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Finding Cliques in Social Networks: A New Distribution-Free Model

Authors Jacob Fox, Tim Roughgarden, C. Seshadhri, Fan Wei, Nicole Wein

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Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, Stanford, CA 94305, USA
Tim Roughgarden
  • Department of Computer Science, Stanford University, Stanford, CA 94305, USA
C. Seshadhri
  • Department of Computer Science, University of California, Santa Cruz, CA 95064, USA
Fan Wei
  • Department of Mathematics, Stanford University, Stanford, CA 94305, USA
Nicole Wein
  • EECS, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Funding

  • Fox, Jacob: Supported by a Packard Fellowship, an NSF Career Award DMS-1352121, and an Alfred P. Sloan Fellowship
  • Roughgarden, Tim: Supported by NSF award CCF-1524062
  • Wein, Nicole: Supported by an NSF Graduate Fellowship

Cite AsGet BibTex

Jacob Fox, Tim Roughgarden, C. Seshadhri, Fan Wei, and Nicole Wein. Finding Cliques in Social Networks: A New Distribution-Free Model. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.55

Abstract

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure - the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a c-closed graph, where for every pair of vertices u,v with at least c common neighbors, u and v are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to c on c-closed graphs. Our results carry over to weakly c-closed graphs, which only require a vertex deletion ordering that avoids pairs of non-adjacent vertices with c common neighbors. Numerical experiments show that well-studied social networks tend to be weakly c-closed for modest values of c.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Graph algorithms
  • social networks
  • fixed-parameter tractability

