A Local Algorithm for Structure-Preserving Graph Cut

Transcription

1 A Local Algorithm for Structure-Preserving Graph Cut Presenter: Dawei Zhou Dawei Zhou* (ASU) Si Zhang (ASU) M. Yigit Yildirim (ASU) Scott Alcorn (Early Warning) Hanghang Tong (ASU) Hasan Davulcu (ASU) Jingrui He (ASU)

2 Outline Background Related Work Proposed HOSPLOC Algorithm Experiments Conclusion - 2 -

3 Networks Network is everywhere Internet Social Network Online Transaction

4 Applications Networks Dense subgraph detection Internet Social Network Online Transaction Emerging Trends Detection Community Detection Money Laundering Detection

5 High-order structures in practice Stars M. Wilson, and C. Nicholas, 2008, October. Topological analysis of an online social network for older adults. In Proceedings of the 2008 ACM workshop on Search in social media (pp ). ACM. - 5Z. -Li, F. Yan, and Y. Jiang, Cross-layers cascade in multiplex networks. Autonomous Agents and Multi- Agent Systems, 29(6), pp

6 High-order structures in practice Triangle C. Klymko, D. Gleich, and T.-G. Kolda Using triangles to improve community detection in directed networks. arxiv preprint arxiv: (2014). J. W. Berry, B. Hendrickson, R. A. LaViolette, and C. A. Phillips, Tolerating the community detection - 6resolution - limit with edge weighting. Physical Review E, 83(5), p

7 High-order structures in practice Loops K.-K. R. Choo Money laundering risks of prepaid stored value cards. Australian Institute of Criminology. (363) B. J. Kloostra, C. Dalvi. and B. N. Behm, Crowe Horwath Llp, System and method for analyzing and - 7dispositioning - money laundering suspicious activity alerts. U.S. Patent Application Arizona 12/258,784. State University

8 Problem definition 1 2 Given: Graph G = (V, E), user-defined structure N. 3 Find: Find a structure-rich dense subgraph that largely preserves the user-defined structures. Example: G Dense Subgraph with rich 3-node lines User-define structure 3-node line - 8 -

9 Outline Background Related Work Proposed HOSPLOC Algorithm Experiments Conclusion - 9 -

10 Related works High-order graph clustering Frequency based methods [Strogatz, 98; Milo et al., 2012; Vinayagama et al., 16] Spectral clustering based methods [Benson et al., 2015; Tsourakakis et al., 2017] Pros: Improving our understanding of highorder organization of complex systems. Co-clustering based methods [Wu et al., 2016] Cons: Expensive time complexity!

11 Related works Local graph clustering Vector based methods [Spielman and Teng, 2004; Andersen et al., 2006] Approximated methods [Gharan and Trevisan, 2012] Randomized methods [Spielman and Teng, 2013; Andersen et al., 2016] Max-Flow based methods [Andersen and Lang, 2008; Orecchia and Zhu, 2014] Spectral clustering based methods [Chung, 2007] Pros: Running time does not depend on the size of the given graph. Cons: Unable to explore the high-order network structures!

12 Local high-order graph clustering? With HOSPLOC you can!

13 Outline Background Related Work Proposed HOSPLOC Algorithm Experiments Conclusion

14 HOSPLOC HO: High-Order SP: Structure-Preserving LO: Local C: Cut

15 Evaluation metric k th -order conductance Definition 1. For any cluster C in graph G and the k th -order structure, the k th -order conductance Φ(C, N) is defined as Φ C, N = cut C, N min μ C, N, μ( ҧ C, N) The number of network structures broken due to the partition of G into C and Cҧ The number of network structures in C. The number of network structures in C. ҧ

16 Evaluation metric Example Cut C Φ C, N = cut C, N min μ C, N, μ( ҧ C, N) 2 nd -order conductance Φ C, N = 1 min{4,11} = 1/4 Graph G 3 rd -order conductance Φ C, N = 2 min{3,34} =

17 Problem definition 1 2 Given: Graph G = (V, E), user-defined structure 3 N, initial vertex v. Find: Local cluster C including or near v that largely preserves user-defined structures N. 4 C = argmin Φ C, N C V Example: 3 1 G 4 2 Dense Subgraph with rich 3-node line User-define structure 3-node line

18 High-order random walks (HRW) Adjacency tensor Definition 2. Given a graph G = (V, E), the k th -order network structure N on G could be represented in a k-dimensional adjacency tensor T as follows Example: For the set of nodes {2, 4, 1} T 1,4,2 = 0 G: N: For the set of nodes {6, 8, 10} T 10,8,6 =

