Skyline Community Search in Multi-valued Networks

Transcription

1 Syline Community Search in Multi-value Networs Rong-Hua Li Beijing Institute of Technology Beijing, China Jeffrey Xu Yu Chinese University of Hong Kong Hong Kong, China ABSTRACT Given a scientific collaboration networ, how can we fin a group of collaborators with high research inicator (e.g., h- inex) an iverse research interests? Given a social networ, how can we ientify the communities that have high influence (e.g., PageRan) an also have similar interests to a specifie user? In such settings, the networ can be moele as a multi-value networ where each noe has ( 1) numerical attributes (i.e., h-inex, iversity, PageRan, similarity score, etc.). In the multi-value networ, we want to fin communities that are not ominate by the other communities in terms of numerical attributes. Most existing community search algorithms either completely ignore the numerical attributes or only consier one numerical attribute of the noes. To capture numerical attributes, we propose a novel community moel, calle syline community, base on the concepts of -core an syline. A syline community is a maximal connecte -core that cannot be ominate by the other connecte -cores in the -imensional attribute space. We evelop an elegant space-partition algorithm to efficiently compute the syline communities. Two striing avantages of our algorithm are that (1) its time complexity relies mainly on the size of the answer s (i.e., the number of syline communities), thus it is very efficient if s is small; an (2) it can progressively output the syline communities, which is very useful for applications that only require part of the syline communities. Extensive experiments on both synthetic an real-worl networs emonstrate the efficiency, scalability, an effectiveness of the propose algorithm. CCS CONCEPTS ormation systems Social networs; KEYWORDS Community search; -core; Syline; Massive graphs This wor partially one while Xiao Xiaoui was with NTU. Permission to mae igital or har copies of all or part of this wor for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. Copyrights for components of this wor owne by others than ACM must be honore. Abstracting with creit is permitte. To copy otherwise, or republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. Request permissions from permissions@acm.org. SIGMOD 18, June 10 15, 2018, Houston, TX, USA 2018 Association for Computing Machinery. ACM ISBN /18/06... $ Lu Qin University of Technology Syney, Australia Lu.Qin@uts.eu.au Xiao Xiaoui National University of Singapore Singapore xxiao@ntu.eu.sg Fanghua Ye Sun Yat-Sen University Guangzhou, China smartyfh@outloo.com Nong Xiao an Zibin Zheng Sun Yat-Sen University Guangzhou, China xiaon6@mail.sysu.eu.cn; zhzibin@mail.sysu.eu.cn ACM Reference Format: Rong-Hua Li, Lu Qin, Fanghua Ye, Jeffrey Xu Yu, Xiao Xiaoui, an Nong Xiao an Zibin Zheng Syline Community Search in Multi-value Networs. In Proceeings of 2018 International Conference on Management of Data (SIGMOD 18). ACM, Yor, NY, USA, 17 pages. 1 INTRODUCTION Many real-worl networs such as social networs consist of community structures. Discovering the communities from a networ is a funamental problem in networ analysis. Recently, a query-epenent community iscovery problem calle community search has attracte much attention in the atabase community ue to a large number of applications [10 13, 15, 22]. The goal of the community search problem is to fin those ensely-connecte subgraphs in a networ that satisfy the query conitions. In many real-worl applications, the noes in a networ are often associate with numerical attributes. Such numerical attributes can be obtaine from the profiles of the noes or the statistical information of the noes compute by ifferent networ analysis methos (e.g., the egree, PageRan, influence, etc.). For example, in the Aminer scientific collaboration networ (aminer.org), each author has several numerical attributes, incluing the number of publishe papers, h-inex, activity, iversity, sociability, etc. Such networ ata is typically moele as a multi-value networ where each noe is associate with ( 1) numerical attributes. Given a multi-value networ, how can we fin the communities that are not ominate by the other communities in terms of numerical attributes? For instance, consier a pair of numerical attributes (h-inex, iversity) in the Aminer scientific collaboration networ. How can we fin a group of collaborators with high h-inex an iverse research interests in the Aminer networ? Similarly, consier two numerical attributes (#papers, activity). How can we fin a community in the Aminer networ so that its members not only have a number of publications, but also they are very active in their research areas in recent years? Most existing community search algorithms either completely ignore the numerical attributes or only consier one numerical attribute of the noes [15], an therefore they cannot be irectly use to answer these questions. A naive approach to aress these questions is escribe as follows. First, we can compute the average value (or other linear combinations) over all numeric attributes for each noe in the multi-value networ. Then, base on the average value of each noe, we apply the previous community search algorithm for one numerical attribute [15] to ientify communities. This naive metho, however, cannot fully capture all the interesting communities in the -imensional attribute space. 457

