strateges reported n the followng sectons are related to sngle objectve approaches. The multobjectve methods are then treated n detal n Secton 7. 4 ENCODING SCHEMES The representaton of a soluton s a crucal part for the success of an algorthm. Several proposals exst to encode the dvson of a network n sub-graphs. These representatons are often adapted from the encodng used to solve the classcal data clusterng problem wth evolutonary methods [61]. 4.1 Label-based representaton In ths knd of encodng a genotype s an nteger vector of sze n, where n s the number of nodes. A poston 1 n corresponds to a node, thus, f k s the number of communtes, each gene can assume a value n the alphabet {1,..., k}. Ths value s the label dentfyng the communty to whch node belongs. For example, consder the network of Fgure 1. Fgure 2 shows the label-based representaton of the dvson of the network nto the three groups {{1, 2, 3}, {4, 5, 6, 7}, {8, 9, 10, 11, 12}}. Label-based representaton has been wdely used for data clusterng [61]. Tasgn and Bngol [106] adopted t for communty structure dentfcaton, makng t also very popular for complex networks. Ths encodng scheme, as observed n [61], s redundant because, f a genotype represents a dvson nto k groups of nodes, there can be k! dfferent chromosomes correspondng to the same partton. The vector [ ] represents the same network dvson of [ ]. More generally, snce the number of communtes can be any number between 1 and n, the sze of the search space can be n n. Thus, for example, [ ] always represents the same soluton. A possble strategy to solve ths problem s to apply a renumberng procedure, as suggested by Falkenauer [34], that s class labels are renumbered startng from the frst avalable label number determned by the orderng of nodes n the chromosome. For nstance, n the chromosome [ ], class label 4 s changed to 1, class label 10 to 2, and class label 6 to 3. Though ths augments the computaton tme, on the other hand the sze of the search space s sensbly reduced. However, none Fg. 3. (a): Locus-based representaton of the network dvson of the example of Fgure 1. (b) Correspondng graph dvson nto three connected components. of the methods that adopt label-based representaton takes nto account the renumberng procedure. Gog et al. [44] proposed enrchng ths representaton by endowng each chromosome wth the value of the best ancestor ndvdual and the value of the best ndvdual obtaned so far. Ancestors are defned as all the ndvduals n prevous generatons that contrbuted to the generaton of the current ndvdual. The genetc materal retaned through ancestors s then exploted to expand the search space, snce recombnaton s performed only between ndvduals havng no common ancestors. (b) 4.2 Locus-based representaton The locus-based adjacency representaton has been orgnally proposed n [85] for data clusterng and exploted by Handl and Knowles [55] nsde a multobjectve clusterng method. In ths graph-based representaton an ndvdual of the populaton conssts of n genes g 1,..., g n and each gene can assume allele values j n the range {1,..., n}. A value j assgned to the th gene s nterpreted as a lnk between the nodes and j of V. Ths nduces a dvson of the network nto connected components, represented through subgraphs, often trees. Consder agan the network of Fgure 1. The network parttoned nto three groups, vsualzed by dfferent colors of the nodes, can be represented, out of the many possble genotypes, by the chromosome reported n Fgure 3(a), that corresponds to the graph dvson gven n Fgure 3(b). In ths representaton a decodng step s necessary to dentfy all the components of the graph, so that nodes partcpatng n the same component are assgned to the same cluster. Ths decodng step can be effcently done n lnear tme by usng the method reported n [23]. A man advantage of ths representaton s that the number k of clusters s automatcally obtaned by the number of components contaned n an ndvdual and determned by the decodng step. It s worth notng that also the locus-based representaton s redundant. However, the complexty of the search space

4 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X reduces from n n of the label-based representaton, to n =1 k where k s the degree of node. Snce often networks are sparse, the soluton space s narrower, thus the locusbased representaton can sensbly mprove the effcency of the evolutonary approach. Locus-based representaton has been frst used n [86] for communty detecton. Snce then, because of the ablty of naturally mappng the communty detecton problem to that of automatcally determnng k sub-graphs (often n the form of sub-trees) of a graph, t has been adopted as a vald alternatve to the label-based presentaton by several authors. Chra and Gog [21], analogously to [44], extended the locusbased representaton of an ndvdual wth the best potental soluton, the ndvdual s best ancestor, and added also the lowest ftness soluton. Ths extra nformaton s exploted to defne a specalzed selecton functon and a collaboratve crossover operator that changes an allele value by takng nto account also the ancestors. The effectveness of ths extenson for both label and locus representatons, however, cannot be proved snce the authors expermented only on two small networks. 4.3 Medod-based representaton The medod-based representaton uses an array of dmenson k, where k, the number of communtes, must be gven as nput parameter. The -th element of the array contans one of the nodes composng a communty. For nstance, a medodbased representaton of the network of Fgure 1 s the array [1 5 10], where 1 s the prototype of communty {1, 2, 3}, 5 of {4, 5, 6, 7}, and 10 of {8, 9, 10, 11, 12}}. Though ths representaton s more effcent n terms of space complexty, t has many drawbacks. Frst of all, k must be known n advance; moreover t s redundant because any element of the communty can be used as medod. Fnally, t needs a decodng step to recover the communtes. Whle for tradtonal clusterng recoverng s obtaned by assgnng a data object to the nearest medod, computed wth respect to a dstance measure such as the Eucldean one, the concept of dstance between nodes s not obvous. Frat et al. [36] dscuss ths problem and show that a dstance measure based on random walks s superor to Eucldean dstance. A random walk from a node to a node j n a graph s a stochastc process modelng the path startng at to reach j by choosng the next neghborng node at random. The authors also pont out that the assgnment of a node to the nearest medod leaves parts of the search space unexplored, thus preventng the achevement of potentally good solutons. They thus propose to extend the medod-based representaton wth excepton-bns, appended to the end of the genome, contanng set of nodes that are not assgned to the nearest medod. However, how many nodes allocate to excepton bns and how many bns should be used has remaned an open problem, thus ths proposal dd not receve much attenton. 4.4 Permutaton-based representaton The representatons descrbed above do not allow a node to partcpate n more than one communty. To overcome ths problem, Lu et al. [77] proposed a new representaton scheme that can generate overlappng communtes. In ths representaton, n the followng denoted permutaton-based, a chromosome A = (A P, A C ) s composed of two components. The frst, A P, s a permutaton of all the nodes {1, 2,..., n} A P = {v π1, v π2,..., v πn } (5) and the second component, A C, s a vector of n elements A C = {c 1, c 2,..., c n } (6) where c denotes the communty of node. In order to obtan A C, the authors adopt a so called decoder, whch actually s an ncremental method that fnds communtes by optmzng the communty ftness functon of Lancchnett et al. [69]. Nodes are examned n the order gven by A P and added to an exstng communty f the ftness functon augments, otherwse a sngleton communty s created. Ths mples that the same node could mprove the ftness of more than one communty, and thus added to many communtes, gvng rse to overlapped communtes. When all nodes have been examned, a mergng phase combnes couples of communtes f they have n common at least 50% of nodes. The decodng step of ths representaton presents two knds of problems. The frst s that at each generaton, n order to obtan A C, an algorthm must be executed. Thus decodng could be computatonally expensve. The second problem s that many sngleton communtes could be generated. Though teratve mergng of communtes can dampen the problem, communtes consttuted by sngle nodes can stll be present. Each of the above representatons has postve and negatve aspects. The label-based one s the most smple but also hghly redundant. The man drawback s that t generates a clusterng dvson C = {C 1,..., C k } such that a communty C could contan nodes not connected to any of the nodes present n C. To overcome ths undesrable behavor, specalzed operators have been suggested [106], but the guarantee that dsconnected nodes wll not be present n communtes cannot be assured. The man dsadvantage of the locus-based representaton s the need of decodng each chromosome before the ftness evaluaton, thus f both the sze of the populaton and the number of nodes are hgh, ths could slow down an algorthm. However, decodng can be effcently performed n O(n log n) tme by usng a dsjont-set data structure, as descrbed n [23]. The medod-based representaton needs the number of communtes as nput parameter. Ths makes t not applcable to real world networks snce ths nformaton s not known n advance. The man weakness of the recently proposed permutaton-based representaton s the choce of the decoder to obtan the assgnment of a node to a communty, wth a consderable ncrease of the computatonal resources. On the other hand, the decoder allows assgnng a node to more than one group, thus enablng overlappng among communtes. 