19 High-order random walks (HRW) Transition tensor Definition 3. Given a graph G = V, E and the adjacency tensor T for the k th -order network structure N, the corresponding transition tensor P could be computed as Example: For the set of nodes {2, 4, 1} P 1,4,2 = 0 G: N: For the set of nodes {6, 8, 10} P 10,8,6 = 1/3-19 -

20 High-order random walks (HRW) k th -order Markov chain interpretation Each vertex represents an individual state Depends on the last k 1 states (nodes) P(i 1,, i k ) = Pr(S t+1 = i 1 S t = i 2,, S t k+2 = i k ) Example: For the set of nodes {2, 4, 1} P 1,4,2 = P r v 1 v 4, v 2 = 0 G: N: For the set of nodes {6, 8, 10} P 10,8,6 = P r v 10 v 6, v 8 = 1/3-20 -

21 High-order random walks (HRW) Using the rank-1 approximation [Li and NG, 2013], the high-order random walks can formulated as q t = Pq (t 1) q (t k+1) Truncated HRW distribution Truncation threshold ε Round small values to 0 to reduce computational load

22 Locally search high-order structure by HRW Vector based graph cut Locally conduct high-order random walks to explore the user-defined structure N. Compute the permutation π of the returned HRW distribution vector q such that: Iteratively check the potential cuts C 1, C 2,, C n 1, where C i = π 1,, π i

23 HOSPLOC Algorithm

24 HOSPLOC Algorithm 3-node line Initialization C = Truncated HRW Dense Subgraph with rich 3-node line Vector-based Graph Cut

25 Theoretical analysis Effectiveness Provide a lower bound of effectiveness. Theorem 4. Let C be a cluster on graph G such that Φ C, N 1 c 2 (l+2). If HOSPLOC runs with starting vertex v Ck,ξ and returns a nonempty set C, then we have μ C C 2 b 1, where b controls the minimum volume of returned cluster C. Efficiency Lemma 5. Given graph G and the k th -order network structure N k 3, the time complexity of HOSPLOC is bounded by O(t max 2 bk φ 2k log3k m)

26 Outline Background Related Work Proposed HOSPLOC Algorithm Experiments Conclusion

27 Data sets Category Network Type Nodes Edges Citation Author Undirected 61, ,074 Paper Undirected 62,602 10,904 Infrastructure Airline Undirected 2,833 15,204 Oregon Undirected 7,352 15,665 Power Undirected 4,941 13,188 Social Epinion Undirected 75, ,837 Review Rating Bipartite 8,724 90,962 Financial PII Multipartite

28 Comparison methods Local graph clustering Nibble [Andersen et al., 2006] Nibble-PageRank [Spielman and Teng, 2013] LS-OQC [Tsourakakis et al., 2013] HOSPLOC We proposed! Global graph clustering NMF [Ding et al., 2008] User-defined High-Order Structure TSC [Benson et al., 2015] Triangle

29 Effectiveness analysis Worse a Φ(C, edge) b Φ(C, triangle) Better Better c Triangle Desnsity Worse 29 -

30 Effectiveness analysis Worse 97% smaller Better Our methods a Φ(C, edge)

31 Effectiveness analysis Worse 12.2% smaller Better Our methods b Φ(C, triangle)

32 Effectiveness analysis Better 80% larger Worse Our methods c Triangle density

33 log t Scalability analysis Polylogarithmic! Linear! Running Time t s (a) The number of vertices v.s running time t log(μ(c)) (b) The lower bound of log(μ(c)) v.s log(t) * Nibble = A special case of HOPLOC

34 Parameter analysis Converge Faster! Worse Better (a) Φ(C, edge) v.s Conductance upper bound φ (b) Φ(C, triangle) v.s Conductance upper bound φ Lower Conductance!

35 Case study 1: Community detection Bipartite Graph 4 th Order Structure Graph Cut Movie Review Network User-defined Structure Miniature of the Identified Cluster Rich 4-node loop!

36 Case study 2: Synthetic ID detection Multipartite Graph 5 th Order Structure Graph Cut PII Network User-defined Structure Miniature of the Identified Cluster Rich 5-node star!

37 Outline Background Related Work Proposed HOSPLOC Algorithm Experiments Conclusion

38 Conclusion Goals: Detect structure-preserving dense subgraph Preserve Structure Time Complexity High-order Clustering Yes Expensive Local Clustering No Scalable HOSPLOC Yes Scalable Solutions: HOSPLOC algorithm Theoretical guaranteed effectiveness. Polylogarithmic time complexity w.r.t # edges. Results: Consistently better than 5 state-of-arts on 6 real networks. Scalability = Polylogarithmic complexity. Sensitivity = Stable. Case studies w.r.t. higher-order (>3) structures

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