2 This is because a community with high average value in each imension coul also be ominate by the other communities (as confirme in our experiments). To fully characterize all those interesting communities, we propose a novel community moel calle syline community base on the concepts of -core [20] an syline [4]. A syline community is a maximal connecte -core (not necessary the maximal -core as efine in [20]) that is not ominate by the other connecte -cores in the -imensional attribute space (the etaile efinition can be foun in Section 2). Except for fining interesting communities in a scientific collaboration networ, our syline community moel can also be use for many other interesting applications, two of which are introuce as follows. Personalize influential community search. In an online social networ, a user may want to iscover the influential communities with similar interests. For example, in the Faceboo social networ, a football-fan user woul lie to fin the influential football-fan groups, as these groups play important roles for football information issemination in the networ. In this application, we can extract two numeric attributes for each user: the influence an similarity (i.e., the similarities between the query user an the other users in the social networ). By iscovering the syline communities on these numeric attributes, we can obtain the communities that are not ominate by the others in terms of both influence an similarity. Therefore, from the syline communities, the query user can get the esire communities that are not only influential, but also have similar interests to him. Close social groups iscovery in LBSN. The locationbase social networ (LBSN) is a special social networ in which each user is associate with a location. To join similar an close social groups, a user in an LBSN may wish to fin the social groups such that they not only have similar interests, but they are also close to him. Similar query may also help the companies to perform mareting or promotion activities. For example, the fast foo company KFC may want to ientify the social groups that are not only intereste in KFC s foo, but they are also close to the location of KFC. In these applications, we can extract two numeric attributes for each user in the LBSN: (1) the similarity between the query an the user, an (2) the istance between the query location an the user s location. By mining the syline communities on these numeric attributes, we are able to obtain the esire social groups. Contributions. In this paper, we formulate an provie efficient solutions for iscovering syline communities in a multi-value networ. The contributions of this paper are summarize below. community moel. We propose the syline community moel which can be applie to iscover the communities that are not ominate by the other communities in a multivalue networ. To the best of our nowlege, the syline community moel is the first community moel for multivalue networs, an our wor is also the first to introuce syline for community moeling. Novel algorithms. We first evelop an efficient algorithm, calle SylineComm2D, to fin all the syline communities in the 2D case, i.e., = 2. The time complexity of SylineComm2D is O(s(m + n)) where s enotes the number of 2D syline communities (i.e., the answer size), an the space complexity of SylineComm2D is O(m + n + s), which is linear w.r.t. the graph an answer size. To hanle the high-imensional case (i.e., 3), we propose a basic algorithm an an elegant space-partition algorithm to fin the syline communities efficiently. Two striing features of the space-partition algorithm are that (1) its worst-case time complexity is epenent mainly on the answer size, thus it is very efficient when the answer size is not very large; an (2) it is able to progressively output the syline communities uring the execution of the algorithm, an therefore it is very useful for applications that only require part of the syline communities. Extensive experiments. We conuct extensive experiments over both synthetic an real-worl networs to evaluate the propose algorithms. The results show that SylineComm2D is very efficient which taes less than 3.5 secons to compute all the syline communities in a real-worl networ with 2.5 million noes an 7.9 million eges. The results also emonstrate the high efficiency an scalability of the spacepartition algorithm. For example, in the same million-scale networ, the space-partition algorithm is able to erive all the syline communities within 500 secons when = 3. In aition, we conuct comprehensive case stuies to evaluate the effectiveness of the propose syline community moel. The results show that many interesting an meaningful communities can be iscovere using our moel that cannot be foun by other methos. 2 PROBLEM STATEMENT We moel a graph with numerical attributes as a multivalue graph G = (V, E, X), where V ( V = n) an E ( E = m) enote the set of noes an eges respectively, an X ( X = n) is a set of -imensional vectors. In a multi-value graph, each noe v V is associate with a -imensional real-value vector enote by X v = (x v 1,, x v ), where X v X an x v i R. For convenience, we refer to the i-th imension (i = 1,, ) as the x i imension. Suppose without loss of generality that on the x i imension, x v i for all v V form a strict total orer, i.e., x v i x u i for any u v. It is important to note that if this assumption oes not hol, we can easily construct a strict total orer by using the noe ientity to brea ties for any x v i = x u i. The -imensional vector X v represents the values of the noe v w.r.t. ifferent numerical attributes. Base on the multi-value graph moel, we stuy the community search problem in a networ with numerical attributes. To moel the structural cohesiveness, we use the wiely-use -core moel to represent the communities [3, 10, 15, 20, 22]. Specifically, enote by δ(v, G) the egree of noe v in the multi-value graph G. Let H = (V H, E H) be an inuce subgraph of G, i.e., V H V an E H = {(u, v) u, v V H, (u, v) E}. A -core H is an inuce subgraph where each noe v H has a egree at least, i.e., δ(v, H). The maximal -core H is a -core such that there is no super -core containing H. For each noe v V, the core number of v is the maximal such that a -core contains v. Note that the maximal -core is not necessarily connecte. To avoi confusion, we refer to a connecte -core as a -ĉore. Clearly, the traitional -core moel cannot capture the -imensional numerical attributes of a community. Li et al. [15] recently propose an influential community moel which can capture the influence of a community. Each noe in their moel is associate with an influence value, an the influence of a community is efine as the minimum influence value among all the values of its members. In the context of multivalue graph, the influential community moel only wors for the = 1 case, because it consiers the one-imensional 458