5 GENETIC OPERATORS 5.1 Crossover Tradtonal crossover operators appled to the detecton of communtes can present several problems, analogous to those

5 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X ponted out n [34] and dscussed n [61] for data clusterng. The knd of problem s related to the representaton used by the method. Medod-based representaton uses one-pont crossover. As regards to the label-based representaton, however, a standard one-pont or two-pont crossover has two man drawbacks. The frst s that t could generate nvald solutons n whch nodes havng no connectons are assgned to the same group,.e. a cluster can contan dsconnected subgroups of nodes. To mtgate ths problem, Tasgn and Bngol [106] proposed one-way crossover, whch s analogous to the group-based crossover descrbed n [34], but whch generates only one offsprng from the two parents. One-way crossover, fxed the roles of the parents between source and destnaton chromosome, selects at random a node n the source, and creates a chld chromosome by transferrng n the destnaton chromosome the communty label of to the node, and to all the nodes havng the same label of n the source. An example of one-way crossover s shown n Fgure 4(a). To better understand ths knd of crossover, a graphcal llustraton can be seen n Fgure 4(b). In ths example, the node 7, whose label s 4, s chosen at random. Thus the chld has the same gene values of the destnaton chromosome, except for postons {6, 7, 8}, snce nodes 6 and 8 have the same label of node 7. The label of these three nodes s changed to 4. A modfed one-way crossover, named two-way, whch generates two offsprng by exchangng the roles of source and destnaton of the parent chromosomes, has been proposed by Gong et al. [48]. The second problem s that the offsprng does not nhert the genetc characterstcs of parents, thus destroyng some buldng blocks already obtaned. Ths problem has been faced by He et al. [58] by ntroducng the defnton of multndvdual ensemble learnng-based crossover operator, that generates an offsprng by usng a herarchcal agglomeratve clusterng method. Ths method starts by assgnng each node to a communty, and teratvely merges the two communtes wth the maxmal ftness value, provded that they contan a couple of nodes belongng to the same cluster n at least an ndvdual, out of the M best chromosomes of the current populaton. Though the authors state that ths knd of crossover mproves the global search capablty of ther method, they do not dscuss the computatonal tme ncrease due to the executon, at each step, of the herarchcal clusterng method that has to take nto account the best network dvsons of the current generaton. Moreover, how many promsng clusterng solutons should be chosen by the current populaton to form the ensemble has not been argued. Standard unform crossover s the knd of crossover that fts well for the locus-based representaton [86]. In fact, t guarantees the generaton of an offsprng that fully explots the genetc nformaton comng from the parents. A bnary mask of length equal to the number of nodes s randomly created, and an offsprng s generated by selectng from the frst parent the genes where the mask s 0, and from the second parent the genes where the mask s 1. Snce the value of a gene at poston s one of the neghbors of node, the effect of unform crossover s to connect a node wth another Source Destnaton Chld (a) (b) Fg. 4. (a) One-way crossover where the random poston 7 s selected. The class label 4 s thus assgned to genes at postons {6, 7, 8}, whch have the same label value 4 of gene 7. (b) Graphcal llustraton of one-way crossover. neghborng node, thus the lnks of the nodes n the network are mantaned n the chld ndvdual. Fgure 5 shows an example of unform crossover. Sh et al. [98] proposed the use of two-pont crossover, but the advantages wth respect to unform crossover have not been nvestgated. Zadeh and Kobt [116] proposed a mult-populaton cultural algorthm that mantans, besdes the populaton space, a belef space havng the role of knowledge repostory made of selected ndvduals havng the best ftness values. New ndvduals are generated by explotng ths belef space. Crossover s thus performed by choosng one parent randomly from the belef space, and the second parent from the ndvduals not appearng n t. Bnomal crossover s a knd of crossover employed n Dfferental Evoluton approaches [26] that generates a new ndvdual u from the target vector x and the mutant vector 1 v as follows: { vj f rand CR or j = j u j = rand otherwse x j where rand s random number between 0 and 1, j rand s an nteger random number between 1 and n, and CR s a control parameter. Ja et al. [63] modfed ths bnomal crossover operator for communty detecton by addng the one-way strategy of Tasgn and Bngol [105]; that s, the communty label s changed not only for node j, but also for all the nodes belongng to the same communty of j. 1. The concept of mutant vector [26] s explaned n the next secton (7)

6 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X Parent Parent Mask Chld (a) (a) (b) Fg. 5. (a) Unform crossover for locus-based representaton. (b) Graphcal llustraton. 5.2 Mutaton The task of mutaton s to modfy gene values to allow the exploraton of the search space towards regons not yet nspected. However, mutaton must not be too destructve and nullfy the process of fndng an optmal soluton. For the label based representaton the smplest strategy s to randomly change the membershp of a node by assgnng t to one of the other exstng communtes [106], [70] (see Fgure 6(a)). The same approach s adopted n the medod-based representaton [36]. A varant adopted by [48] s to assgn a node to the cluster of one of ts neghbors, whle n [58] the majorty label of the neghbors s adopted. The rand/1 mutaton strategy of dfferental evoluton [26] has been employed by Ja et al. n [63]. Ths strategy randomly selects three ndvduals x r1, x r2, x r3 from the populaton and generates the mutant ndvdual v as v = x r1 + F (x r2 x r3 ) (8) where F s a real number between 0 and 1. Each element of the mutant vector s then checked to contan one of the allowed labels,.e. an nteger number n the nterval [1, n]. If ths constrant s volated, a functon that takes back the label n the correct range values s appled. In the locus based representaton, chosen at random a node whose allele value s j, the neghbor node j s substtuted wth another node among ts neghbors [86]. Ths smple, but very effectve method, causes ether the splt of a communty or the unon of two communtes, thus modfyng the communty structure. Ths knd of mutaton can be seen n Fgure 6(b). Jn et al. [64] ntroduced the concept of margnal node, that s a node n a chromosome, wth locus-based representaton, (b) Fg. 6. (a) Mutaton, for label-based representaton, of the offsprng of Fgure 4 where node 1 s moved from cluster 2 to cluster 5. (b) Mutaton, for locus-based representaton, of the offsprng of Fgure 5 where node 12 s dsconnected from node 8 and connected to node 11. that never appears as an allele value. Mutaton s performed only on these nodes. The allele value of a margnal node s changed to another neghbor j f the ftness of the communty C to whch j belongs has the best ncrease wth respect to the other communtes, when s added to C. The same local search mutaton s adopted n [76]. 5.3 Populaton Intalzaton The ntal populaton s generally generated by assgnng random values to each ndvdual. Such a strategy, however, gves ntal dvsons of the network of poor qualty, wth true communtes hghly mxed. For label-based representaton, Tasgn and Bngol [106] suggested choosng some nodes and assgnng ther communty label to all ther neghbors. Ths approach nduces the generaton of small ntal communtes that can mprove the convergence of the method. Gong et al. [48] used the same strategy and suggested applyng t to 20% of ndvduals. He et al. [58] proposed a Markov random walk method based on the probablty that an agent can reach a node j from a node n a number of steps. In the locus-based representaton, assgnng to a gene one of ts neghbors s a smple approach that guarantees an ntal dvson of the network n connected groups of nodes [86]. Lu et al. [76], analogously to [58], adopted a Markov random walk strategy. 5.4 Local search operators Genetc operators often can produce solutons that assgn nodes to the wrong communty. In order to mprove the

