Cohesive Subgraph Mining on Attributed Graph

Transcription

1 Cohesive Sbgraph Mining on Attribted Graph Fan Zhang, Ying Zhang, L Qin, Wenjie Zhang, Xemin Lin QCIS, University of Technology, Sydney, University of New Soth Wales fanzhang.cs@gmail.com, {Ying.Zhang, L.Qin}@ts.ed.a, {zhangw, lxe}@cse.nsw.ed.a Abstract Finding cohesive sbgraphs is a fndamental graph problem with a wide spectrm of applications. In this paper, we investigate this problem in the context of attribted graph, where each vertex is associated with content (e.g., geo-locations, tags and eywords). To properly captre the cohesiveness of the vertices in a sbgraph from both graph strctre and vertices attribte perspectives, we advocate a novel cohesive sbgraph model, namely (,r)-core. In particlar, we adopt the poplar concept of -core as the strctral constraint sch that each vertex in a (,r)-core connects to at least other vertices. Meanwhile, there is a ser defined similarity threshold (i.e., similarity constraint) to ensre that the contents of the vertices in the same (,r)-core are similar to each other. We aim to devise efficient algorithms to enmerate all maximal (,r)-cores and find the maximm (,r)- core, where both problems are shown to be NP-hard. Effective end efficient prning techniqes are proposed to significantly redce the search space of two algorithms. Novel doble -core based pper bond comptation method is also devised to enhance the performance of the maximm (,r)-core comptation algorithm. According to the different natre of two mining algorithms, we devise effective search orders. Or comprehensive experiments on real-life data demonstrate that the maximal/maximm (,r)- cores enable s to find interesting cohesive sbgraphs, and the performance of two mining algorithms is significantly improved by or effective prning techniqes and smart search orders. I. INTRODUCTION Graph as an expressive data strctre is poplarly sed in a wide spectrm of applications. In many real-life graphs, besides the topological strctre of the graph, the attribte is associated with vertices to present their properties. For instance, the geo-locations of the sers are recorded in geosocial networs, and a set of eywords can be employed to describe ser s personal interests. Sch extension of the graph is nown as attribted graph. In recent years, there is a srge of interest to stdy a variety of graph problems sch as clstering and patten matching in the context of attribted graph where both graph strctre and vertex attribte are considered. In this paper, we stdy the problem of mining cohesive sbgraphs, namely (,r)-cores, on attribted graph. In general, we aim to identify sbgraphs whose vertices are closely related from both strctre and attribte perspectives. Regarding graph strctre, we adopt the -core model [] where each vertex connects to at least other vertices in the sbgraph (strctral constraint). The concept of -core has strong theoretical fondation in social stdy [2], [], and has been widely sed in many applications. From the attribte perspective, a set of vertices is cohesive if their pair-wise similarities are not smaller than a given threshold r (similarity constraint). Ths, given a nmber and a similarity threshold r, we say a connected sbgraph is a (,r)-core if and only if it satisfies both strctral and similarity constraints. To avoid the redndant reslts, we aim to enmerate the maximal (,r)-cores where a (,r)-core Fig.. Motivating Example is maximal if none of its spergraphs is a (,r)-core. Moreover, we are also interested in the maximm (,r)-core which has the largest size among all (,r)-cores. We motivate or wor with the following example. Example (Geo-social Graph): Sppose an organization or company wants to sponsor a nmber of grops to be continosly involved in a particlar activity sch as grop stdy and team collaboration, sally there are two ey criteria to evalate the goodness of the grop: ser engagement and ser similarity. In general, we hope each ser in a grop is liely to remain engaged and is similar to each other. As illstrated in Fig., we can model the IEEE members as a geo-social graph where the location of each individal is represented by a 2 dimensional point and there is a lin between two members if they are friends. According to the social stdy (e.g., [2], []), the incentive of a ser to remain in the grop can be natrally measred by the nmber of friends within the same grop. Moreover, grop members shold be physically close to each other sch that they can reglarly visit each other and do the face-to-face stdy together. Conseqently, two possible grops G 4 and G are not good candidate grops. Althogh the people in G 4 live in the same area, they hardly now each other and hence may be discoraged to be engaged. On the other hand, it is difficlt for people in G to meet each other althogh they are happy to see many friends in the same grop. The maximal (,r)-cores (i.e., G and G 2 in the example) can effectively provide nice candidate grops becase each people has at least friends in the same grop and their mtal distances are bonded. Note that althogh G is also a (,r)-core, it is less interesting becase it is flly contained by a larger grop G. Moreover, it is also desirable to now the largest possible nmber of people engaged in the same candidate grop (i.e., the size of the maximm (,r)-core). With similar argment, (,r)-core mining problem stdied in this paper can also help

2 to identify potential grops for some activities (e.g., grop based prodct promotion) in online social networ based on their friendships and personal interests. The above examples imply an essential need to efficiently enmerate the maximal (,r)-cores and find the maximm (,r)-core. As a matter of fact, sch a demand stems from many real applications sch as social networ, collaboration networ, and genetic interaction networ. In this paper, we formlate the problem of cohesive sbgraph mining and develop efficient and effective algorithms. Note that althogh there are many other cohesive/dense sbgraph models sch as cliqe [4], qasi-cliqe [], - trss [6] and densest sbgraph [7], [8], we find that the - core has been widely adopted as a poplar model to captre the cohesiveness of the sbgraph in graph analysis, which is theoretically nderpinned (e.g., game theory [2], []). Conseqently, we consider these cohesive/dense sbgraph models as a ftre extension of this wor. Challenges and Contribtions A straightforward soltion to or two mining problems is to first compte the -core of the graph and chec the strctral and similarity constraints for each possible sbgraph of the connected -core component. Althogh the linear algorithm for -core comptation [9] is readily available, it is costprohibitive to enmerate the possible sbgraphs. In this paper, we show that the problems of enmerating maximal (,r)-cores and finding maximm (,r)-core are both NP-hard becase of the two constraints involved. Conseqently, it is critical to devise efficient heristics to solve the problems on large scale attribted graph. Since or search process is essentially a bactracing algorithm following the branch and bond paradigm, we develop efficient techniqes