3 (1D) numerical attribute of a community. In this paper, we focus on the community search problem for the -imensional case ( 1), where each noe is associate with real values. Below, we first give a novel efinition to characterize a community for the -imensional case ( > 1). The syline community moel. Note that it is nontrivial to generalize the existing community moel for the 1D case (i.e., the influential community moel) [15] to the - imensional case ( > 1). The efinition in [15] of the influential community moel is base on the comparison of the influence values of the communities. However, unlie the 1D case, it is not easy to compare two communities because each community may have ( > 1) values w.r.t. ifferent imensions. As a result, the influential community moel cannot be irectly extene to the -imensional case ( > 1). To overcome this issue, we introuce the omination relationship between two communities, which will be use to efine our syline community moel. Let H = (V H, E H ) be an inuce subgraph of G. Following the efinition of the influence value of a community in [15], we efine the value of H on the x i imension (for i = 1, 2,, ) as f i (H) = min v VH {x v i }. (1) Below, we briefly iscuss why we use the min operator in Eq. (1) to efine f i(h). The motivation of this efinition is the same as that of the influential community moel [15]. By using the min operator, we can ensure that all the members in H on the x i imension have a value no less than f i (H). That is to say, if f i (H) is large, each noe in H must have a large value on the x i imension. This is very useful for excluing outliers in H (i.e., the noes with small values on the x i imension). For example, consier the case of the Aminer scientific collaboration networ. Assume that the x i imension enotes the h-inex of the authors an we want to fin a group of collaborators with high h-inex. Clearly, we can use the above efinition of f i (H) to measure the research impact of a group of collaborators. Definition 1. Let H = (V H, E H ) an H = (V H, E H ) be two communities. If f i (H) f i (H ) for all i = 1,,, an there exists f i (H) < f i (H ) for a certain i, we call that H ominates H, enote by H H. Intuitively, an interesting community in a multi-value graph G shoul be a cohesive subgraph which also cannot be ominate by other communities. For example, in the Aminer networ, assume that we consier two numerical attributes: h-inex an iversity. In this example, we may want to fin a community that is not ominate by other communities in both h-inex an iversity. Base on this intuition, we use the concepts of -core [20] an syline [4] to efine a new community moel in the multivalue graph, calle syline community. To the best of our nowlege, we are the first to use the concept of syline for community moeling. In our moel, we mae use of -core to represent the cohesive property of a community, as it is successfully use for community search applications [10, 11, 15, 22]. Definition 2. Given a multi-value graph G = (V, E, X) an an integer. A syline community with a parameter is an inuce subgraph H = (V H, E H, X H ) of G such that it satisfies the following properties. Cohesive property: H is a -ĉore (i.e., H is a connecte -core); Syline property: there oes not exist an inuce subgraph H of G such that H is a -ĉore an H H ; v1 v2 v5 v3 V4 v6 v1 v2 v3 v4 v5 v6 X1 X2 X Figure 1: Running example Maximal property: there oes not exist an inuce subgraph H of G such that (1) H is a -ĉore, (2) H contains H, an (3) f i(h ) = f i(h) for all i = 1,,. Note that without the maximal property, there coul be a large number of syline communities with the same f values on the imensions. The maximal property in Definition 2 ensures that a syline community is not containe in a larger syline community with the same f values on the imensions, an therefore avoi reunancy. It is worth noting that the -ĉore in our efinition is not necessarily the maximal -core as efine in [16, 20], an the number of -ĉores coul be exponentially large. The following example illustrates the efinition of the syline community. Example 1. Consier the graph shown in Fig. 1. The left panel is a graph with 6 noes, an the right panel shows the values of these noes in three ifferent imensions. Suppose for instance that = 2. Then, by Definition 2, H 1 = {v 1, v 2, v 3 } is a syline community with values f(h 1 ) = (8, 14, 3), because there oes not exist a 2-ĉore that can ominate it, an it is also the maximal subgraph that satisfies the cohesive an syline properties. Similarly, H 2 = {v 2, v 4, v 5, v 6} is a syline community with f(h 2) = (6, 8, 4). The subgraph H 3 = {v 4, v 5, v 6 } is not a syline community, because it is containe in H 2 = {v 2, v 4, v 5, v 6 } which has the same f values as H 3. The subgraph H 4 = {v 2, v 3, v 4, v 5, v 6 } is also not a syline community, as f(h 4) = (6, 8, 3) is ominate by H 1 an H 2. Discussions. Apart from the min operator use in E- q. (1), another two possible operators may be max an sum. These two operators, however, are not appropriate for syline community moeling. The reason is that unlie the min operator, these two operators are monotonic w.r.t. the community size, i.e., a community has a larger f value than its sub-communities. As a result, the answers are always the set of maximal -ĉores of the original multi-value graph, which are inepenent of the numerical attributes of the noes in the graph. The syline community search problem. Given a multivalue graph G = (V, E, X) an an integer, the problem is to fin all the syline communities from G with the parameter. More formally, let H be the set of all -ĉores in G. We aim to compute a subset R of H which is efine as R = {H H H, H H : H H, H H f(h) = f(h )}, where H H enotes that H is a subgraph of H an H H, an f(h) = f(h ) means that f i (H) = f i (H ) for i = 1,,. Note that if = 1, there is only one syline community by Definition 2. Moreover, we can easily show that when = 1, the syline community search problem is equivalent to the problem of fining the top-1 influential community [15]. Therefore, if = 1, we can use the algorithms propose in [15] to fin the syline community efficiently. However, 459