7 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X qualty of the communty dvson, a number of heurstcs have been proposed. Tasgn and Bngol [106] proposed a clean-up process at the end of each generaton that chooses a number of nodes and computes the communty varance for such nodes. The communty varance of a node s defned as CV () = f (,j) (,j) E k (9) where k s the degree of node, and f (,j) s 0 f and j belong to the same communty, 1 otherwse. Communty varance s thus the rato between the number of dfferent communtes among and ts neghbors { 1,..., k }, and the number of ts neghbors. If ths value s above a fxed threshold, then and all ts neghborng nodes are assgned to the communty contanng the hghest number of nodes, among {, 1,..., k }. Otherwse, no acton s performed. The authors argue that communty varance nduces connected nodes to belong to the same group; however, how many nodes should undergo ths check, how to select them, and whch threshold value should be used have not been dscussed. A dfferent strategy s proposed by L et al. [70] consstng n makng n l copes of a chromosome, then, for each ndvdual, a row j s chosen at random from the adjacency matrx, and the communty label of j s assgned to all ts neghbors. The chromosome s then substtuted by the best, n terms of modularty value, among the n l copes. Ths process s repeated for each ndvdual n the populaton. Also n ths case, whch s the best value to use for n l s an open problem. Gong et al. [48] perform a local search at the end of each generaton, after crossover and mutaton, only on the ndvdual wth the best ftness value. Chosen a node belongng to a communty C r of the clusterng C = {C 1,..., C k }, determned by such an ndvdual, t s deleted from C r and assgned to another cluster C s C. The new partton wth the modfed communtes s called a neghbor of C. The local search procedure fnds all the neghbor parttons of the best ndvdual and, f one of them has a ftness value hgher than that of C, t substtutes C wth ths new one. Ths approach, as the authors also observe, s senstve to the startng pont and requres more computatonal effort. However, the authors reported better results when applyng ths strategy, though they do not say how much the computatonal demand ncreased. Shang et al. [95] observed that a local search based on hllclmbng can prevent exploraton of parts of the search space and gve poor local optmal solutons. Thus, they proposed the smulated annealng method [94] and showed that t can mprove the capablty of the genetc algorthm to fnd hgh qualty solutons. 6 FITNESS FUNCTIONS The choce of the ftness functon s another crtcal step for obtanng good solutons. In the context of communty detecton the most popular functon s modularty, orgnally ntroduced by Newman and Grvan n [43], [83] to evaluate clusterng results, and then used as crteron to optmze n [82]. More formally, modularty s defned as follows: Q = 1 ( A j k ) k j δ(c, C j ) (10) 2m 2m j where A s the adjacency matrx of the graph, m s the number of edges, k and k j are the degrees of nodes and j respectvely, and δ s the Kronecker functon whch yelds one f and j are n the same communty, zero otherwse. Let C 1 and C 2 be two dsjont subsets of the vertex set V, C 1 = V C 1, L(C 1, C 2 ) = A j, L(C 1, C 1 ) = C 1,j C 2 A j. Snce only the pars of vertces belongng to the C 1,j C 1 same cluster contrbute to the sum, modularty can be rewrtten as k ( ) 2 L(C, C ) Q = 2m L(C, V ) (11) 2m =1 where k s the number of communtes. The frst term of each summand s the fracton of edges nsde a communty, whle the second one s the expected value of the fracton of edges that would be n the communty f the network where a random one wth the same expected vertex degree. Values hgher than 0.3 ndcate good communty structure. Extensons to modularty to deal wth weghted and drected networks have been proposed by Arenas et al. [11]. Let W be the weghed adjacency matrx of a graph, then: Q = 1 2w ( j W j wout 2w wj n ) δ(c, C j ) (12) where w out = W j, wj n = W j, and 2w = W j. j j Shen et al. [96] proposed an extenson to modularty for overlappng communtes that takes nto account the number of communtes a node belongs to. The extended modularty EQ s defned as: EQ = 1 1 [A vw k vk w 2m O v O w 2m ] (13) v C,w C where O v s the number of communtes to whch v partcpates. Fortunato and Barthélemy [40] ponted out that the optmzaton of modularty has a resoluton lmt that depends on the total sze of the network and the nterconnectons of the modules. Moreover, the formula does not take nto account the sze of communtes. Ths mples that parttons obtaned by the maxmzaton of modularty could not dscover small groups, hdden wthn large communtes havng hgher modularty value. A modfcaton of modularty to overcome ths problem has been proposed n [73] wth the concept of modularty densty, defned as: D = k =1 L(C, C ) L(C, C ) C (14) The frst term s the average nner degree of a communty C, whch s twce the number of edges n C dvded ts

8 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X number of nodes, whle the second s the average out degree of C, that s the number of edges havng a node nsde C and the other node outsde C, dvded by the number of nodes of C. The authors prove that modularty densty has a number of advantages wth respect to modularty, such as detectng communtes of dfferent sze. A qualty measure of a communty C that maxmzes the n-degree of the nodes belongng to C has been defned n [86] as follows. score(c) = ( C 1 C ) α A j j C C,j C A j (15) where α s a postve real-valued resoluton parameter controllng the sze of the communtes, C s the cardnalty 1 of C, C A j s the fracton of edges connectng node j C to the other nodes n C, and A j s the double of the,j C number of edges connectng vertces nsde C,.e the number of 1 entres n the adjacency sub-matrx of A correspondng to C. The communty score of a clusterng C = {C 1,... C k } s defned as k CS = score(c ) (16) The concept of communty ftness P(C) of a communty C has been ntroduced n [69] as P(C) = C (k n (C) (C) + k out (17) (C)) α k n where α s a resoluton parameter. When k out (C) = 0, P(C) reaches ts maxmum value for a fxed α. In the lterature many other scorng functons, such as conductance, expanson, cut rato, have been defned to capture the concept of communty [110], and classfed wth respect to ther characterstcs of beng based on ether nternal or external connectvty, on a combnaton of both, and on a network model. These other crtera dd not receve much attenton as functons to optmze, probably because of obtanng solutons of lower qualty when compared to modularty. Modularty [83], and ts extensons [11], [96], are based on the dea that a random graph does not present communty structure. Thus, the exstence of communtes can be uncovered by a comparson between the edge densty of a group of nodes and the expected densty of ths group of nodes f they were attached randomly. Though ths qualty functon s one of the most popular functons, because of the resoluton lmt problem, t may be based towards network parttons wth small communtes merged nto larger communtes [40]. Communty score reles on nternal connectvty, and communty ftness on both nternal and external connectvty. Both functons have ntroduced a resoluton parameter α that allows the exploraton of communty structure at dfferent levels of granularty, thus overcomng the resoluton lmt problem of modularty. However, whch value of α gves the best partton s not an easy task, also because, as wll be dscussed n Secton 8, a formal defnton of communty does not exst [41]. In the next secton, multobjetve approaches that ntegrate these ftness functons to unvel dfferent aspects of communty structure, are descrbed. 