to redce the search space of two mining problems. Following is the smmary of or principle contribtions. We advocate a novel cohesive sbgraph model for attribted graph, namely (,r)-core, to captre the cohesiveness of the sbgraph from both graph strctre and vertices attribte perspectives. We show the problems of enmerating maximal (,r)-cores and finding maximm (,r)-core are both NP-hard. (Section II) We develop efficient algorithm to enmerate maximal (,r)- cores with novel candidate prning, early termination and maximal checing techniqes. (Section IV) We also develop efficient algorithm to find the maximm (,r)-core. Particlarly, a novel doble -core based approach is proposed to derive tight pper bond for the size of the candidate soltion. (Section V) Based on some ey observations, we propose three search orders for enmerating maximal (,r)-cores, checing maximal (,r)-cores, and finding maximm (,r)-core algorithms. (Section VI) Or empirical stdies on real-life data demonstrate that interesting cohesive sbgraphs can be identified by maximal (,r)-cores and maximm (,r)-core. The extensive performance evalation shows that the techniqes proposed in this paper can significantly improve the performance of two mining algorithms. (Section VII) Section III presents a naive soltion for the two problems. Notation Definition G a simple attribted graph S,J,R indced sbgraph or corresponding vertices, v vertices in the attribted graph sim(, v) similarity between and v deg(, S) nmber of adjacent vertex of in S deg min(s) minimal degree of the vertices in S DP (, S) nmber of dissimilar vertexes of w.r.t S DP (S) nmber of dissimilar pairs of S SP (, S) nmber of similar vertexes of w.r.t S C candidate vertices set in the search M vertices chosen so far in the search E relevant exclsive vertices set in the search R(M,C) maximal (,r)-cores derived from M C TABLE I. THE SUMMARY OF NOTATIONS Section VIII reviews the related wor and Section IX concldes the paper. II. PRELIMINARIES In this section, we first formally introdce the concept of (,r)-core. Then we show the two problems are NPhard. Table I smmarizes the mathematical notations sed throghot this paper. A. Problem Definition We consider an ndirected, nweighted, and simple attribted graph G = (V, E, A), where V (G) (resp. E(G)) represents the set of vertices (resp. edges) in G, and A(G) denotes the attribte of the vertices. By sim(, v), we denote the similarity/distance of two vertices, v in V (G) which is derived based on their corresponding attribte vales (e.g., sers geo-locations and interests) sch as Jaccard similarity and Eclidean distance. For a given similarity threshold r, we say two vertices are dissimilar (resp. similar) if sim(, v) < r (resp. sim(, v) r). For a vertex and a set S of vertices, DP (, S) (resp. SP (, S)) denotes the nmber of other vertices in S which are dissimilar (resp. similar) to regarding the given similarity threshold r. Meanwhile, we se DP (S) denote the nmber of dissimilar pairs in S. We se S G to denote that S is an indced (attribted) sbgraph of G where E(S) E(G) and A(S) A(G). By deg(, S), we denote the nmber of adjacent vertices of in V (S). Then, deg min (S) is the minimal degree of the vertices in V (S). In this paper, we may se S to denote a set of vertices V (S) when the context is clear. Now we formally introdce two constraints, namely strctral constraint and similarity constraint, which describe the cohesiveness of the vertices of an attribted sbgraph from graph strctre and vertices attribte perspectives, respectively. Definition : Strctral Constraint. Given an integer, a sbgraph S satisfies the strctral constraint if deg(, S) for every vertex V (S),i.e., deg min (S). Definition 2: Similarity Constraint. Given a similarity threshold r, a sbgraph S satisfies the similarity constraint if DP (, S) = 0 for every vertex V (S); that is, there is no dissimilar pair of vertices among V (S) (i.e., DP (S) = 0). Following the convention, when the distance metric (e.g., Eclidean distance) is employed, we say two vertices are similar if their distance is smaller than the given distance threshold.

3 Besides the strctral and similarity constraints, we also enforce that the cohesive sbgraph is a connected graph. Below is the formal definition of (,r)-core. Definition : (,r)-core. Given a connected sbgraph S G, S is a (,r)-core if S satisfies both strctral and similarity constraints. In this paper, we aim to find all maximal (,r)-cores and following is the formal definition. Definition 4: Maximal (,r)-core. Given a connected sbgraph S G, S is a maximal (,r)-core if S is a (,r)-core of G and there exists no (,r)-core S of G sch that S S. We also investigate the problem of finding maximm (,r)- core, which is defined as follows. Definition : Maximm (,r)-core. Let R denote all (,r)-cores of the attribted graph G, a (,r)-core S G is maximm if V (S) V (S ) for every S R. Problem Statement. Given an attribted graph G, an integer and a similarity threshold r, we aim to develop efficient algorithms for the following two fndamental problems: (i) enmerate all maximal (,r)-cores in G; (ii) find the maximm (,r)-core in G. Example 2: In Fig., all vertices are from the -core with =. When we set the distance threshold r to 2 m, G, G 2 and G are three (,r)-cores. Besides, G and G 2 are maximal (,r)-cores while G is flly contained by G. Moreover, G is also the maximm (,r)-core. B. Problem Complexity We can compte -core in linear time by recrsively removing the vertices with degree less than [9]. Nevertheless, two problems stdied in this paper are NP-hard de to the additional similarity constraint. Theorem : The problems of enmerating all maximal (,r)-cores and finding maximm (,r)-core are NP-hard. Proof: Given a graph G(V, E), we constrct an attribted graph G (V, E, A) as follows. We have V (G ) = V (G) and G is a complete graph. More specifically, A() = {e} for each V (G ) where e is the adjacent edges of in G. Sppose the Jaccard similarity is employed and we set sim(, v) = A() A(v) A() A(v) for vertices and v in V (G ). Let similarity threshold r = ɛ where ɛ is an infinite small positive nmber, we have sim(, v) r if the edge (, v) E(G), and otherwise sim(, v) = 0 < r. Since G is a complete graph, i.e., every sbgraph S G with S satisfies the strctral constraint, the problem of deciding whether there is a -cliqe on G can be redced to the problem of finding a (,r)-core on G with r = ɛ, and hence can be solved by the problem of enmerating all maximal (,r)-cores or finding maximm (,r)-core. Conseqently, Theorem holds de to the NP-hardness of the -cliqe problem [4]. A. Naive Soltion III. WARM UP For the ease of nderstanding, we start with a straightforward set enmeration approach where the psedo-code Algorithm : EnmerateMKRC(G,, r) Inpt : G : attribted graph, : degree threshold, r : similarity threshold Otpt : M : Maximal (,r)-cores for each edge (, v) in E(G) do 2 Remove edge (, v) from G If sim(, v) < r; S -core(g); 4 R := ; for each connected sbgraph S in S do 6 NaiveEnm(, S); 7 for each R in R do 8 if there is a R R s.t. R R then 9 R := R \ R; 0 retrn R Algorithm 2: NaiveEnm(M, C) Inpt : M : chosen vertices, C : candidate vertices Otpt : R : (,r)-cores if C = and deg min(m) and DP (M) = 0 then 2 R := R R If R is a connected graph; else 4 choose a vertex in C; NaiveEnm(M, C \ ); /* Expand */; 6 NaiveEnm(M, C \ ); /* Shrin */; is given in Algorithm. At initial stage (Line -2), we remove the edges in E(G) whose corresponding vertices are dissimilar, and then compte the -core of the graph G which reslts in a set S of connected sbgraphs. For each connected sbgraph S S, Procedre NaiveEnm (Line 6) identifies all possible (,r)-cores by enmerating and validating all indced sbgraphs of S. Lines 7-9 eliminate the non-maximal (, r)- cores by checing all (,r)-cores. In NaiveEmn procedre (Algorithm 2), two vertices sets M and C are incrementally maintained to eep the chosen vertices and candidate vertices dring the search, respectively. As shown in Fig. 2, the enmeration process corresponds to a binary search tree with 2 S leaf nodes where each leaf node represents a sbset of S. In each non-leaf node, there are two branches where the chosen vertex will be extended to M (expand branch) and be exclded from M (shrin branch), respectively. Fig. 2. Example of the Search Tree Algorithm Correctness. We can safely remove the dissimilar edges (i.e., their corresponding vertices are dissimilar) since they will not be considered in the (,r)-core comptation de to the similarity constraint. Becase of the strctral constraint and the fact that the sbgraphs S obtained at Line are disjoint, for any (,r)-core R in G, there is one and only one connected sbgraph S from S with R S. Conseqently, we

4 can identify all (,r)-cores by enmerating and validating all possible sbsets of each S in Algorithm 2. Since all possible sbsets of S (i.e., 2 S leaf nodes) are enmerated in the corresponding search tree, any (,r)-core R from the sbgraph S will be accessed exactly once dring the search. Together with the strctral/similarity constraints and maximal property validation, we can otpt with all maximal (,r)-cores. Algorithm can immediately find the maximm (,r)-core by retrning the maximal (,r)-core with the largest size. B. Analysis It is cost-prohibitive to enmerate all sbsets of each - core sbgraph S in Algorithm. Since the search process is essentially a bactracing tree search, we need to develop new techniqes to redce the search space. Below, we briefly discss how to improve the performance of two mining algorithms from the following five perspectives. Redcing candidate size. In Algorithm, each vertex chosen from the candidate C leads to two branches which will explicitly inclde and exclde from the corresponding sb-tree searches. In this paper, we aim to redce the candidate size by explicitly/implicitly exclde vertices in C. Early termination. In Algorithm, we chec the maximal property after the candidate set C is empty. Nevertheless, if we can predicate that all possible (,r)-cores derived in the sbtree are not maximal, the crrent search can be terminated. For instance, sppose we find that an vertex is exclded regarding the crrent search node, bt it can still contribte to every possible (,r)-core obtained from the sbtrees. This implies that we cannot come p with the maximal (,r)-core. In this paper, we careflly maintain some exclded vertices, especially the ones being explicitly exclded at the shrin branch, sch that the search may be terminated early. Efficient maximal chec. The performance of the maximal chec in Algorithm (Lines 7-9) may significantly drop when the nmber of possible (,r)-cores increases. Similar to the above early termination techniqe, we shold chec the maximal property by extending the crrent soltion R based on the exclded vertices. Tight pper bond for the core size. Regarding the problem of maximm (,r)-core, we shold tilize the largest size of the (,r)-cores seen so far to terminate search of some nonpromising sbtrees. To this end, it is critical to estimate the maximal size of the candidate soltion. Good Search Order. In Algorithm, we do not consider the visiting order of the vertices chosen from C as well as the order of two branches. Or empirical stdy shows that an improper search order may reslt in very poor performance, even if all of other advanced techniqes are employed. Considering the different natre of the problems, we shold devise effective search orders for enmeration, finding maximm, and checing maximal algorithms. IV. ENUMERATE MAXIMAL (K,R)-CORES In this Section, we propose the prning techniqes for enmeration algorithm. Specifically, Section IV-A, Section IV-B, and Section IV-C introdce the candidate redcing, early termination, and maximal chec techniqes, respectively. Then the advanced enmeration procedre is presented in Section IV-D. Section IV-E pts the above techniqes together to enmerate all maximal (,r)-cores. Note that we defer the discssion of the search orders of the algorithms to Section VI. A. Redcing Candidate Size In this Sbsection, we present prning techniqes to explicitly/implicitly exclde some vertices from C. () Eliminate Candidates. Intitively, when a vertex in C is assigned (i.e., expand branch) to M or discarded (i.e., shrin branch), we shall recrsively remove some non-promising vertices from C de to the strctral and similarity constraints. Formally speaing, let R(M, C) denote the set of maximal (,r)-cores derived based on M C, we say a vertex C is non-promising if R(M, C) = R(M, C \ ); that is, will contribte to none of the maximal (,r)-cores in the sbtree of the search. Following two prning rles are immediate based on the definition of (,r)-core. Theorem 2: Strctral based prning. We can discard a vertex in C if deg(, M C) <. Theorem : Similarity based prning. We can discard a vertex in C if DP (, M) > 0. Candidate Prning Algorithm. If the chosen vertex is extended to M (i.e., from the expand branch), we first apply the similarity prning rle (Theorem ) to exclde vertices in C which are dissimilar to. Otherwise, none of the vertices will be discarded by similarity constraint when we follow the shrin branch. De to the removal of vertices from C (expand branch) or (shrin branch), we condct the strctral based prning by compting -core for vertices in C M. Note that the search terminates if any vertex in M is discarded. It taes O( C ) time to find dissimilar vertices of from C. De to the -core comptation, the strctral based prning taes linear time to the nmber of edges in the indced graph of M C. Example : In Fig., we have M = { 0 } and C = {,..., 9 }. Sppose is chosen from C. Following the expand branch, will be extended to M and then 9 will be prned de to the similarity constraint. The we need remove 8 as deg( 8, M C) <. Regarding the shrin branch, is explicitly discarded, which in trn leads to the deletion of de to strctral constraint. After applying the candidate prning, following two important invariants always hold at each search node nless the search is terminated. Similarity Invariant. We have DP (, M C) = 0 for every vertex M () That is, M satisfies the similarity constraint regarding M C. Degree Invariant. We have deg min (M C) (2) That is, M and C together satisfy the strctral constraint.