4 when > 1, the problem becomes much harer an the algorithms propose in [15] cannot be use. Below, we iscuss the challenges of our problem. Challenges. The challenges to solve our problem are t- wofol. First, the number of -ĉores (i.e., connecte - cores) in a multi-value networ can be exponentially large. Thus, it is intractable to enumerate all the -ĉores to i- entify the syline communities. Secon, unlie the traitional -imensional syline computation problem [4], the -imensional ata points in our problem, which correspon to the -ĉores, are not given. As a result, it is challenging to evise an efficient algorithm to etect the syline communities without enumerating all the -ĉores. In the following sections, we will evelop several efficient algorithms to overcome these challenges. 3 ALGORITHM FOR = 2 In this section, we propose an efficient algorithm to fin all syline communities in the 2D case (i.e., = 2). The algorithm for the 2D case will be use as the builing-bloc when we process the > 2 case. In the rest of this paper, we assume without loss of generality that the values of the noes on all imensions are positive (i.e., x u i > 0 for all u V an i = 1,, ). For example, in the Aminer networ, the numerical attributes such as h-inex an the number of papers are positive. Note that if this assumption oes not hol, we can revise x u i by x u i = x u i min v V {x v i } + ϵ > 0 for all i = 1,, an u V which oes not affect the correctness of all the propose algorithms (ϵ is a positive constant). Recall that in the 2D case each syline community H has two values (f 1 (H), f 2 (H)). If H =, we efine f i (H) = 0 for i = 1, 2. For each syline community H, we mainly focus on evising an algorithm to compute (f 1(H), f 2(H)), because we can easily extract the community from G base on these two values. The basic iea of our algorithm is as follows. First, we only consier the x 2 imension in graph G, an compute the maximal f 2 value, enote by f2, among all the -ĉores. We fin a maximal -ĉore (enote by H) which achieves f2 by recursively eleting the noe with the smallest x 2 value until the graph contains no -core. Note that the maximal -ĉore H may not be a syline community. This is because H coul be ominate by a community H which has the same f 2 value, but a larger f 1 value than H. However, such a community H must be containe in H, because it has the same f 2 value as H, which is maximal over all the -ĉores. Therefore, to fin a syline community, we can apply the same proceure to compute the maximal f 1 value, enote by f1, over all the connecte sub--cores containe in H. The resulting -ĉore enote by H 1 must be a syline community with values (f 1(H 1), f 2(H 1)), where f 1(H 1) = f1 an f 2(H 1) = f2. This is because f2 is maximal among all the -ĉores, thus no other -ĉore that can ominate it on the x 2 imension. On the other han, f1 is maximal over all the -ĉores with the same f2 value, thus no -ĉore exists that can ominate it. S- ince the above recursive proceure ensures that the resulting -ĉore is maximal, it must be a syline community. Using the above metho, we can fin one syline community which has the maximal f 2 value of all the syline communities. The challenge is how to fin the other syline communities. We can tacle this challenge base on the following result. All the proofs in this paper can be foun in the Appenix. Lemma 1. Let H 1 with values (f 1 (H 1 ), f 2 (H 1 )) be the syline community that has the maximal f 2 value over all the syline communities. The noes in G whose x 1 values are no larger than f 1 (H 1 ) cannot then be containe in the other syline communities. Base on Lemma 1, we can shrin the graph G by removing all the noes whose x 1 values are no larger than f 1 (H 1 ). We invoe the above proceure in the reuce graph to fin the next syline community H 2. It shoul be note that H 2 must be ifferent from H 1, because its f 1 value is larger than f 1(H 1). We can iteratively invoe this proceure to fin all the syline communities until the reuce graph contains no -core. Algorithm 2 implements the above proceure. In Algorithm 2, I enotes the set of constraints. Initially, I = {x 1 > 0, x 2 > 0}, which means that no constraint is active (because x u i > 0 for all u V an i = 1, 2 by our assumption). F enotes the set of fixe noes. For the 2D case, there is no nee to fix noes, thus F =. However, for the > 2 case, we will use the set F to maintain the fixe noes (see Sections 4 an 5), which cannot be elete by the algorithm. To fin all the 2D syline communities, we can invoe SylineComm2D(G, {x 1 > 0, x 2 > 0}, ). The etaile algorithm is escribe as follows. First, Algorithm 2 invoes Algorithm 1 to calculate the maximal f 2 over all the syline communities (line 1). Specifically, Algorithm 1 first eletes all the invali noes (i.e., shrins the graph, line 1 in Algorithm 1), an then computes the maximal -core H (line 2 in Algorithm 1). The algorithm then recursively eletes the noe with the smallest x 2 value until H = (lines 6-12 in Algorithm 1). The algorithm returns the maximal f 2 value over all the -ĉores subject to the constraints I. After etermining f 2, Algorithm 2 iteratively computes the syline communities in lines 2-5. In line 3, Algorithm 2 first refines I by the constraint x 2 f 2. Here we use a notation to enote the refinement operator. In particular, if I = {x 1 > 0, x 2 > 0}, then Ĩ = I {x 2 f 2 } = {x 1 > 0, x 2 f 2 }, because the constraint x 2 > 0 in I is refine by x 2 f 2. Then, Algorithm 2 calls Algorithm 1 with the refine constraints Ĩ to calculate the maximal f 1 value (line 3). It shoul be note that in Algorithm 1, the constraint x 2 f 2 ensures that all the noes with x 2 values smaller than f 2 are elete. Therefore, the obtaine f 1 value in line 3 (Algorithm 2) is the maximal f 1 value over all the -ĉores with the same f 2 value. By efinition, there is a syline community that has values (f 1, f 2). In line 4, the algorithm as (f 1, f 2 ) into the answer set. Subsequently, in line 5, the algorithm refines the constraint by (x 1 > f 1 ), because noes with x 1 values no larger than f 1 cannot be inclue in the uniscovere syline communities (see Lemma 1). Then, the algorithm calculates the maximal f 2 value subject to the refine constraints Ĩ. After obtaining f2, the algorithm iteratively applies the same proceure to compute the next syline community. The algorithm terminates when f 2 = 0, which means that no -core exist that satisfies the refine constraints. The correctness of Algorithm 2 is shown in the following theorem. Theorem 1. Algorithm 2 correctly computes all the 2D syline communities. We analyze the time an space complexity of Algorithm 2 in the following theorem. Theorem 2. Let s be the number of syline communities in G. Then, the worst case time an space complexity of Algorithm 2 is O(s(m + n)) an O(m + n + s) respectively. 460