7 MULTIOBJECTIVE OPTIMIZATION The approaches descrbed so far optmze only one of the objectve functons reported n the prevous secton. Though these sngle-objectve methods have obtaned very good results on both artfcal and real world networks, the ntutve noton of communty that the number of edges nsde a communty should be much hgher than the number of edges connectng to the remanng nodes of the graph, has two dfferent objectves: maxmzng the nternal lnks and mnmzng the external lnks. Thus, the communty detecton problem s naturally formulated wth multple competng objectves. The frst proposal of usng a multobjectve framework to uncover communty structure has been presented by Pzzut n [87], [89]. In partcular, the method maxmzes the communty score (formula (16)) and mnmzes the communty ftness (formula (17)), and uses as multobjectve framework the Nondomnated Sortng Genetc Algorthm (NSGA-II) proposed by Deb et al. n [28]. NSGA-II bulds a populaton of competng ndvduals and ranks them on the bass of nondomnance. The soluton of the Pareto front havng the hghest value of modularty s chosen as fnal result. It s worth to outlne that a man characterstc of the multobjectve approach s that the set of Pareto optmal solutons reveals the herarchcal organzaton of the network, where solutons wth a hgher number of groups are ncluded n solutons havng a lower number of communtes. Ths pecularty gves a great chance to analyze the network at varous herarchcal levels and study communtes wth dfferent modular levels. A varaton to ths method has been proposed by Agrawal [2]. The objectves to mnmze are { fq = 1 Q f QCS = f Q + 10 (18) (1 CS) where Q s the modularty (formula (11)) and CS s the communty score (formula (16)). The weght 10, as the authors state, has been obtaned emprcally. Sh et al. [101], [99] observed that the modularty formula Q = k =1 L(C,C ) 2m ( L(C,V ) 2m ) 2 s composed of two terms, where the left term consders the number of nternal lnks of communtes, thus t should be maxmzed, whle the rght one should be mnmzed because t ncludes the connectons wthn dfferent communtes. To obtan the frst objectve, communtes should be densely connected, to obtan the second one, the network should be dvded n many groups wth small total degree. In order to mnmze two objectves, the frst term s redefned as ntra(c) = 1 k =1 L(C, C ) 2m (19)

9 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X and nter(c) = k ( ) 2 L(C, V ) (20) =1 2m These two objectves balance the tendency of each other s to ncrease or decrease the number of communtes. If the number of communtes ncreases, the number of edges nsde each communty dmnshes, thus the frst term of modularty dmnshes, consequently ntra(c) augments, whle nter(c) dmnshes. When, nstead, the number of communtes dmnshes, nter(c) ncreases, snce the nter-connectons between communtes ncreases. Usng them as the two objectves to optmze thus, as the authors state, avods convergence to trval solutons. Regardng the model selecton from the Pareto front, the authors use two approaches: one chooses the soluton havng the maxmum modularty value, the other ntroduces the concept of Max-Mn dstance between models. Ths strategy generates a random network N wth the same scale of the real network N, and obtans the Pareto front CF of N. Then the dstance between the two Pareto front solutons s computed as dst(c, C ) = (ntra(c) ntra(c ) 2 + (nter(c) nter(c ) 2 (21) where C and C are solutons from the real and the random Pareto front, then S Max Mn = maxarg{mn{dst(c, C ) C CF }} Gong et al. [51] followed a smlar approach to that of Sh et al. [99] by splttng the modularty densty formula n two. Thus, the frst term, called Negatve Rato Assocaton (NRA) s NRA = k =1 L(C, C ) C and the second term, called Rato Cut (RC), s RC = k =1 L(C, C ) C (22) (23) Wu and Pan [108] proposed enrchng a multobjectve evolutonary algorthm wth a local search procedure to mprove the soluton. They adopt the Nondomnated Neghbor Immune algorthm (N N IA) [50] as optmzaton mechansm, label-based representaton of ndvduals along wth one-way crossover and neghbor-based mutaton, and the nter(c) and ntra(c) objectve functons of Sh et al. [99]. The local search procedure s executed after the applcaton of crossover and mutaton operators to the current nondomnated ndvduals of the Pareto front, and uses a label propagaton rule to change class membershp of nodes. A multobjectve evolutonary algorthm that obtans both separated and overlappng communtes has been proposed by Lu et al. [77]. The man novelty of ths approach s the ntroducton of the permutaton-based representaton descrbed n Secton 4.4. To obtan both separated and overlappng communtes the authors optmze three functons: f qualty (A) = ( k P(C) )/k =1 f separated (A) = V overlappng k f overlappng (A) = mn k V overlappng c k (24) where P(C) s the communty ftness of [69] (formula (17)), V overlappng s the set of nodes belongng to more than one communty, and k c s the number of edges connectng node wth communty c. Ths method uses the NSGA-II framework and apples nether crossover nor mutaton, but only the reverse operator on the permutaton component A P (formula (5)) of a chromosome. Multobjectve evolutonary approaches, analogously to sngle objectve ones, are able to dscover communty structures of qualty comparable wth, or even better than, those obtaned by computatonal methods not based on evolutonary computaton. The choce of the objectves to optmze should take nto account the suggestons gven by Sh et al. n [100], where a comparson of several objectve functons n a multobjectve framework has been performed. Eleven functons have been consdered, and a correlaton analyss revealed that couples of negatvely correlated objectves gve better results of postvely correlated ftness functons. The authors expermented that negatve correlaton has opposte nfluence on the number of communtes, thus t enhances dversty and avods obtanng trval solutons. Optmzng pars of postvely correlated objectves, nstead, s equvalent to a sngle objectve approach, thus t does not yeld any beneft to the algorthm. It s worth pontng out that nether of the above methods performs a correlaton analyss among the objectves, also because many methods are antecedent to ths analyss. 8 SINGLE OBJECTIVE VERSUS MULTIOBJEC- TIVE The concept of communty n a network s based on the dea that nternal connectons are dense, whle few tes should exst wth the rest of the graph. A formal defnton of communty, however, does not exst. Wasserman and Faust [107] defned four general propertes that cohesve groups of nodes should satsfy to be consdered communtes: complete mutualty, closeness or reachablty, frequency of nternal tes, relatve te frequences among group members versus non-members. The qualty of a communty can be defned wth respect to one, or more than one, of these propertes, and t measures how well the propertes are satsfed. Sngle objectve methods optmze a sngle property, whle multobjectve approaches smultaneously optmze competng objectves [22]. The two approaches present advantages and dsadvantages. Sngle objectve optmzaton dentfes a sngle best soluton that gves nsghts on the graph organzaton, however ths soluton could be based towards a partcular structure nherent nsde the crteron to optmze. Optmzng multple objectves, on the other hand, allows a smultaneous evaluaton of communty