5 M : C : E: Fig.. Prning Candidates Fig. 4. Retaining Candidates Fig.. Early Termination Fig. 6. Maximal Chec (2) Retain Candidates. In addition to explicitly prning some non-promising vertices, we may implicitly redce the candidate size by not choosing some vertices from C. In this paper, we say a vertex is similarity free w.r.t C if is similar to all vertices in C, i.e., DP (, C) = 0. By SF (C) we denote the set of similarity free vertices in C. Intitively, similarity free vertices are welcomed in the sense that they never violate the similarity constraint in the following sbtree search. The theorem below indicates that we do not need to explicitly choose vertices in SF (C) dring the search. Theorem 4: Given that the prning techniqes are applied in each search step, we only need to consider to choose vertices from SF (C). Moreover, M C is a (,r)-core if we have C = SF (C). Proof: For every vertex SF (C), we have DP (, M C) = 0 de to the similarity invariant of M (Eqation ) and the definition of SF (C). Let M and C denote the corresponding chosen set and candidate after is chosen for expansion. Similarly, we have M 2 and C 2 if goes to the shrin branch. We have M 2 M and C 2 C, becase there is no discarded vertices when is extended to M while some vertices may be eliminated de to the removal of in the shrin branch. This implies that R(M 2, C 2 ) R(M, C ). Conseqently, we do not need to explicitly discard as shrin branch of is seless, and hence we can simply retain in C in the following comptation. On the other hand, C = SF (C) implies every vertex in M C satisfies the similarity constraint. Moreover, also satisfies the strctral constraint de to the degree invariant (Eqation 2) of M C. Conseqently, M C is a (,r)-core. Note that a vertex SF (C) may be discarded in the following search de to the strtral constraint. Otherwise, it is moved to M when the condition SF (C) = C holds. For each vertex in C, we can pdate DP (, C) in a passive way when its dissimilar vertices is eliminated from the comptation. Ths it taes O(n d ) time in the worst case where n d denote the nmber of dissimilar pairs in C. Remar : With similar rationale, we can move a vertex directly from C to M if it is similarity free (i.e., SF (C)) and is adjacent to at least vertex in M. As this validation rle is trivial, it will be sed in this paper withot frther mentioning. Example 4: In Fig. 4, sppose we have M = { 0, 2,, 7 }, C = {,, 4, 6, 8, 9 }. In or following rnning examples, and 9 are only dissimilar pairs in the example, and we have SF (C) = {, 4, 6, 8 }. Note that we may directly move { 4, 6 } to M since deg( 4, M) and deg( 6, M). B. Early Termination Trivial Early Termination. There are two trivial early termination rles. As discssed in Section IV-A, we immediately terminate the search if any vertex in M is discarded de to the strctral constraint. On the other hand, the search is terminated if M is disconnected to C. They will be applied in this paper withot frther mentioning. In addition to identifying the sbtree which cannot come p with (,r)-core, we frther redce the search space by identifying the sbtrees which cannot lead to maximal (,r)- core. Intitively, we aim to find some discarded vertices sch that they can immediately integrate to every (,r)-core derived from this search node and hence lead to a larger (,r)-core. In this way, we may safely ct-off the sbtree. By E, we denote the related exclded vertices set for a search tree node where the discarded vertices dring the search are ept if they are similar to M, i.e., DP (v, M) = 0 for every v E and E (M C) =. Moreover, We se SF C (E) denote the similarity free vertices in E w.r.t the set C; that is, DP (, C) = 0 for every SF C (E). Similarly, by SF C E (E) we denote the similarity free vertex in E w.r.t the set E C. Theorem : Early Termination Rle. We terminate the crrent search if one of the following two conditions hold: (i) there is a vertex SF C (E) with deg(, M) ; (ii) there is a set U SF C E (E), sch that deg(, M U) for every vertex U. Proof: (i) We show that every (,r)-core derived from crrent M and C (i.e., R(M, C)) can come p with a larger (,r)-core by attaching the vertex. For any R R, we have deg(, R) becase deg(, M) and M V (R). also satisfies the similarity constraint based on the facts that SF C (E) and R M C. Conseqently, V (R) {} is a (,r)-core. (ii) We have the correctness of condition (ii) with the similar rationale. The ey idea is that for every U, satisfies the strctral constraint becase deg(, M U) ; and also satisfies the similarity constraint becase U SF E C (E) implies that DP (, U R) = 0. Early termination chec. It taes O( E ) time to chec the condition (i) of Theorem by one scan of the vertices in SF C (E) 2. Regarding the condition (ii), we may condct - core comptation on M SF C E (E) to see if a sbset of SF C E (E) are inclded in the -core. The time complexity is O(n e ) where n e is the nmber of edges in the indced graph of A where A = M C E. 2 Note that, instead of one vertex, we may achieve better prning effect by trying a sbset of vertices. Bt this is not cost-effective.