5 Algorithm 1 DimMax(G, I, F, ) Input: A multi-value graph G, constraints I, fixe noes set F,. Output: The maximal value on the -th imension. 1: G elete all the noes in G that violate the constraints I; 2: H compute the maximal -core in G; 3: if F then 4: H the maximal -core in H that contains F; 5: Compute f (H) base on Eq. (1); 6: f f (H); visit(u) 0 for all u H; 7: while H o 8: Let u H be the smallest-value noe on the x imension; 9: flag 1; flag DFS(u); 10: if flag = 0 then brea; 11: if F = then 12: H the maximal -core in H that contains F; 13: f max{f, f (H)}; 14: return f ; 15: Proceure DFS (u) 16: if u F then return 0; {// the fixe noe cannot be elete} 17: visit(u) 1; 18: Let N(u, H) be the neighborhoo of u in H; 19: Let δ(u, H) be the egree of u in H; 20: for all v N(u, H) an visit(v) = 0 o 21: δ(v, H) δ(v, H) 1; 22: if δ(v, H) < then DFS(v); 23: Delete u from H; 24: return 1; Algorithm 2 SylineComm2D(G, I, F) Input: A multi-value graph G, constraints I, fixe noes set F. Output: Syline Communities in G. 1: f 2 DimMax (G, I, F, 2); R ; 2: while f 2 > 0 o 3: Ĩ I {x 2 f 2 }; f 1 DimMax (G, Ĩ, F, 1); 4: R R {(f 1, f 2 )}; 5: Ĩ I {x 1 > f 1 }; f 2 DimMax (G, Ĩ, F, 2); 6: return R; Note that in the 2D case, the total number of syline communities s is boune by n, because the number of f 2 values of the syline communities is boune by n. Thus, the time an space complexity of Algorithm 2 is also boune by O(n(m + n)) an O(m + n) respectively. In our experiments, we will show that s is typically very small, thus our algorithm is very efficient in practice. 4 THE BASIC ALGORITHM FOR 3 Recall that Algorithm 2 can iteratively compute all the 2D syline communities once it has foun the first syline community. To fin the first syline community, Algorithm 2 computes the maximal f 2 value, an applies a similar proceure to etermine the f 1 value. Unfortunately, this iea oes not wor in the case of 3. This is because for the 3 case, we o not now how to etermine the remaining values (f 1 an f 2 ) of a syline community after computing the maximal f 3 value. Furthermore, even if we can fin the first syline community for the 3 case, it is still quite nontrivial to fin all the remaining syline communities. Below, we evelop a basic algorithm to tacle these challenges base on an in-epth analysis of the syline community moel. For convenience, we first evise a basic algorithm to hanle the 3D case (i.e., = 3), an then we exten this algorithm to hanle the > 3 case. 4.1 Hanling the = 3 case The imension reuction iea. Our algorithm is base on a imension reuction iea which involves three steps. First, we erive all the possible f 3 values that the syline communities may have on the x 3 imension. Secon, for each possible f 3 value, enote by f 3, we fin all the 2D syline communities on the x 1 an x 2 imensions such that the f 3 values of these syline communities equal f3. Here we refer to a syline community base on the x 1 an x 2 imensions as a 2D syline community, an all those base on three imensions as 3D syline communities. Finally, we merge the resulting syline communities for all possible f 3 values, an invoe a traitional syline algorithm [4, 14] to etermine all the 3D syline communities. Let F 3 be the set of all the possible f 3 values. For the first step, a naive solution is to set F 3 to be the set of all the x 3 values of the noes in G, because the f 3 values of all the syline communities must tae from the set of all the x 3 values of noes. The secon step can be implemente as follows. We remove all the noes whose x 3 values are smaller than f3, an fix the noe u with x u 3 = f3 (a fixe noe enotes that the noe cannot be elete by the algorithm). Note that only one noe u with x u 3 = f3 can be fixe, because all the x 3 values form a total orer by our assumption. We invoe SylineComm2D with constraint I = {x 3 f3 } an fixepoint set F = {u} to compute the 2D syline communities on the x 1 an x 2 imensions. It can be easily shown that the resulting communities are 2D syline communities (on the x 1 an x 2 imensions) with f 3 values equaling f3. An improve implementation. The naive implementation is inefficient because it nees to invoe the SylineComm2D algorithm F 3 = n times. Here we propose an improve implementation base on an interesting connection between our problem an the influential community search problem [15]. Recall that the influential community moel is tailore to the networ with only one numerical attribute [15]. In a multi-value networ with numerical attributes, the influential community can be efine on each imension x i (i = 1,, ). Specifically, on the x i imension, a community H is calle an influential community [15] if (1) it is a connecte -core (i.e., -ĉore), an (2) there oes not exist a -ĉore H such that H contains H an f i(h ) = f i(h) (see Eq. (1)). Let T i be the set of values of all the influential communities efine on the x i imension. T i can be compute using the influential community search algorithm escribe in Algorithm 3 [15]. In particular, Algorithm 3 iteratively eletes the smallest-value noe on the x i imension, an recors the f i values of the current maximal -ĉore which correspons to the value of an influential community [15]. Note that both the syline communities an influential communities are -ĉores satisfying a maximal property; an both the f i values of the syline communities an the values of the influential communities on the x i imension (i.e., T i) are efine by the min operator (Eq. (1)). Intuitively, the f i values of the syline communities shoul be containe in T i because T i consists of all the possible values of the maximal -ĉores efine by the min operator. Inee, Lemma 2 shows that the f i values of the syline communities must be taen from T i. Lemma 2. For each imension x i (i = 1,, ), the f i values of all the syline communities are containe in the set T i which is compute by Algorithm 3. Base on Lemma 2, we can set F 3 = T 3 in our algorithm. Since T 3 n an can be substantially smaller than n in practice [15], this improve implementation is much more efficient than the naive implementation. Our algorithm is epicte in Algorithm 4. In line 1, we compute F 3 by invoing Algorithm 3 base on the x 3 imension. In lines 2-7, we calculate the 2D syline communities for each f 3 F 3. The algorithm first fixes the noe u that 461