10 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X structure from dfferent perspectves, but then t s the user s responsblty to choose a soluton. Consder for example the toy network of Fgure 1. By maxmzng modularty the soluton obtaned dvdes the network nto the three groups {{1, 2, 3}, {4, 5, 6, 7}, {8, 9, 10, 11, 12}}. However, by optmzng the two objectves of communty score and communty ftness, we obtan two solutons. One s the same dvson nto three communtes, the other one merges the frst two communtes gvng {{1, 2, 3, 4, 5, 6, 7}, {8, 9, 10, 11, 12}}. As can be observed from Fgure 1, ths second soluton s actually a possble and plausble soluton that gves a dfferent vew of group organzaton. It s worth pontng out that, for sngle layer networks the choce of sngle or multple objectves can depend on the applcaton doman. However, for other types of network models, such as multlayer networks, descrbed n the followng, many objectve methods seem to ft better. For example, the evoluton of dynamc networks wth temporal smoothness s well represented as a multobjectve problem optmzng snapshot qualty and temporal cost, as wll be clear n Secton SIGNED NETWORKS Sgned networks are an extenson of networks to model the relatonshps between ndvduals that, actually, can be ether postve or negatve, such as lke-dslke, frends-enemes. Postve lnks denote frendly relatons, whle negatve lnks represent antagonstc relatons. Detectng communty structure on these knds of networks s an mportant research topc snce t allows us to determne nstablty nsde relatonshps, and, consequently, to predct changes n group organzaton. In order to deal wth sgned networks, Gomez et al. [46] extended modularty as follows: Q S = 1 2m + + 2m,j V ( A,j ( a+ a+ j 2m + a ) a j 2m ) δ(c, C j ) (25) where A s the weghted adjacency matrx assocated wth the graph G = (V, E, W ) modelng a sgned network, m + and m are the number of postve and negatve entres n A, a + and a are the postve degree and the negatve degree of node, respectvely. A sgned verson of the toy network of Fgure 1, along wth the correspondng adjacency matrx, s shown n Fgure 7. A multobjectve approach that detects communtes n a sgned network has been proposed by Amelo and Pzzut n [5], [8]. The goal of obtanng communtes havng dense ntraconnectons and most edges wthn clusters postve, whle sparse nter-connectons and most of these edges negatve, s acheved by optmzng the concepts of sgned modularty and frustraton, ntroduced by Dorean and Mrvar [30]. Frustraton F (C) of a communty C s defned as the sum of the number of negatve edges between nodes nsde the same communty and the number of postve edges between nodes nto dfferent communtes. F (C) =,j V A,j δ(c, c j ) + A +,j (1 δ(c, c j )) (26) A = Fg. 7. An example of sgned network wth the correspondng adjacency matrx. Dashed edges denote negatve connectons. L et al. [71] performed a comparatve analyss of four evolutonary and memetc algorthms. EA-SN adopts a label based representaton, one-way crossover, a mutaton operator that randomly changes the neghbor of a node wth one of ts postvely connected nodes, and sgned modularty as objectve functon; CSA-SN expands the clonal expanson operator of [53] to sgned networks; EA HC -SN and CSA HC -SN ntegrate the hll clmbng strategy of [49] n a multobjectve algorthm that optmzes sgned modularty and modularty densty, extended for sgned networks. The authors showed that CSA HC -SN performs better than the other methods. Lu et al. [75] used the same representaton scheme proposed n [77] to defne a multobjectve evolutonary method to fnd communtes n sgned networks. The two objectves to optmze are based on the concepts of postve and negatve cluster smlarty between two neghborng nodes, ntroduced by Huang et al. [62], and extended to sgned lnks. The objectves to maxmze are the followng: ( k P C n =1 P C n +P C out f pos n (C = {C 1,..., C k }) = 1 m ( k ) f neg out (C = {C 1,..., C k }) = 1 N C out m =1 N C n +N C out (27) where P C C n and Pout are the postve nternal and external smlarty of a communty, whle N C C n and Nout are the negatve nternal and external smlarty. The smlarty between two nodes and j s defned as ψ(x) s sgned (, j) = x Γ() x Γ() Γ(j) w 2 (, x) x Γ(j) ) w 2 (j, x) where { 0 f w(, x) < 0 and w(j, x) < 0 ψ(x) = w(, x) w(j, x) otherwse (28)

11 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X (a) (busness relaton) (b) (marrage relaton) Fg. 8. An example of a multlayer network wth two elementary layers. Notce that, n [8] a correlaton analyss of sgned modularty and frustraton revealed that these two objectves are negatvely correlated, whle a correlaton analyss of the two objectves employed by [75] produced a postve correlaton value. 10 MULTILAYER NETWORKS The representaton of complex networks wth graphs consstng of sngle statc connectons between couples of nodes has been unversally adopted by researchers for many years. Recently, however, the need of rcher models able to represent the varety of nterconnectons of real-world systems has led to the nvestgaton of networks wth multple types of connectons, the so-called multlayer networks [27], [68], [12]. Each layer represents a combnaton of dfferent features of the network, called aspects or facets. Thus, for each aspect a, there s a set of elements L a, where each element s called an elementary layer. A layer wll then be obtaned by a combnaton of elementary layers from all the aspects. More formally, a multlayer network M s defned as a quadruple [68]: M = (V M, E M, V, L) where V s the set of nodes, L = {L a } l a=1 s a sequence of sets of elementary layers L a, V M V L 1... L l contans only the set of combnatons of nodes and elementary layers effectvely present n a layer, E M V M V M s a set of couples of possble combnatons. Nodes could be connected to any other both nsde the same layer and across layers. When the network has only one aspect wth multple types of edges and the same set of nodes, the network s called multplex or multrelatonal. An example of a multplex network, taken from [107], havng two types of relatonshps, namely busness and marrage, regardng Florentne famles, can be seen n Fgure 8. Notce that the connectons between the same nodes appearng n both layers are mplct. Though the nterest n multlayer networks s rapdly ncreasng, there are stll few approaches that detect communtes n these knds of networks [80], [104]. As ponted out n [68], the development of communty detecton methods for multlayer networks s just at the begnnng. Also, the concept of communty s not well-defned. Mucha et al. [80] generalzed modularty for multlayer networks, whle Tang et al. [104] ntroduced the noton of shared latent communty structure, that s a dvson of nodes that optmzes the same crteron for each dmenson. As regards evolutonary methods, there are few proposals. In [6] multplex networks are consdered by extendng both the locus-based representaton and modularty. The extended representaton s such that an ndvdual I = {I 1,..., I d } of the populaton s composed by a set of d elements I s, 1 s d, each element I s beng the locus-based representaton of the correspondng layer s. The concept of modularty for multlayer networks s defned by combnng the modularty values computed for each layer n such a way that the value for each layer s nfluenced by the values of all the other layers. The man drawbacks of ths method are the computaton tme needed to compute the ftness functon and the space requrements. Other proposals concentrated manly on the dynamc aspect of networks. In fact, a dynamc or temporal network can be consdered as a multlayer network restrcted to two aspects. The frst aspect L 1 = {T 1,..., T T } represents the temporal nformaton,. e. the tme n whch a connecton between two nodes occurred, whle the second one L 2 = {D 1,..., D d }, gves the type of nteracton among nodes. The set of combnatons of a fxed elementary layer T t L 1 wth all the elementary layers D j L 2, j = 1,..., d, wll be called multplex (or multdmensonal) network at tme t, and denoted as T t = {N1, t N2, t..., Nd t t }, where each N s the network representng one of the elementary layers of L 2. A temporal or dynamc multlayer network s defned as a sequence DM = {T 1,..., T T } of networks, where each T t, t = 1,..., T s a snapshot of the network at tme t, referred as tmestamp or tmestep. In ths context, there are two types of proposals. In the former [38], [66], [52], [39], methods consder only one type of nteracton of the aspect L 2,.e. d = 1, n the latter d > 1 [9]. All these methods are based on the concept of evolutonary clusterng ntroduced by Chakrabart et al. n [18] for data clusterng. Evolutonary clusterng s a framework assumng that abrupt changes of clusterng n a short tme perod are not desrable, thus t smooths each communty over tme. For smoothng, a cost functon composed by two sub-costs, snapshot cost (SC) and temporal cost (T C), s defned. The snapshot cost SC measures how well a communty structure CC t represents the data at tme t. The temporal cost T C

