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Graph Homomorphism Revisited for Graph Matching

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Graph Homomorphism Revisited for Graph Matching Graph Homomorphism Revisited for Graph Matching Wenfei Fan1,2 Jianzhong Li2 Shuai Ma1 Hongzhi Wang2 Yinghui Wu1 1University of Edinburgh 2Harbin Institute of Technology wenfei@inf.,shuai.ma@,y.wu-18@sms. ed.ac.uk {lijzh, wangzh}@hit.edu.cn { } A B Abstract G p G In a variety of emerging applications one needs to decide whether books audio books sports digital a graph G matches another Gp, i.e., whether G has a topological textbooks abooks albums categories booksets CDs structure similar to that of Gp. The traditional notions of graph DVDs homomorphism and isomorphism often fall short of capturing the features genres structural similarity in these applications. This paper studies revi- artsschool audiobooks sions of these notions, providing a full treatment from complexity albums to algorithms. (1) We propose p-homomorphism (p-hom) and 1-1 Figure 1: Graphs representing online stores p-hom, which extend graph homomorphism and subgraph isomor- phism, respectively, by mapping edges from one graph to paths in are also carried by the store G, and G and Gp can be navigated another, and by measuring the similarity of nodes. (2) We intro- similarly, i.e., if a site for selling item a can be reached from a site duce metrics to measure graph similarity, and several optimization for item b in Gp by following hyperlinks, then the site for item a problems for p-hom and 1-1 p-hom. (3) We show that the deci- can also be reached from the site for b in G. sion problems for p-hom and 1-1 p-hom are NP-complete even for When graph homomorphism or subgraph isomorphism is used to DAGs, and that the optimization problems are approximation-hard. measure graph similarity, G does not match Gp. Indeed, (a) nodes (4) Nevertheless, we provide approximation algorithms with prov- in G may not find a node in G with the same label, e.g., audio; and able guarantees on match quality. We experimentally verify the worse still, (b) there exists no sensible mapping from Vp to V that effectiveness of the revised notions and the efficiency of our algo- maps edges in Gp to edges in G accordingly. rithms in Web site matching, using real-life and synthetic data. However, a page checker (e.g., [8, 29]) may find connections between pages in Gp and those in G based on their functionality: 1. Introduction A 7→ B, books 7→ books, audio 7→ digital, textbooks 7→ school, abooks 7→ audiobooks, albums 7→ albums The notions of graph homomorphism and subgraph isomorphism That is, the store indeed has the capability of . While the [9] can be found in almost every graph theory textbook. Given G Gp edges in Gp are not preserved by the similarity relation, each edge two node-labeled graphs G1 = (V1, E1) and G2 = (V2, E2), the in Gp is mapped to a path in G, e.g., the edge (books, textbooks) problem of graph homomorphism (resp. subgraph isomorphism) is in Gp is mapped to the path books/categories/school in G. This to find a (resp. 1-1) mapping from V1 to V2 such that each node in tells us that G preserves the navigational structure of Gp. Hence G V1 is mapped to a (resp. distinct) node in V2 with the same label, should logically be considered as a match of Gp. ✷ and each edge in E1 is mapped to an edge in E2. These conventional notions are, however, often too restrictive These highlight the need for revising the conventional notions for graph matching in emerging applications. In a nutshell, graph of graph matching. In response to these, several extensions of the matching is to decide whether a graph G matches another graph conventional notions have been studied for graph matching [10, 11, Gp, i.e., whether G has a structure similar to that of Gp, although 14, 24, 32]. However, a formal analysis of these extensions is not not necessarily identical. The need for this is evident in, e.g., Web yet in place, from complexity bounds to approximation algorithms. anomaly detection [23], search result classification [25], plagiarism detection [20] and spam detection [3]. In these contexts, identi- Contributions. We propose several notions to capture graph struc- cal label matching is often an overkill, and edge-to-edge mappings tural similarity that encompass the previous extensions, and pro- only allow strikingly similar graphs to be matched. vide a full treatment of these notions for graph matching. (1) We introduce p-homomorphism (p-hom) and 1-1 p-hom in Sec- Example 1.1: Consider two online stores depicted in Fig. 1 as tion 3. These notions extend graph homomorphism and subgraph graphs Gp = (Vp, Ep) and G = (V, E). In these graphs, each isomorphism, respectively, by (a) incorporates similarity metrics to node denotes a Web page for sale of certain items, as indicated measure the similarity of nodes, as opposed to node label equal- by its label; and the edges denote hyperlinks. One wants to know ity; and (b) mapping edges in a graph to paths in another, rather whether G matches Gp, i.e., whether all the items specified by Gp than edge-to-edge mappings. In contrast to previous extensions, one can use node similarity to assure, e.g., that two Web pages are Permission to make digital or hard copies of all or part of this work for matched only when they have similar contents [29] or play a simi- personal or classroom use is granted without fee provided that copies are lar role (as a hub or authority [6]). Edge-to-path mappings allow us not made or distributed for profit or commercial advantage and that copies to match graphs that have similar navigational structures but can- bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific not be identified by the conventional notions of graph matching. In permission and/or a fee. Articles from this volume were presented at The addition, these notions can be readily extended to deciding whether 36th International Conference on Very Large Data Bases, September 13•17, two graphs are similar to each other in a symmetric fashion. 2010, Singapore. Proceedings of the VLDB Endowment, Vol. 3, No. 1 (2) To provide a quantitative measure of graph similarity, we de- Copyright 2010 VLDB Endowment 2150•8097/10/09... $ 10.00. velop two metrics, also in Section 3, based on (a) the maximum number of nodes matched, and (b) the maximum overall similar- graphs based on simulation [17, 12], subgraph isomorphism (com- ity when the weights of nodes are taken into account, respectively. mon maximum subgraph) [27, 30], or edit distance [31] (see [9, These metrics give rise to two natural optimization problems, re- 26] for surveys). Graph simulation considers edge-preserving re- ferred to as the maximum cardinality problem and the maximum lations instead of functions from one graph to another. Graph edit similarity problem, respectively, for each of p-hom and 1-1 p-hom. distance is essentially based on subgraph isomorphism. (c) Vertex In particular, the maximum common subgraph problem [19] is a similarity: it builds a matrix of node similarity based on fixpoint special case of the maximum cardinality problem for 1-1 p-hom. computation [6, 21] and the roles of nodes in the structures of the (3) We establish complexity bounds of the decision problems and graphs (e.g., hubs or authorities [6]). As pointed out by [25, 30], the optimization problems for p-hom and 1-1 p-hom, in Section 4. We feature-based approach does not observe global structural connec- show that the problems for determining p-hom and 1-1 p-hom, tivity, and is often less accurate than the structure-based measure. as well as the maximum cardinality problem and the maximum As observed by [4, 23], vertex similarity alone does not suffice to identify accurate matches since it ignores the topology of graphs similarity problem, are all NP-complete, even for directed acyclic by and large. For Web site matching in particular, it is essential graphs (DAGs). Worse still, the optimization problems are hard to to consider how pages are linked to each other. One cannot match approximate: unless P = NP, it is beyond reach in practice to ap- proximate the problems within O(1/n1−ǫ) of the optimal solutions two sites with different navigational structures even if most of their for any constant ǫ. All proofs are given in the appendix. pages can be matched pairwise. Further, vertex similarity requires fixpoint operations and is often too expensive to compute on large (4) Nevertheless, we provide in Section 5 approximation algo- graphs. As opposed to previous approaches, we introduce (1-1) rithms for finding mappings with the maximum cardinality or the p-hom to capture both structural similarity by enforcing edge-to- maximum similarity, for p-hom and 1-1 p-hom. These algorithms path mappings, and the contents of individual nodes by incorporat- possess performance guarantees on match quality: for any graphs ing node similarity. In addition, we provide maximum cardinality G1 and G2, the solutions found by the algorithms are provable to be and maximum overall similarity metrics to quantitatively measure 2 within a polynomial O(log (n1n2)/(n1n2)) of the optimal solu- graph similarity, which have not been studied by previous work. tions, where n1 (resp. n2) is the number of nodes in G1 (resp. G2). A number of graph matching algorithms have been developed (5) Using Web site matching as a testbed, we experimentally evalu- (see [9] for a survey).
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