6 Algorithm 3 [15]Comm(G, ) Input: A multi-value graph G,. Output: All the f values. 1: H the maximal -core of G; T ; 2: while H o 3: Let H be the maximal connecte component of H with smallest f value, enote by f ; 4: T T {f }; Let u H be the noe that x u = f ; 5: CommDFS(u); 6: return T ; 7: Proceure CommDFS(u) 8: for all v N(u, H) o 9: Delete ege (u, v) from H; 10: if δ(v, H) < then CommDFS(v); 11: Delete noe u from H; Algorithm 4 3D(G, I, F) Input: A multi-value graph G, constraints I, fixe noes set F. Output: Syline Communities in G. 1: F 3 Comm(G, 3); R ; 2: for all f 3 F 3 o 3: Let u be the noe that x u 3 = f 3; F F {u}; 4: Ĩ I {x 3 f 3 }; 5: T SylineComm2D(G, Ĩ, F); {// Compute syline communities base on the first two imensions.} 6: for all (f 1, f 2 ) T o 7: R R {f 1, f 2, f 3 }; 8: return Syline(R); Algorithm 5 HighD(G, I, F, ) Input: A multi-value graph G, constraints I, fixe noes set F. Output: Syline Communities in G. 1: if = 3 then return 3D(G, I, F); 2: F Comm(G, ); R ; 3: for all f F o 4: Let u be the noe with x u = f ; F F {u}; 5: Ĩ I {x f }; 6: T HighD(G, Ĩ, F, 1); 7: for all (f 1,, f 1 ) T o 8: R R {f 1,, f 1, f }; 9: return Syline(R); x u 3 = f 3 (line 3), because the noe u must be containe in all the 2D syline communities whose values on the x 3 imension are equal to f 3. In line 4, the algorithm refines the constraint I by {x 3 f 3 } which inicates that the noes whose x 3 values are smaller than f 3 will be remove. The algorithm then invoes Algorithm 2 to compute the 2D syline communities base on the x 1 an x 2 imensions (line 5). Lastly, the algorithm combines the results (lines 6-7), an applies a traitional syline algorithm to etermine all the 3D syline communities (line 8). We analyze the correctness an complexity of Algorithm 4 in Theorem 3. Theorem 3. Algorithm 4 correctly fins all the 3D syline communities, an the worst-case time an space complexity of Algorithm 4 is O(n 2 (m + n)) an O(n 2 ) respectively. 4.2 Hanling the > 3 case We generalize Algorithm 4 to hanle the > 3 case in Algorithm 5. The general proceure is very similar to Algorithm 4. The main ifference is that the algorithm recursively invoes itself with a parameter 1 to compute the ( 1)- imensional syline communities (line 6). The recursive proceure terminates when = 3 (line 1), because we invoe Algorithm 4 to compute the 3D syline communities. The correctness analysis of Algorithm 5 is also similar to that of Algorithm 4, thus we omit the etails for brevity. Below, we analyze the complexity of Algorithm 5. Theorem 4. For 3, the worst-case time an space complexity of Algorithm 5 is O(n 1 (m+n+( 1) log 3 n)) an O(n 1 ) respectively. Note that the above complexity is the worst-case complexity. In practice, since the number of syline communities in the -imensional space is much smaller than O(n 1 ) an also is very small (e.g., 5), our basic algorithm wors for many real-worl networs as shown in the experiments. 4.3 A pruning rule We present a simple but efficient pruning rule to spee up the basic algorithms. When we fix the noe u with x u = f in both Algorithm 4 (line 3) an Algorithm 5 (line 4), we can use the -imensional values of noe u for pruning, i.e., X u = {x u 1,, x u }. Since all the ( 1)-imensional syline communities compute by fixing u must contain u, the values of u form an upper boun of all those syline communities. Therefore, when fixing u, we first chec whether u is ominate by the alreay compute syline communities. If this is the case, we o not nee to recursively invoe the algorithm to compute the ( 1)-imensional syline communities (line 5 in Algorithm 4 an line 6 in Algorithm 5), because those communities are efinitely ominate by the alreay compute syline communities. Using this pruning rule, we can save a number of recursive calls in the basic algorithms. To implement this pruning rule, we first sort the set F in a ecreasing orer, an then compute the syline communities for each f F following this orer. When we fix u (line 3 in Algorithm 4 an line 4 in Algorithm 5), we chec whether (x u 1,, x u 1) is ominate by the alreay compute ( 1)-imensional syline communities. If this is the case, u is ominate because the f values of all the alreay compute ( 1)-imensional syline communities are larger than x u, an thus there is no nee to recursively invoe the algorithm to calculate the ( 1)-imensional syline communities with fixe u. 5 THE SPACE-PARTITION METHOD Although the pruning rule significantly accelerates the basic algorithm, it is still inefficient for the > 3 case because it nees to compute a large number of invali syline communities. In this section, we propose a more efficient algorithm base on a novel space-partition iea. The worst-case time complexity of our new algorithm relies mainly on the number of syline communities, i.e., the answer size, thus it is very efficient if the answer size is not very large. Unlie the basic algorithm, the new algorithm outputs the syline communities progressively, an no invali syline community is generate. This progressive feature is very useful when the applications only nee to compute part of the syline communities. Below, we first consier the = 3 case, an then we exten our algorithm to hanle the > 3 case. 5.1 Hanling the = 3 case The ey iea. The basic iea of our new algorithm is that we compute the syline communities following the ecreasing orer of the f 3 values of the 3D syline communities. In other wors, we first compute the set of 3D syline communities that have the largest f 3 value, an then calculate the 3D syline communities having the secon-largest f 3 value, etc. The challenge is how to implement this proceure without computing invali syline communities. Our solution is etaile as follows. Let H be the set of 3D syline communities that have the maximum f 3 value. H can be easily compute by the following proceure. First, we invoe the DimMax algorithm 462