12 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X measures how smlar the communty structure CC t s wth the prevous clusterng CC t 1. A specalzed verson of ths functon n the context of dynamc networks has been ntroduced n [74] as follows: cost = α SC + (1 α) T C (29) where α s an nput parameter fxed by the user to emphasze one of the two objectves. When α = 1 the approach returns the clusterng wthout temporal smoothng. When α = 0, however, the same clusterng of the prevous tme step s produced,.e. CC t = CC t 1. Thus, a value between 0 and 1 s used to control the preference degree of each sub-cost. In [38], [39] the detecton of communty structure wth temporal smoothness has been formulated as a multobjectve optmzaton problem where the frst objectve s the maxmzaton of the snapshot qualty, and the second objectve s the mnmzaton of the temporal cost. Several ftness functons have been expermented to optmze snapshot qualty, such as modularty, communty score, conductance, and normalzed cut. The Normalzed Mutual Informaton, a well known entropy measure n nformaton theory [25], has been employed as second objectve to mnmze the temporal cost T C. NMI(CC t, CC t 1 ) measures the smlarty between the communty structure CC t at the current tme step t and the prevous one CC t 1. The normalzed mutual nformaton N M I(A, B) of two parttons A and B, s defned as: 2 c A cb =1 j=1 C jlog(c jn/c.c.j) NMI(A, B) = ca =1 C.log(C./N) + c B j=1 C.jlog(C.j/N) where C s the confuson matrx whose element C j s the number of nodes of the communty A A that are also n the communty B j B, c A (c B ) s the number of groups n the parttonng A (B), C. (C.j ) s the sum of the elements of C n row (column j), and N s the number of nodes. If A = B, NMI(A, B) = 1. If A and B are completely dfferent, NMI(A, B) = 0. A man advantage of ths approach s that the parameter α, that must trade-off the beneft of mantanng a consstent clusterng over tme (temporal cost) wth the cost of devatng from an accurate representaton of the current data (snapshot cost), s automatcally determned durng the computaton of the non-domnated solutons. A varaton of ths approach, wth the same objectve functons of modularty and N M I, that uses as multobjectve optmzaton method the Nondomnated Neghbor Immune NNIA algorthm [50] has been proposed by of Gong et al. [52]. Moreover, the same authors [78] extend the framework of multobjectve evolutonary algorthm based on decomposton [51] to deal wth dynamc networks by optmzng agan modularty and NMI. Chen et al. [20] use the same framework by changng the frst objectve wth modularty densty. A multobjectve method based on mmgrant schemes, that replaces a proporton of the populaton wth the am of mantanng populaton dversty and adaptng to changes, has been proposed by Km et al. [66]. The method ntroduces three (30) mmgrant strateges nsde the multobjectve evolutonary algorthm NSGA-II [28] to deal wth networks that can ncrease the number of edges and/or nodes wth tme. A chromosome, usng the locus-based representaton, s extended wth new genes f the number of nodes augments. The objectve functons to optmze are the mn-max cut ntroduced n [29] and the global slhouette ndex [93]. A comparson among the three mmgrant schemes has been performed on a synthetc data set. However, no comparson wth classcal communty detecton methods s present, thus the capablty of the approach to dscover hgh qualty clusters s not known. Moreover, as the authors pont out, the method s applcable only to networks that grow, but no node or edge can dsappear, whch s not a realstc scenaro. In [6] the evolutonary clusterng framework s modfed by ntroducng the concepts of facet qualty FQ, and sharng cost SQ. Facet qualty guarantees that the clusterng found for the -th dmenson under consderaton maxmzes the qualty functon as much as possble, whle the sharng cost means that the clusterng of the current facet agrees as much as possble wth the clusterng obtaned for the prevously consdered -1 dmensons. In [9] an extenson that encompasses both tme and multple dmensons s defned. In ths extended framework, a shared communty structure among the networks N t of T t s obtaned by teratvely optmzng both facet qualty and sharng cost. The communty structure obtaned for the last layer d s consdered the best sharng communty structure among the d layers. Let CC t 1,..., CC t d be the communty structures obtaned for each elementary layer of a network T t = {N1, t N2, t..., Nd t}, at tmestamp t. The concept of shared communty structure ntroduced n [104] s formalzed as follows: CC = {C 1,..., C k } s a shared communty structure of T t f the two functons are maxmzed: f q (CC, N t ), = 1,..., d (31) f s (CC, CC t ), = 1,..., d (32) where (31) s the qualty functon computed on the network N t by usng the communty structure CC, and (32) s a functon that computes the smlarty between CC and the communty structure obtaned for N t by maxmzng f q, ndependently from the other layers. f q and f s can be any functons computng the qualty of a clusterng and the smlarty between two clusterngs, respectvely. Thus, the method searches for a communty structure CC that maxmzes a ftness functon on each elementary layer N t, whle takng nto account the smlarty wth the clusterng obtaned on the other layers. Ths framework s then utlzed between couples of consecutve tmestamps t 1 and t, by resortng to the dynamc evolutonary approach where the temporal cost T C s guaranteed by consderng the smlarty between the communty structure CC t 1 obtaned for the prevous tmestamp and that found for the frst elementary layer CC t 1 of the current tmestamp.

13 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X In the last few years a lot of effort n defnng effcent and effcacous methods for communty detecton has been drected to fndng dsjont communtes. However, n real world networks the membershp of an entty to many groups s very common, thus the nterest n defnng methods for fndng overlappng communtes has been growng. It s worth pontng out that the representaton schemes descrbed n Secton 6, except for the permutaton-based representaton, do not allow a node to be a member of more than one communty, thus only recently a number of evolutonary computaton methods, both sngle-objectve and multobjectve, have been proposed to fnd overlappng communtes. In [88] the concept of lne graph has been exploted to defne a lnk clusterng method that detects overlappng communtes by parttonng the set of lnks, rather than the set of nodes. The lne graph L(G) of an undrected graph G s another graph L(G) such that each vertex of L(G) represents an edge of G, and two vertces of L(G) are adjacent f and only f ther correspondng edges share a common endpont n G. A lne graph represents the adjacency between edges of G. By applyng a communty detecton method to the lne graph generates an overlappng dvson of the orgnal nteracton network, thus allowng nodes to be present n multple communtes. An example of network and the correspondng lne graph s shown n Fgure 9. A communty dvson of the lne graph nto the two clusters C 1 = {(1, 2), (1, 3), (1, 4), (3, 4)}, C 2 ={(3, 5), (3, 6), (5, 6)}, gves the dvson {{1, 2, 3, 4}, {3, 5, 6}} of the orgnal graph n whch node 3 appears n both the clusters. The method descrbed n [88] adopts the locus-based representaton on the lne graph. Ths means that each gene corresponds to an edge of G, and the value t contans s one of the neghborng edges, that s an edge havng a node n common. The algorthm fnds a communty structure of the lne graph L(G) and evaluates ts qualty by computng the communty score of the correspondng dvson of the orgnal graph G. Sh et al. [97] proposed a smlar method that clusters lnks, whch s equvalent to usng the lne graph snce two edges are connected only f they share a node. However, ther method uses as ftness functon the concept of partton densty proposed by Ahn et al. [3], whch s based on the number of lnks, thus ts evaluaton can be done on the orgnal graph. Let {P 1,..., P C } be the partton of the lnks n C subsets. Each subset P c has m c = P c lnks and n c = ej P c {, j} nodes. The lnk densty of P c s defned as D c = m c (n c 1) n c (n c 1)/2 (n c 1) = 2 m c (n c 1) (n c 2)(n c 1) (33) Fg. 9. An example network wth 6 nodes and 7 edges, and the correspondng lne graph wth 7 nodes and 11 edges. 11 OVERLAPPING COMMUNITY DETECTION D c s thus the normalzaton of the number of lnks m c by the mnmum and maxmum number of possble l lnks between n c connected nodes. It s assumed that D c = 0 when n c = 2. The partton densty P D s the average of the D c, weghted by the fracton of present lnks: P D = 2 m c m c m c (n c 1) (n c 2)(n c 1) (34) Another proposal that clusters the set of lnks by optmzng the two objectve functons of modularty densty D (formula (14)) and extended modularty EQ (formula (13)) has been proposed by Du et al. [32]. Yuxn et al. [115], nstead, consder the communty ftness, and defne the negatve ftness sum (NFS) and the unftness (US) of a communty structure by substtutng the numerator of communty ftness (formula (17)) wth the sum of external degrees. Let unft(c) = C (k n (C) (C) + k out (35) (C)) α k out be the external connecton densty of a communty C C = {C 1,... C k }, then the two modfed objectve functons are the followng: NF S = k P(C) C C US = unf t(c) (36) C C To mprove the convergence, the algorthm adopts an ntalzaton strategy that expands a seed node by mergng adjacent edges untl the communty ftness mproves. Ths process s repeated untl all edges are assgned to a communty. 12 OTHER BIO-INSPIRED APPROACHES In recent years, bo-nspred computaton has attracted the nterest of many researchers n several felds to solve optmzaton problems. The basc prncple of these methods s self-organzaton, that s f a system s allowed to evolve for a