6 Example : In Fig., sppose we have M = { 0,, 7 }, C = {, 8 } and E = {, 2, 4, 6, 9 }. Recall that and 9 are the only dissimilar pair in this rnning example. Therefore, we have SF C (E) = {, 2, 4, 6, 9 } and SF C E (E) = { 2, 4, 6 }. According to Theorem (i), the search is terminated becase there is a vertex 4 SF C (E) with deg( 4, M). We may also terminate the search according to Theorem (ii) becase we have U = { 2, 6 } SF C E (E) sch that deg( 2, M U) and deg( 6, M U). C. Maximal Chec In Algorithm (Lines 7-9), we need validate the maximal property based on all (,r)-cores of G. The cost significantly increases with the nmber and the average size of the (,r)- cores. Similar to the early termination techniqe, we have the following rle to chec maximal property. Theorem 6: Checing Maximal. Given a (,r)-core R, we claim that R is a maximal (,r)-core if there doesn t exist a non-empty set U E sch that R U is a (,r)-core, where E is the exclded vertices set when R is generated. Proof: According to the definition of the exclded vertices set E, it contains all discarded vertices which are similar to M. For any (,r)-core R which flly contains R, we have R E R becase R = M and C = ; that is, the vertices otside of E R cannot contribte to R. Therefore, we can safely claim that R is maximal if we cannot find R among E R. Example 6: In Fig. 6, we have C =, M = { 0, 2, 4,, 6, 7 } and E = {,, 8, 9 }. Here M is a (,r)- core, bt we can frther extend and to M, and come p with a larger (,r)-core. Hence, M is not a maximal (,r)-core. Since the maximal chec algorithm is similar to or advanced enmeration algorithm, we delay the details of the algorithm to Section IV-D. Remar 2: As a matter of fact, the early termination techniqe can be regarded as a lightweight version of maximal chec, which attempts to terminate the search before a (,r)- core is constrcted. D. Advanced Enmeration Method Algorithm : AdvancedEnm(M, C, E) Inpt : M : chosen vertices set, C : candidate vertices set, E : relevant exclded vertices set Otpt : R : maximal (,r)-cores Update C and E based on candidate prning techniqes (Theorem and Theorem 2); 2 Retrn If crrent search can be terminated (Theorem ); if C = SF (C) (Theorem 4) then 4 M := M C; R := R M If ChecMaximal(M, E ) (Theorem 6); 6 else 7 a vertex in C \ SF (C) (Theorem 4); 8 AdvancedEnm(M, C \, E); 9 AdvancedEnm(M, C \, E ); Note that we cannot confirm the maximal property based on the (,r)-cores seen so far. In Algorithm, we present the psedo code of or advanced enmeration algorithm which integrates the techniqes proposed in previos Sbsections. We first apply the candidate prning algorithm in Section IV-A to eliminate some vertices based on strctral/similarity constraints. Note that the search immediately terminates if any vertex in M is eliminated. Besides C, we also pdate E by inclding discarded vertices and removing the ones which are not similar to M. Then Line 2 may terminate the search based on or early termination rle. If the condition C = SF (C) holds, M C is a (,r)-core according to Theorem 4, and we can condct maximal chec (Lines -). Otherwise, Lines 7-9 choose one vertex from C \ SF (C) and contine the search following two branches, where three set M, C and E are pdated accordingly. Algorithm 4: ChecMaximal(M, C) Inpt : M : chosen vertices, C : candidate vertices Otpt : ismax : tre if M is a maximal (,r)-core Update C based on similarity and strctral constraint; 2 if M is a (,r)-core then Exit the algorithm with ismax = false If M < M ; 4 else if C > 0 then a vertex in C; 6 ChecMaximal(M, C \ ); 7 ChecMaximal(M, C \ ); Chec Maximal Algorithm. According to Theorem 6, we need to chec if some vertices in E can be inclded into crrent (,r)-core, denoted by M, and lead to a larger (,r)-core. This can be regarded as the process of frther exploring the search tree by treating E as candidate C (Line of Algorithm ). Algorithm 4 presents the psedo code of or maximal chec algorithm. Similar to the enmeration algorithm, we need to pdate C according to strctral and similarity constraints. Once M is a (,r)-core with size larger than M, we can claim M is not a maximal (,r)-core. Otherwise, we need to explore two possible sbtrees and confirm M is a maximal (,r)-core if we eventally cannot find a larger (,r)-core. E. Enmerate All Maximal (,r)-cores To enmerate all maximal (,r)-cores of G, we need to replace the NaiveEnm procedre (Line 2) in Algorithm by or advanced enmeration method (Algorithm ). Moreover, the naive maximal chec process (Line 7-9) is not necessary since maximal chec is already condcted by or enmeration procedre (Algorithm ). Algorithm Correctness. Section III-A shows the correctness of the Algorithm where none of or prning techniqes is applied. Section IV-A confirms that or candidate size redcing and early termination techniqes can safely exclde some vertices from frther comptation; that is, we can reach all maximal (,r)-cores on the prned search tree. Each nonmaximal (,r)-cores will be discarded by either maximal chec techniqe or early termination techniqe. Moreover, the search order of the algorithms will not affect their correctness. Ths, the correctness of or enmeration algorithm follows. Time Complexity. Let n e and n d denote the total nmber of edges and dissimilar pairs in M C E. According to analysis of candidate size redcing and early termination techniqes, it