7 (Algorithm 1) with constraint I = {x 1 > 0, x 2 > 0} to erive the largest f 3 value in G, enote by f3. Then, we fix the noe u with x u 3 = f3 an invoe SylineComm2D with constraint I = {x 1 > 0, x 2 > 0} an fixe-point set F = {u} to compute the 2D syline communities on the x 1 an x 2 imensions. Clearly, all the resulting communities are vali 3D syline communities having the largest f 3 value. Since f3 is maximum, the f 3 values of the remaining 3D syline communities in G must be smaller than f3. Hence, the (f 1, f 2 ) values of the remaining 3D syline communities cannot be ominate by those of the syline communities in H. By the syline property, all the (f 1, f 2 ) values of the 3D syline communities in H form a staircase-lie shape in the 2D space. For ease of unerstaning, we use an example shown in Fig. 2(a) to illustrate our iea. In this example, we have three 3D syline communities in H = {H 1, H 2, H 3}, an the label enotes the 3D syline communities on the x 1 an x 2 imensions. Clearly, the space below the staircaselie shape is ominate by the syline communities in H which can be safely prune. The (f 1, f 2 ) values of the remaining 3D syline communities must be locate on the top of the staircase-lie shape. Obviously, the maximum f 3 value of the remaining 3D syline communities is the secon-largest f 3 value over all the 3D syline communities. However, it is challenging to erive the maximum f 3 value of the remaining 3D syline communities. This is because (1) the (f 1, f 2) values of the remaining 3D syline communities are locate on the top of the staircase-lie shape which forms an irregular 2D space (see Fig. 2(a)), an (2) we cannot irectly apply DimMax to compute the maximum f 3 value given that the (f 1, f 2 ) values are locate in such an irregular 2D space. To overcome this challenge, we propose a space-partition approach. The ey step of our approach is to partition the irregular 2D space (the 2D space on the top of the staircaselie shape) into several overlappe regular 2D subspaces, in which the maximum f 3 value can be compute by DimMax. Formally, the regular 2D space is efine as follows. Definition 3. Given two imensions x 1 an x 2, a 2D space is calle a regular 2D space if an only if it can be represente by a pair of constraints (x 1 > f 1, x 2 > f 2 ), where (f 1, f 2) is a 2D point. Note that the above efinition of the regular 2D space can be irectly extene to the high-imensional case. Again, we use the example shown in Fig. 2 to illustrate the spacepartition iea. In this example, the irregular 2D space in Fig. 2(a) is ivie into four overlappe regular subspaces as shown in Fig. 2(b) where each 2D point C i correspons to a regular subspace. For a regular 2D space represente by (x 1 > f 1, x 2 > f 2), we can compute the maximum f 3 value in that space by invoing DimMax with constraint I = {x 1 > f 1, x 2 > f 2}. As a result, we are able to erive the maximum f 3 value in the irregular 2D space, enote by f 3, using such a spacepartition metho. Furthermore, we can also ientify the regular 2D subspaces in which the maximum f 3 value achieves f 3. After obtaining f 3 an the corresponing regular 2D subspaces, the SylineComm2D algorithm can be use to compute the 2D syline communities in that regular 2D subspace. We claim that the compute 2D syline communities are also the 3D syline communities. The reasons are as follows. First, the (f 1, f 2 ) values of these 2D syline communities cannot be ominate by the previously compute syline communities (i.e., H), because they are locate on the top of the staircase-lie shape forme by the alreay compute x2 0 H1 H2 H3 x1 (a) A syline example (b) Corner points an subspaces Figure 2: Illustration of the space-partition iea C1 x2 Algorithm 6 The Space-Partition Framewor 1: Let P be the initial 2D space represente by (x 1 > 0, x 2 > 0); 2: R ; 3: while P o 4: S partition P into a set of overlappe regular subspaces; 5: (f3, T ) ientify the largest f 3 value (f3 ) an the corresponing regular subspaces (T ) in S by the DimMax algorithm; 6: H compute the set of 2D syline communities in T by SylineComm2D; 7: R R H; 8: P prune the 2D space ominate by H in P ; 9: return R; 3D syline communities (base on the x 1 an x 2 imensions). Secon, since our algorithm computes the 3D syline communities following the ecreasing orer of the f 3 values, the f 3 values of the uniscovere 3D syline communities must be smaller than f 3. As a result, all the compute 2D syline communities are vali 3D syline communities. Once we obtain a set of new 3D syline communities, we can iteratively use the same space-partition metho to compute the remaining 3D syline communities. The general framewor of our space-partition metho is shown in Algorithm 6. To implement our framewor, the remaining question is how can we ivie the irregular 2D space into several overlappe regular 2D subspaces? Below, we efine two important concepts calle MIN syline an corner point which will be use to partition the irregular 2D space. Definition 4. Let L be a set of -imensional points. The MIN syline of L, enote by A, contains all the points in L that satisfy the following conition. For any point x = (x 1,, x ) A, there oes not exist a point y = (y 1,, y ) L\A such that y i x i for all i = 1,, an y i < x i for a certain i = 1,,. Definition 5. Let R be a set of syline points in the - imensional space. Let B be the set of all the cross points in the bounary of the -imensional staircase-lie shape forme by the syline. The corner point set C is the MIN syline compute over all the cross points in B. Reconsier the graph shown in Fig. 2(a). There are seven cross points in the bounary of the staircase-lie shape (incluing three syline points). We compute the MIN syline over all the cross points. Clearly, we can obtain four corner points which are labele by in Fig. 2(b). Note that the coorinates of the corner points can be etermine by the (f 1, f 2 ) values of the 3D syline communities. For example, in Fig. 2(b), the coorinates of the corner point C 3 can be etermine by the 3D syline communities H 2 an H 3, which are (f 1(H 2), f 2(H 3)). Base on the corner points, we can easily ivie the irregular 2D space into several overlappe regular 2D subspaces as illustrate in Fig. 2(b). Note that each corner point correspons to a regular 2D subspace. For the corner point C 3 = (f 1 (H 2 ), f 2 (H 3 )) for example, the corresponing regular 2D subspace can be represente by (x 1 > f 1 (H 2 ), x 2 > f 2 (H 3 )). 0 C2 C3 C4 x1 463