14 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X suffcently long perod, self-organzed structures may emerge [113]. In the last decade, a relevant number of these new metaheurstc algorthms have been employed for communty detecton, ncludng swarm ntellgence [114], n partcular Partcle Swarm Optmzaton (PSO) [65] and Ant Colony Optmzaton (ACO) [31], Frefly [111] and Bat [112] algorthms. Partcle Swarm Optmzaton. PSO s an optmzaton technque based on the swarm behavor of brd and fsh schoolng [65]. Each partcle s characterzed by two components: the poston vector x and the velocty vector v. Partcles are attracted towards the best poston g of the swarm, and ts personal best poston x, whle movng randomly at the same tme. The new velocty and poston vectors are updated as v t+1 = v t + αɛ 1 (g x t ) + βɛ 2 (x x t ) (37) x t+1 = x t + v t+1 (38) where α and β are acceleraton parameters, and ɛ 1, ɛ 2 are two random vectors takng values n the range [0,1]. Ca et al. [15] apples the partcle swarm method to detect communtes n sgned networks by optmzng the sgned modularty. The poston vector represents a partton of the network where x s the communty label of node. The update rules of the partcle status are redefned to ft n the dscrete context as follows: v t+1 = Γ(ωv t + αɛ 1 (g x t ) + βɛ 2 (x x t )) (39) x t+1 = x t Θv t+1 (40) where ω s an nerta weght [102] that, when ts value s hgh, t s better for global search, whle, when small, for local search, s the xor operator. The Γ functon assgns 1 to v f x 1, 0 otherwse. The operator Θ s a neghbor operator that updates the poston of a node by consderng ts neghbors. The same method s appled for unsgned networks n [14], and for sgned networks n Gong et al. [47]. In ths latter paper the problem has been formulated as a multobjectve optmzaton problem where the objectve functons are obtaned by extendng the Negatve Rato Assocaton (NRA) and Rato Cut (RC), ntroduced n [51]. Thus, the sgned network clusterng problem s reformulated as the mnmzaton of the objectves: SRA = k SRC = k =1 =1 L + (C,C ) L (C,C ) C L + (C,C ) L (C,C ) C (41) where L + (C, C ) = A j, A j > 0 and C,j C L (C, C ) = A j, A j < 0. C,j C A multobjectve varant of these methods, also for sgned networks, has been proposed by L et al. [72]. The objectve functons are the same of Gong et al. [47] (formula (41)). The man dfferences wth Ca et al. [15] and Gong et al. [47] are the defnton of the Γ functon, and a replacement operaton that substtutes only a subset of the solutons n the new generaton. The new Γ functon s defned as: { 1 f rand(0, 1) 1/(1 + e Γ(y) = y ) 0 othewse (42) Ant Colony Optmzaton. ACO mmcs the foragng behavor of ants [31]. Ant movement s controlled by pheromone, whch evaporates over tme, and ts concentraton s an ndcator of the qualty of the soluton. In these algorthms there are two mportant ssues: the probablty of choosng a route and the evaporaton rate of pheromone. The probablty of choosng a route from node to node j s gven by the rule: p j = φ α j dα j n,j=1 φα j dα j (43) where α > 0, β > 0 are the nfluence parameters, φ j s the pheromone concentraton of the route between and j, and d j s a heurstc functon that reflects the tendency of selectng the edge from to j. Chen et al. [19] proposed an algorthm based on ant colony optmzaton that adopts the concept of assocate degree between nodes as a heurstc functon. Let A = (A j ) be the adjacency matrx of the network and A k = (A k j ) the number of k-step paths connectng two nodes. The assocate degree s defned as: d j = k 1 A 1 j + k 2 A 2 j k p A p j (44) where p s a postve constant nteger, k, = 1,..., p are coeffcents. The pheromone updatng s then performed accordng to the formula: φ j (t + 1) = ρφ j (t) + m φ k j(t) (45) k=1 m s the number of ants, φ k j = C Q(S k), wth C a constant, and Q(S k ) the modularty value of the soluton S k, f and j are n the same communty, 0 otherwse. A dfferent heurstc nformaton, based on the Pearson correlaton, has been proposed by Honghao et al. [60]. Gven two nodes and j, the Pearson correlaton s defned as: l V C(, j) = (A l µ )(A jl µ j ) (46) nσ σ j where A l s the lth element of the th row n the adjacency matrx, µ the average and σ the standard devaton. Then d j = e C(,j) (47) The formula for pheromone updatng uses the best modularty value obtaned at the current teraton, and only edges whose nodes belong to the current best soluton receve ths value. Frefly algorthm. Ths metaheurstc method s based on the flashng patterns and behavor of frefles [111]. It assumes that frefles are unsexual, they are attracted to other frefles proportonally to ther brghtness, the brghtness s determned

15 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X by the landscape of the objectve functon. The movement of a frefly s defned as x t+1 = x t + β 0 e γr2 j (x t j x t ) + αɛ t (48) where the second term n the formula s the attractveness functon, wth β 0 the attractveness when the dstance r = 0. The thrd term s a randomzaton wth parameter α, and ɛ t s a vector of random numbers. Amr et al. [10] adopted ths approach to desgn a multobjectve method that optmzes communty score and communty ftness, by ntroducng some varatons to mprove solutons. They mantan an external repostory to store the non-domnated solutons and apply a nchng mechansms to preserve dversty. Moreover, they assume that the frefles can have dfferent sex, and the parameter α s dynamcally tuned by usng a chaotc sequences, as proposed n [17], nstead of random ones. Bat Algorthm. Ths approach s nspred by the behavor of bats and ther capablty of echolocaton, a type of sonar, that allows them to detect prey and to avod obstacles [112]. Bats emt sound waves whose loudness gradually reduces whle frequency of emsson gets faster, as the dstance to the prey s closer. If x s the poston of the bat at tme t, f the frequency varyng n the nterval [f mn, f max ], v the velocty,.e. the change degree of ts poston, r the emsson rate and A the loudness, these values are updated wth the rules: f = f mn + (f max f mn )ɛ, v t+1 = v t + (x t x )f (49) where x s the current best soluton. x t+1 = x t +v t, A t+1 = αa t, r t = r 0 [1 exp( βt)] (50) Hassan et al. [57] observed that the bat algorthm cannot drectly be appled for communty detecton. Thus, a dscretzaton and redesgn of the bat movement s necessary before usng the approach. Let the vector state of an artfcal bat be x = (x 1,..., x n ), the velocty vector v = (v 1,..., v n ), and g(x ) the group assgnment of node. The dstance to the current best soluton x s computed as : { d = (x x 1 f g(x ) g(x ) = ) 0 f g(x ) = g(x ) (51) Then the new poston value s computed as { x new x = f v 1 othewse x (52) Another dscrete bat algorthm has been proposed by Song et al. [103] to dscover communtes by makng dscrete the values of x and v. The new dscrete velocty formula s defned as follows. Let Sg(v t) = 1/(1 + exp( vt )) be the sgmod functon, and rand a random number n the range (0,1), then v t = 1 f Sg(vt ) > rand, 0 otherwse. 13 CONCLUSION Evolutonary computaton has been successfully appled to many real-world problems as an optmzaton technque, and showed to be compettve also for the study of complex networks. The paper presented an up-to-date revew on evolutonary methods for communty detecton. Though research n ths context s rather recent, there has been a surge of nterest and many methods have been proposed to deal wth complex networks. A man contrbuton of the survey s that t systematzes the several approaches presented n the lterature by provdng the basc common prncples for the desgn of methods that solve the problem of uncoverng communty structure. In partcular, the most popular representaton schemes, along wth the crossover and mutaton operators apt for them are descrbed n detal, by dscussng advantages or drawbacks of each, and the most common ftness functons adopted by methods are also analyzed. A categorzaton n sngle objectve and multple objectves optmzaton has been gven. Though many surveys for communty detecton are avalable n the lterature [42], [92], [41], [24], [84], [109], [79], [91], [56], [1], [7], [90], specfc revews for evolutonary based approaches are few [16]. The paper sensbly extends the work of Ca et al. [16] by ncludng multlayer networks, and by gvng a more detaled descrpton of ndvdual