7 taes O(n e + n d ) times at each search node 4 in the worst case. Another isse is the comptation of the dissimilar pairs. To spport general similarity metric, we do not consider the indexing of the vertices attribte (e.g., R-tree and M-tree). Ths, we need to materialize the dissimilar pairs based on pair-wise comptation. In or implementation, we compte the pair-wise similarity of the vertices for the following search when the first vertex is inserted into M (i.e., M = ) with time complexity O(n v + n 2 p) where n v is the nmber of vertices in S (Line of Algorithm ) and n p is the nmber of similar pairs of in S. V. FIND MAXIMUM (K,R)-CORE In this Section, we first introdce the pper bond based algorithm to find maximm (,r)-core. Then a novel doble -core approach is proposed to derive tight pper bond of the (,r)-core size. Algorithm : FindMaximm(M, C, E) Inpt : M : chosen vertices set, C : candidate vertices set, E : relevant exclded vertices set Otpt : R : the largest (,r)-core seen so far Update C and E; Early terminate if possible; 2 if KRCoreSizeUB(M, C) R then if C = SF (C) then 4 R := M C; A. Algorithm else choose a vertex in C \ SF (C); if Expansion is preferred then FindMaximm(M, C \, E); FindMaximm(M, C \, E ); else FindMaximm(M, C \, E ); FindMaximm(M, C \, E); Algorithm presents the psedo code of finding the maximm (,r)-core, where R denote the largest (,r)-core seen so far. Compared with the enmeration algorithm (Algorithm ), there are three main differences. (i) Line 2 terminates the search if we find the crrent search is non-promising based on the pper bond of the core size, denoted by KRCore- SizeUB(M,C). (ii) We do not need to validate the maximal property. (iii) Besides the order of vertex visiting, the order of the two branches also matters for qicly identifying the large (,r)-core (Lines 6-2). We delay the discssion to Section VI. To find the maximm (,r)-core in G, we need to replace the NaiveEnm procedre (Line 2) in Algorithm by the above method (Algorithm ), and remove the naive maximal chec part (Line 7-9) of Algorithm. Moreover, in order to qicly find a (,r)-core with large size, we start the algorithm from the sbgraph S which has the vertex with highest degree. The maximm (,r)-core is identified when Algorithm terminates. 4 We can regard the maximal chec processing as the contine of the search by attaching vertices of E to C. Algorithm Correctness. Since Algorithm is essentially an enmeration algorithm with an pper bond based prning techniqe, the correctness of this algorithm is immediate if KRCoreSizeU B(M, C) at Line 2 is calclated correctly. Time Complexity. As shown in the Section V-B, we can efficiently compte the core size pper bond in O(n e + n s ) time where n s is the nmber of similar pairs w.r.t M C E. Conseqently, for each search node the time complexity of the maximm algorithm is same to that of enmeration algorithm. On the other hand, the nmber of search nodes of the maximm algorithm is mch less than that of enmeration algorithm in practice. This is becase the former can not only frther prne sbtrees by core size pper bond bt also avoid the checing maximal process. B. Size Upper Bond of (,r)-core We se R to denote the (,r)-core derived from M C. Then, M + C is obviosly an pper bond of R. However, it is very loose becase it does not consider the similarity constraint. Let G denote a new graph which connects the similar vertices of V (G), namely similarity graph. By J and J, we denote the indced sbgraph of vertices M C from graph G and similarity graph G, respectively. Clearly, we have V (J) = V (J ). Becase J is a cliqe if M is a (,r)-core, we can apply the maximm cliqe size estimation techniqes on J to derive the pper bond of R. K-core and color based methods [0] are two state-of-the-art techniqes for maximm cliqe size estimation. Below, we tae -core based method as an example. -core based pper bond. Let max denote the maximal vale sch that -core of J is not empty. Since a cliqe is also a ( )-core, this implies that we have R max +. Therefore, we may apply the existing -core decomposition approach [9] to compte the maximal core nmber (i.e., max ) on the similarity sbgraph J. At the first glance, both strctral and similarity constraints are sed in the above method becase J itself is a -core (strctral constraint) and we consider the max -core of J (similarity constraint). Nevertheless, we observe that the bond can be tighter if we enforce that two cores on J and J share the same set of vertices. Doble -core based pper bond. We first introdce the concept of doble -core, denoted by (, )-core. Definition 6: (, )-core. Given a graph G and a set of vertices U with U V (G), we say U is a (, )-core of G if deg(, U) and SP (, U) for every U. Based on the fact that a (,r)-core R is also a (, )-core with = R according to the definition of (,r)-core, we have the following theorem. Theorem 7: Let J denote the indced sbgraph based on M and C. If there is a (,r)-core R in J, then there exists a (, )-core with R. Theorem 7 shows that we can derive the pper bond for any possible (,r)-core R in J based on the largest possible vale, denoted by max, for (, )-core of R. Ths, we need to

8 Fig (a) Graph (J) Upper Bond Examples (b) Similarity Graph (J ) develop an efficient algorithm to chec if there is a (, )-core with R. Algorithm 6 shows the details of pper bond (i.e., max) comptation. We se deg[] and deg sim [] to denote the degree and similarity degree (i.e., the nmber of similar pairs from ) of w.r.t M C, respectively. Meanwhile, NB[] (resp. NB sim []) denote the set of adjacent (resp. similar) vertices of. The ey idea is to recrsively mar the r max vale of the vertices ntil we reach the maximal possible vale. Line sorts all vertices based on the increasing order of their similarity degrees. In each iteration, the vertex with the lowest similarity degree already reaches its maximal possible (Line ). Then Line 4 invoes the procedre DobleKcore- Update to remove and decrease the degree (resp. similarity degree) of its neighbors (resp. similarity neighbors) at Lines 9- (resp. Lines 2-). Note that we need to recrsively remove vertices with degree smaller than (Line ) in the procedre. At Line, we need to reorder the vertices in H since their similarity degree vales may be pdated. According to Theorem 7, + is retrned at Line 6 as the core size pper bond. Example 7: In Fig. 7, we have C = {, 2,, 4, } and M = { 0 }. Fig. 7(a) shows edges in J (i.e., indced sbgraph from M C) and Fig. 7(b) shows edges (i.e., similarity pairs) in the similarity graph J. Initially, we have deg sim [ 0 ] =, deg sim [ ] = 4, deg sim [ 2 ] =, deg sim [ ] =, deg sim [ 4 ] = and deg sim [ ] = 4 in similarity graph. Applying -core decomposition on similarity graph J, we get a pper bond since max = 4. Regarding the doble -core techniqe (Algorithm 6), we can find a (,)- core in J with for vertices { 0, 2,, 4 }, and there is no (,4)-core. Ths, we come p with the pper bond 4, which is tighter than. Time Complexity. Let n e and n s denote the nmber of edges in the graph J and similarity graph J, respectively. In Algorithm 6, each edge in J or J will be visited at most once for the pdate of degrees and similarity degrees. Moreover, we can se an array H to maintain vertex where H[i] eep the vertices with similarity degree i. Then the sorting of the vertices can be done in O( J ) time. Conseqently, the time complexity of the algorithm is O(n e + n s ). Algorithm Correctness. Let max() denote the largest vale can contribte to (, )-core of J. By H j, we represent the vertices {} with max() j according to the definition of (, )-core. Then we have H j H i for any i < j. This implies that a vertex on H i with max() = i will not contribte to H j with i < j. Ths, we can prove the correctness by indction. Algorithm 6: DobleKcoreBond(M, C) Inpt : M : vertices chosen, C : candidate vertices Otpt : max : the pper bond for M C H := vertices in M C with increasing order of their similarity degrees; 2 for each H do := deg sim(); 4 DobleKcoreUpdate(,, H); reorder H accordingly; 6 retrn + 7 DobleKcoreUpdate(,, H); 8 Remove from H; 9 for each v NB sim[] H do 0 if deg sim[v] > then deg sim[v] := deg sim[v] ; 2 for each v NB[] H do deg[v] := deg[v] ; 4 if deg[v] < then DobleKcoreUpdate(v,, H); VI. A. Important Measrements SEARCH ORDER In this paper, we need to consider two inds of search orders: (i) vertex visiting order: the order of which vertex is chosen from candidate set C and (ii) branch visiting order: the order of which search branch (expansion or shrining branch) goes first. It is difficlt to find simple heristics or cost fnctions for two problems stdied in this paper becase, generally speaing, finding a maximal/maximm (,r)-core can be regarded as an optimization problem with two constraints. On one side, we need to diminish the dissimilar pairs to satisfy the similarity constraint. This implies that we need to eliminate a considerable nmber of vertices from C. On the flip side, the strctral constraint and the maximal/maximm property is in favor of larger nmber of edges (vertices) in M C; that is, we prefer to eliminate less nmber of vertices from C. To accommodate the above observations, we propose three measrements where M and C denote the pdated M and C after a chosen vertex is extended to M or discarded. : the change of the nmber of dissimilar pairs, where = DP (C) DP (C ) DP (C) Note that we have DP (, M C) = 0 for every M according to the similarity invariant (Eqation ). 2 : the change of the nmber of edges, where 2 = E(M C) E(M C ) E(M C) Recall that E(V ) denote the nmber of edges in the indced graph from the vertices set V. deg(, M C): Degree. We also consider the degree of the vertex as it may reflect its importance. In or implementation, we choose the vertex with highest degree at the initial stage (i.e., M = ). () (4)

9 B. Finding Maximm (,r)-core Since the size of the largest (,r)-core seen so far is critical to redce the search space, we aim to qicly identify the (,r)- core with larger size. One may lie to careflly discard vertices sch that the nmber of edges in M will be redced slowly (i.e., only prefer smaller 2 vale). However, as shown in or empirical stdy, this may reslt in poor performance becase it sally taes many search steps to satisfy the strctral constraint. On the contrary, we may easily fall into the pitfall of finding (,r)-cores with small size if we are only een on removing dissimilar pairs (i.e., only favor larger vale). In or implementation, we se a catios greedy strategy where a parameter λ is sed to mae a trade-off. In particlar, we se λ 2 to measre the goodness of a branch for each vertex in C \ SF (C). In this way, each candidate has two scores. Then the vertex with the highest score will be chosen and its branch with higher score will be explored first (Line 6-2 in Algorithm ). For time efficiency, we only explore vertices within two hops from the candidate vertex when we compte its and 2 vales. It taes O(n c (d 2 + d 2 2)) time where n c denote the nmber of vertices in C\SF (C), and d (resp. d 2 ) stands for the average degree of the vertices in J (resp. J ). C. Enmerating (,r)-cores The ordering strategy of this Sbsection is different than that of finding maximm in two perspectives. (i) We observe that has mch higher impact than 2 in the enmeration problem, so we adopt the -then- 2 strategy; that is, we prefer the larger, and the smaller 2 is considered if there is a tie. This is becase the enmeration algorithm is not een on (,r)-core with very large size since it eventally needs to enmerate all maximal (,r)-cores. Moreover, thans to the early termination techniqe proposed in Section IV-B, we can avoid exploring many non-promising sbtrees misled by the greedy heristic. (ii) We do not need to consider the order of two branches becase we need to explore both branches eventally. Ths, we se the score smmation of two branches to evalate the goodness of a vertex. The complexity of this ordering strategy is the same as the one in Section VI-B. D. Checing Maximal The search order of checing maximal is rather different than that of enmeration and maximm algorithms in the sense that it is more cost-effective for checing maximal algorithm to find a smaller (,r)-core which flly contains the candidate (,r)-core. To this end, we adopt a short-sighted greedy heristic. In particlar, we choose the vertex with the largest degree and the expand branch is always prefered as shown in Algorithm 4. By continosly maintaining a priority qee, we can fetch the vertex with the highest degree in O(log C ) time. VII. PERFORMANCE EVALUATION This Section evalates effectiveness and efficiency of or algorithms throgh comprehensive experiments. Dataset Nodes Edges Avg.Degree Brightite 8,227 88,80. DBLP,66,98 2,922, Gowalla 96,9 90, Poec,62,80 0,622,64 7. TABLE II. STATISTICS OF DATASETS A. Experimental Setting Algorithms. To the best of or