8 Algorithm 7 3D(G, I, F) Input: A multi-value graph G, constraints I, fixe noes set F. Output: Syline Communities in G. 1: Result R ; Priority Queue Q ; C {(0, 0)}; 2: if DimMax(G, I, F, 3) > 0 then 3: Q.Push((0, 0), DimMax(G, I, F, 3)); 4: while Q o 5: c 3 Q.MaxVal(); S ; 6: while Q.MaxVal() = c 3 o 7: ((c 1, c 2 ), c 3 ) Q.Pop(); {// c 3 is the priority of (c 1, c 2 ) Q} 8: Ĩ I {x 1 > c 1, x 2 > c 2 }; 9: Let u be the noe that x u 3 = c 3; F F {u}; 10: S tmp SylineComm2D(G, Ĩ, F); 11: S S S tmp ; 12: for all (c 1, c 2 ) S o 13: R R {(c 1, c 2, c 3 )}; 14: for all s S o C UpateCornerPoints(C, s, 2); 15: for all (c 1, c 2 ) C o 16: Ĩ I {x 1 > c 1, x 2 > c 2 }; 17: if (c 1, c 2 ) / Q an DimMax(G, Ĩ, F, 3) > 0 then 18: Q.Push((c 1, c 2 ), DimMax(G, Ĩ, F, 3)); 19: return R; Algorithm 8 UpateCornerPoints(C, s, ) Input: Output: Corner Points C, Syline Point s = (x 1,..., x ),. Upate Corner Points by Aing s. 1: for i = 1 to o 2: 3: C i ; for all (c = (x 1,..., x )) C s.t. x j < x j for 1 j o 4: C C \ {c}; replace x i with x i in c; 5: 6: C i C i {c}; C i Syline(C i,, MIN); {// compute by classic syline algorithms} 7: return C C i... C ; Implementation etails. The etaile implementation of our algorithm is shown in Algorithm 7. In Algorithm 7, we use a priority queue Q to maintain all the regular 2D subspaces where the priority of the subspace is the maximum f 3 value in that subspace. Specifically, in the priority queue Q, we use a pair ((c 1, c 2 ), c 3 ) to enote a regular 2D subspace, where (c 1, c 2 ) enotes the corner point corresponing to the regular 2D subspace an c 3 is the priority of that subspace (i.e., the maximal f 3 value in that subspace). Initially, the algorithm pushes the initial regular 2D space into Q (lines 1-3). Then, the algorithm iteratively computes the syline communities base on the best-first strategy (lines 4-18). Note that the algorithm can erive syline communities following a ecreasing orer of the f 3 values base on the best-first strategy. In each iteration, the algorithm first fins the maximum priority from Q an sets c 3 as the maximum priority (line 5). The algorithm then iteratively pops the regular 2D space whose priority equals c 3 from Q (line 7). For a poppe regular 2D space ((c 1, c 2 ), c 3 ), the algorithm refines the constraint I by {x 1 > c 1, x 2 > c 2 } (line 8), an fixes the noe u with x u 3 = c 3 (line 9). The algorithm invoes SylineComm2D with the refine constraint an fixe noe u to compute the 2D syline communities (line 10). All the compute 2D syline communities are recore in S (lines 10-11). Since all the compute 2D syline communities must be 3D syline communities by the best-first strategy, the algorithm as all these compute 2D syline communities into the answer set R (lines 12-13). The algorithm then upates the corner points base on the newly-calculate syline communities in this iteration (line 14). To compute the corner points, we evise an incremental algorithm which is epicte in Algorithm 8. Specifically, for x2 0 x1 (a) Before upating (b) After upating Figure 3: Illustration of the corner points upating each syline community s S, the algorithm incrementally upates the previously-compute corner points set C (line 14 in Algorithm 7) by invoing Algorithm 8. Clearly, if the previous-compute corner point c is completely ominate by the syline point s, this corner point must be below the staircase-lie shape forme by the upate syline after aing s. Here we call a point x = (x 1,, x ) completely ominating a point y = (y 1,, y ) if an only if x i > y i for all i = 1,,. For example, consier the corner points shown in Fig. 3(a). The re enotes the newly-ae syline point s. In this example, there is one corner point that is completely ominate by s. Let C be the set of corner points completely ominate by s. We remove all the corner points in C, because these corner points are no longer the cross points. The completely-ominate corner points in C can be use to compute the new cross points generate by aing s. For each ominate corner point c C, we obtain a cross point by replacing the x i coorinate of c with that of s an eeping the other coorinates of c unchange. Clearly, for each completely-ominate corner point, we obtain new cross points. After obtaining all the cross points, we compute the MIN syline to get the upate corner points. Reconsier the example shown in Fig. 3. In this example, we obtain two cross points which are also the corner points as shown in Fig. 3(b). Algorithm 8 etails this proceure. In Algorithm 8, we compute the MIN syline in each imension (line 7 in Algorithm 8), because the cross points generate in ifferent imensions cannot be ominate w.r.t. each other. Moreover, the remaining corner points in C (the corner points that are not completely ominate by s ) cannot be ominate by the newly-compute corner points. Thus, the algorithm outputs the union of all corner points, forming a MIN syline. After upating C, Algorithm 7 pushes the newly-generate regular spaces into Q (lines 15-18), an then iteratively computes the syline communities base on the best-first strategy, until Q = an the algorithm terminates. The correctness of Algorithm 7 is analyze in Theorem 5. Theorem 5. Algorithm 7 correctly computes all the 3D syline communities. We analyze the complexity of Algorithm 7 in Theorem 6. Theorem 6. Let s be the number of 3D syline communities. The worst-case time an space complexity of Algorithm 7 is O(s 2 (m + n)) an O(m + n + s) respectively. An improve 3D algorithm. Due to the overlappe spacepartition metho, a syline community may be recompute in Algorithm 7 if its (f 1, f 2 ) values are locate in two regular 2D subspaces with the same priority (see lines 6-11 in Algorithm 7). To avoi such reunant computations, we propose an improve algorithm to ensure that no syline community will be recompute. The syline community clearly cannot be recompute in two regular 2D subspaces with ifferent priorities in Algorithm 7, thus we nee to avoi reunant computations x2 0 x1 464