representaton and assocated operators. To summarze the approaches descrbed n the paper, Table 1 reports the sngle objectve methods, and Table 2 the multobjectve ones. For each approach, the knd of representaton, crossover, mutaton, ftness functon employed, and f overlappng s allowed, are reported. When present, local search strateges adopted to mprove the methods are ncluded. For the multobjectve methods, also the type of network and the multobjectve evolutonary optmzaton method used are added. Moreover, Table 3 summarzes the other bo-nspred approaches. The tables, for all the methods, report also the real-world networks and/or the knd of synthetc dataset used for evaluatng the qualty of results. Fnally, Table 4 contans the lst of all these networks, along wth the web address from whch t s possble to download the network. Lnks to the source codes for the methods, when avalable, are reported n the References Secton. The revew hghlghted that, though there s a lot of work on networks representng a sngle type of nteracton, further nvestgaton s necessary as regards overlappng communty detecton and multlayer networks. In fact, new representaton schemes are desrable to effcently deal wth overlappng communtes, and novel deas to tackle the dynamc and multrelatonal aspects of networks. Another aspect that t s worth pontng out s that evolutonary algorthms are tme consumng, thus, though they are compettve wth the non-evolutonary approaches as regards the qualty of the obtaned soluton, they are not able to cope wth large networks, very common n the current bg data era, where networks wth mllons of nodes are generated. The need of developng parallel mplementatons, consderng the nbult parallel characterstcs of evolutonary methods, to accelerate the response tmes s an mportant ssue to make them comparable wth the other methods avalable n the lterature. Moreover, more effcent representatons, such as varable length chromosomes, should be nvestgated to reduce both tme and space requrements. The survey can be a startng pont for researchers nterested

16 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X TABLE 1 A summarzaton of sngle-objectve methods. METHOD REPR. FITNESS CROSSOVER MUTATION LOCAL SEARCH Tasgn and Bngol [106] (2007) Frat et al. [36] (2007) medod par-wse dstances sum OVERLAP NETWORKS label Q one-way random clean-up no ZKC, KPB, GN two-pont random - no synthetc Gog et al. [44] (2007) label Q collaboratve random - no ZKC Pzzut [86] (2008) locus CS unform neghbor best no ZKC, BD, ACF, KPB, GN Pzzut [88] (2009) locus CS unform neghbor best yes ZKC, BD, ACF, KPB, GN L et al. [70] (2009) label Q one-way random n l copes no ZKC, BD, LM, Ucnet, Pajek He et al. [58] (2009) label Q multndvdual majorty neg. label - no ZKC, ACF, GN Sh et al. [98] (2009) locus Q two-pont random - no ZKC, ACF, CN Jn et al. [64] (2010) locus Q unform neghbor margnal node no ZKC, BD, ACF, KPB, JM, WA, SFI Chra and Gog [21] (2011) locus CS collaboratve random - no ZKC, BD, KPB Gong et al. [49] (2011) label D two-way neghbor neghbor label Gong et al. [48] (2012) label D two-way neghbor label no ZKC, BD, ACF, KPB,LFR - no ZKC, BD, ACF, KPB, LFR Ja et al. [63] (2012) label Q bnary rand/1 clea-up no ZKC, ACF, GN Shang et al. [95] (2013) label Q two-way random smulated annealng Lu et al. [76] (2013) locus Q unform neghbor nsde mutaton Sh et al. [97] (2013) locus Q unform neghbor nsde mutaton no no yes ZKC, BD, ACF, KPB, LFR GN, LFR, ZKC, BD, ACF, KPB, JM, WA, SFI ZKC, BD, ACF, KPB, WA, LM, PG, LFR Zadeh et al. [116] (2015) locus CS unform neghbor no no ZKC, BD, KPB n approachng the problem of communty detecton wth computatonal models nspred by evoluton n nature. The knowledge of a dfferent computatonal paradgm wth respect to tradtonal approaches can be benefcal to explore new strateges and prncples to deal wth ths problem. ACKNOWLEDGMENT Ths work has been partally supported by MIUR D.D. n , under the project BA2KN OW P ON03P E REFERENCES [1] Charu Aggarwal and Karthk Subban. Evolutonary network analyss: A survey. ACM Comput. Surv., 47(1):10:1 10:36, May [2] Rohan Agrawal. B-objectve communty detecton (BOCD) n networks usng genetc algorthm. In Proceedngs of the 4th Internatonal Conference on Contemporary Computng, IC3 2011, Noda, Inda, August 8-10, 2011., pages 5 15, [3] Yong-Yeol Ahn, James P. Bagrow, and Sune Lehmann. Lnk communtes reveal multscale complexty n networks. Nature, 466: , [4] Réka Albert and Albert lászló Barabás. Statstcal mechancs of complex networks. Rev. Mod. Phys, 74(1):47 97, [5] Alessa Amelo and Clara Pzzut. Communty mnng n sgned networks: a multobjectve approach. In Advances n Socal Networks Analyss and Mnng 2013, ASONAM 13, Nagara, ON, Canada - August 25-29, 2013, pages 95 99, [6] Alessa Amelo and Clara Pzzut. Communty detecton n multdmensonal networks. In 26th IEEE Internatonal Conference on Tools wth Artfcal Intellgence, ICTAI 2014, Lmassol, Cyprus, November 10-12, 2014, pages , [7] Alessa Amelo and Clara Pzzut. Overlappng Communty Dscovery Methods: A Survey. In Socal Networks: Analyss and Case Studes, pages Sprnger, Venna, [8] Alessa Amelo and Clara Pzzut. An evolutonary and local refnement approach for communty detecton n sgned networks. Internatonal Journal on Artfcal Intellgence Tools, 25(4):1 44, [Code avalable at: [9] Alessa Amelo and Clara Pzzut. Evolutonary clusterng for mnng and trackng dynamc multlayer networks. Computatonal Intellgence, 33(2): , [Code avalable at: [10] Babak Amr, Laquat Hossan, John W. Crawford, and Rolf T. Wgand. Communty detecton n complex networks: Mult-objectve enhanced frefly algorthm. Knowl.-Based Syst., 46:1 11, [11] Alex Arenas, J. Duch, A. Fernandez, and S. Gomez. Sze reducton of complex networks preservng modularty. New Journal of Physcs, 9:176, [12] Federco Battston, Vncenzo Ncosa, and Vto Latora. Structural measures for multplex networks. Phys. Rev. E, 89(3):032804, [13] Hans-Georg Beyer and Hans-Paul Schwefel. Evoluton strateges a comprehensve ntroducton. Natural Computng, 1(1):3 52, [14] Qng Ca, Maoguo Gong, Lja Ma, Shasha Ruan, Fuyan Yuan, and Lcheng Jao. Greedy dscrete partcle swarm optmzaton for largescale socal network clusterng. Inf. Sc., 316: , [15] Qng Ca, Maoguo Gong, Bo Shen, Lja Ma, and Lcheng Jao. Dscrete partcle swarm optmzaton for dentfyng communty structures n sgned socal networks. Neural Netw., 58:4 13, October 2014.

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19 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. X, NO. X, X [107] S. Wasserman and K. Faust. Socal Network Analyss Methods and Applcatons. Cambrdge Unversty Press, [108] Peng Wu and L Pan. Mult-objectve communty detecton based on memetc algorthms. PLOS One, 10(5):e , [109] Jeru Xe, Stephen Kelley, and Boleslaw K. Szymansk. Overlappng communty detecton n networks: the state of the art and comparatve study. ACM Computng Survey, 45(4), [110] Jaewon Yang and Jure Leskovec. Defnng and evaluatng network communtes based on ground-truth. Knowl. Inf. Syst., 42(1): , January [111] Xn-She Yang. Frefly algorthms for multmodal optmzaton. In Proceedngs of the 5th Internatonal Conference on Stochastc Algorthms: Foundatons and Applcatons, SAGA 09, pages , [112] Xn-She Yang. A new metaheurstc bat-nspred algorthm. In Nature Inspred Cooperatve Strateges for Optmzaton (NICSO 2010), pages 65 74, [113] Xn-She Yang. Nature-Inspred Optmzaton Algorthms. Elsever Scence Publshers B. V., Amsterdam, The Netherlands, The Netherlands, 1st edton, [114] Xn-She Yang, Zhhua Cu, Renbn Xao, Amr Hossen Gandom, and Mehmet Karamanoglu. Swarm Intellgence and Bo-Inspred Computaton: Theory and Applcatons. Elsever Scence Publshers B. V., Amsterdam, The Netherlands, The Netherlands, 1st edton, [115] Zhao Yuxn, L Shenghong, and Jn Feng. Overlappng communty detecton n complex networks usng mult-objectve evolutonary algorthms. Computatonal and Appled Mathematcs, 36(1): , [116] Pooya Moradan Zadeh and Zad Kobt. A mult-populaton cultural algorthm for communty detecton n socal networks. Proceda Computer Scence, 52: , Clara Pzzut receved the Master s degree n Mathematcs from Unversty of Calabra, Italy, and a Ph.D n Scence from the Radboud Unverstet Njmegen, NL. She s senor researcher at the Insttute of Hgh Performance Computng and Networkng (ICAR) of the Italan Natonal Research Councl(CNR), where she leads the Smart Data and Models research laboratory. From 1995 to 2010 she was contract professor n the department of Computer Scence at the Unversty of Calabra. Her research nterests nclude evolutonary computaton, knowledge dscovery n databases, data mnng, data streams, bonformatcs, socal network analyss and mnng.