nowledge, there is no existing wor investigating maximal (,r)-core enmeration and finding maximm (,r)-core problems. In this paper, we implement and evalate following algorithms. AdvEnm. The advanced enmeration algorithm proposed in Section IV-E which applies all advanced prning techniqes inclding candidate size redcing techniqes (Theorem 2, and 4 in Section IV-A), early termination techniqe (Theorem in Section IV-B) and checing maximal techniqe (Theorem 6 in Section IV-C). Moreover, its best search order ( -then- 2, in Section VI-C) is sed. BasicEnm The basic enmeration algorithm proposed in Algorithm with strctral and similarity constraints based prning techniqes (Theorem 2 and in Section IV-A). The best search order is also applied. AdvMax. The advanced finding maximm (,r)-core algorithm proposed in Section V-A with doble -core pper bond techniqe (Algorithm 6). Its best search order (λ 2, in Section VI-B) is applied. BasicMax. The AdvMax algorithm with the naive core size pper bond: M + C. Datasets. For real datasets are deployed in or experiments. The original data of DBLP is downloaded from ni-trier.de/ and the others are downloaded from stanford.ed/. In DBLP, we consider each athor as a vertex with attribte of attended conferences and pblished jornals list. There is an edge for a pair of athors if they have at least one co-athored paper. We tae Weighted Jaccard Similarity of the corresponding attribtes to measre the similarity between two athors. In Poec, we consider each ser as a vertex with personal interests. We tae Weighted Jaccard Similarity as similarity metric. And there is an edges between two sers if they are friends. share similar personal interests. In Gowalla and Brightite, we consider each ser as a vertex with location information. The graph is constrcted based on the friendship information. We se Eclidean Distance of two locations to measre the similarity of two sers. Table II shows statistics of for datasets. Parameters. We condct experiments in different settings inclding strctral constraint and similarity threshold r. We set reasonable positive integers to, which varies from to. Regarding Gowalla and Brightite, we se Eclidean distance as the distance threshold r, ranging from m to 00 m. It is difficlt for sers to set a proper similarity threshold r over DBLP and Poec becase their pair-wise similarity distribtions are highly sewed. Therefore, we se the thosandth of the pair-wise similarity distribtion with decreasing order which grows from to (i.e., the similarity threshold vale drops). Regarding the search orders of the AdvMax and BasicMax algorithms, we set λ to by defalt. We do not specifically evalate the strctral/similarity based candidate prning techniqes becase they are indispensable for the baseline algorithm.

10 (a) Enmeration Fig. 9. Case Stdy on Gowalla (=0, r=0m) Fig. 8. (b) Maximm Case Stdy on DBLP (=, r= ) All programs are implemented in standard C++ and compiled with G++ in Linx. All experiments are performed on a machine with Intel Xeon 2.GHz CPU and Redhat Linx System. We evalate the performance of an algorithm by the rnning time. To better evalate the difference of the algorithms, we set the time cost to if an algorithm cannot terminate within one hor. We also report the nmber of (,r)- cores and their average and maximm sizes. B. Effectiveness We condct two case stdies on DBLP and Gowalla to demonstrate the effectiveness of or (,r)-core model. We observe that, compared with -core, (,r)-core enables s to find more valable information with the additional similarity constraint on the vertices attribte. DBLP. Fig. 8(a) and (b) show two examples of DBLP with = and r= 6. In Fig. 8(a), all athors come from the same -core based on their co-athorship information (strctral constraint) alone. While there are two (,r)-cores with one common athor named Steven P. Wilder, if we also consider their research bacgrond (similarity constraint). After searching internet, we find that a large nmber of athors in the left (,r)-core are bioinformatician from the Eropean Bioinformatics Institte (EBI) while many of the athors in the right (,r)-core are from the Wellcome Trst Centre. Moreover, It trns ot that Dr. Wilder got his Ph.D. from the Wellcome Trst Centre for Hman Genetics, University of Oxford in 2007, and then has been wored at EBI ever since. Crrently he is a bioinformatician with research focs on genome analysis. Fig. 8(b) depicts the maximm (,r)-core of DBLP with 49 athors. We find that they have intensively co-athored many papers related to a project named Ensembl ( which is one of the well 6 To avoid the noise, we enforce that there are at least three co-athored papers between two connected athors in the case stdy. nown genome browsers. It is very interesting that, althogh the size of maximm (,r)-core changes when we vary and r vales, the athors remaining in the maximm (,r)-core are closely related to the project. Gowalla. Fig. 9 illstrates a set of Gowalla sers who are from the same -core with = 0. By setting r to 0 m, there are two grops of sers each of which is a maximal (,r)-core. We cannot identify them by strctral constraint or similarity constraint alone. We observe that the maximm (,r)-core in Gowalla always appears at Astin when 6. The we realize that this is becase the headqarter of Gowalla is located at Astin. We also report the nmber of (,r)-cores, the average size and maximm size of (,r)-cores on Gowalla and DBLP. Fig. 0(a) and (b) show that both maximm size of (,r)-cores and the nmber of (,r)-cores are mch more sensitive to the change of r or on two datasets, compared with the average size. #(,r)-cores Maximm Size Average Size Size 2K.6K.2K 0.8K 0.4K Fig. 0. 2K r (m) C. Efficiency (a) Gowalla, = (,r)-core Statistics.6K.2K 0.8K 0.4K Nmber of (,r)-cores Size 0.6K 0.K 0.4K 0.K 0.2K 0.K 0 2K 0K 8K 6K 4K 2K (b) DBLP, r= In this Sbsection, we evalate the efficiency of the techniqes proposed in this paper, where time costs of the algorithms are reported Fig.. BasicEnm BE+CR BE+CR+ET AdvEnm r (m) (a) Gowalla, = Evalate Prning Techniqes Nmber of (,r)-cores (b) DBLP, r=