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References

  1. Enron email dataset. URL: https://www.cs.cmu.edu/~./enron/.
  2. A. Abraham and R. Sandler. An Introduction to Finite Projective Planes. Dover, 2015. Google Scholar
  3. R. Albert, H. Jeong, and A.-L. Barabási. Error and attack tolerance of complex networks. Nature, 406:378-382, 2000. Google Scholar
  4. N. Alon, S. Arora, R. Manokaran, D. Moshkovitz, and O. Weinstein. Inapproximability of densest κ-subgraph from average case hardness. Unpublished manuscript, 2011. Google Scholar
  5. J. Balogh, R. Morris, and W. Samotij. Independent sets in hypergraphs. J. Amer. Math. Soc., 28(3):669-709, 2015. Google Scholar
  6. A.-L. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286:509-512, 1999. URL: http://dx.doi.org/10.1126/science.286.5439.509.
  7. A. Bhaskara, M. Charikar, E. Chlamtac, U. Feige, and A. Vijayaraghavan. Detecting high log-densities: an O(n ^1/4) approximation for densest k-subgraph. In Proceedings of the 2010 ACM Symposium on Theory of Computing, pages 201-210. ACM, 2010. Google Scholar
  8. M. Bloznelis and V. Kurauskas. Clustering function: a measure of social influence. CoRR, abs/1207.4941, 2012. URL: http://arxiv.org/abs/1207.4941,
  9. M. Borassi, P. Crescenzi, and L. Trevisan. An axiomatic and an average-case analysis of algorithms and heuristics for metric properties of graphs. arXiv preprint arXiv:1604.01445, 2016. Google Scholar
  10. P. Brach, M. Cygan, J. Łącki, and P. Sankowski. Algorithmic complexity of power law networks. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1306-1325. SIAM, 2016. Google Scholar
  11. A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener. Graph structure in the web. Computer Networks, 33:309-320, 2000. Google Scholar
  12. D. Chakrabarti and C. Faloutsos. Graph mining: Laws, generators, and algorithms. ACM Computing Surveys, 38(1), 2006. URL: http://dx.doi.org/10.1145/1132952.1132954.
  13. D. Chakrabarti, Y. Zhan, and C. Faloutsos. R-MAT: A recursive model for graph mining. In SIAM Conference on Data Mining, pages 442-446, 2004. URL: http://siam.org/proceedings/datamining/2004/dm04_043chakrabartid.pdf.
  14. F. Chung and L. Lu. The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA, 99(25):15879-15882, 2002. URL: http://dx.doi.org/10.1073/pnas.252631999.
  15. F. Chung and L. Lu. Connected components in random graphs with given degree sequences. Ann. Comb., 6:125-145, 2002. URL: http://dx.doi.org/10.1007/PL00012580.
  16. A. Conte, R. De Virgilio, Antonio Maccioni, M. Patrignani, and R. Torlone. Finding all maximal cliques in very large social networks. In EDBT, pages 173-184, 2016. Google Scholar
  17. M. Cygan, F. V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms, volume 3. Springer, 2015. Google Scholar
  18. N. Du, B. Wu, L. Xu, B. Wang, and X. Pei. A parallel algorithm for enumerating all maximal cliques in complex network. In Data Mining Workshops, 2006. ICDM Workshops 2006. Sixth IEEE International Conference on, pages 320-324. IEEE, 2006. Google Scholar
  19. V. Dujmović, G. Fijavž, G. Joret, T. Sulanke, and D. R. Wood. On the maximum number of cliques in a graph embedded in a surface. European J. Combin., 32(8):1244-1252, 2011. Google Scholar
  20. D. Eppstein, M. Löffler, and D. Strash. Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time, pages 403-414. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. URL: http://dx.doi.org/10.1007/978-3-642-17517-6_36.
  21. D. Eppstein and D. Strash. Listing all maximal cliques in large sparse real-world graphs. Experimental Algorithms, pages 364-375, 2011. Google Scholar
  22. E. M. Eschen, C. T. Hoàng, J. P. Spinrad, and R. Sritharan. On graphs without a C₄ or a diamond. Discrete Appl. Math., 159(7):581-587, 2011. Google Scholar
  23. M. Faloutsos, P. Faloutsos, and C. Faloutsos. On power-law relationships of the internet topology. In Proceedings of SIGCOMM, pages 251-262, 1999. Google Scholar
  24. A. Ferrante, G. Pandurangan, and K. Park. On the hardness of optimization in power law graphs. In Proceedings of Conference on Computing and Combinatorics, pages 417-427, 2006. Google Scholar
  25. A. Ferrante, G. Pandurangan, and K. Park. On the hardness of optimization in power-law graphs. Theoret. Comput. Sci., 393(1):220-230, 2008. URL: http://dx.doi.org/10.1016/j.tcs.2007.12.007.
  26. S. Fortunato. Community detection in graphs. Physics Reports, 486:75-174, 2010. Google Scholar
  27. J. Fox and F. Wei. On the number of cliques in graphs with a forbidden subdivision or immersion, 2016. Google Scholar
  28. J. Fox and F. Wei. On the number of cliques in graphs with a forbidden minor. J. Combin. Theory Ser. B, 126:175-197, 2017. URL: http://dx.doi.org/10.1016/j.jctb.2017.04.004.
  29. Jacob Fox, Tim Roughgarden, C. Seshadhri, Fan Wei, and Nicole Wein. Finding cliques in social networks: A new distribution-free model. arXiv preprint arXiv:1804.07431, 2018. Google Scholar
  30. Z. Füredi and M. Simonovits. The History of Degenerate (Bipartite) Extremal Graph Problems, pages 169-264. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39286-3_7.
  31. L. Gąsieniec, M. Kowaluk, and A. Lingas. Faster multi-witnesses for boolean matrix multiplication. Information Processing Letters, 109(4):242-247, 2009. Google Scholar
  32. M. Girvan and M. Newman. Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA, 99(12):7821-7826, 2002. URL: http://dx.doi.org/10.1073/pnas.122653799.
  33. R. Gupta, T. Roughgarden, and C. Seshadhri. Decompositions of triangle-dense graphs. SIAM J. Comput., 45(2):197-215, 2016. Google Scholar
  34. M. Jha, C. Seshadhri, and A. Pinar. Path sampling: A fast and provable method for estimating 4-vertex subgraph counts. In World Wide Web (WWW), pages 495-505, 2015. Google Scholar
  35. J. M. Kleinberg. Navigation in a small world. Nature, 406:845, 2000. Google Scholar
  36. J. M. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of the Symposium on Theory of Computing, pages 163-170, 2000. Google Scholar
  37. J. M. Kleinberg. Small-world phenomena and the dynamics of information. In Advances in Neural Information Processing Systems, volume 1, pages 431-438, 2002. Google Scholar
  38. R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal. Stochastic models for the web graph. In Proceedings of Foundations of Computer Science, pages 57-65, 2000. URL: http://dx.doi.org/10.1109/SFCS.2000.892065.
  39. J. Leskovec, D. Chakrabarti, J. M. Kleinberg, C. Faloutsos, and Z. Ghahramani. Kronecker graphs: An approach to modeling networks. J. Mach. Learn. Res., 11:985-1042, 2010. URL: http://jmlr.csail.mit.edu/papers/v11/leskovec10a.html.
  40. J. Leskovec, K. Lang, A. Dasgupta, and M. Mahoney. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Math., 6(1):29-123, 2008. Google Scholar
  41. H. Lin, C. Amanatidis, M. Sideri, R. M. Karp, and C. H. Papadimitriou. Linked decompositions of networks and the power of choice in Polya urns. In Proceedings of the Symposium on Discrete Algorithms, pages 993-1002, 2008. Google Scholar
  42. M. Mitzenmacher, J. Pachocki, R. Peng, C. Tsourakakis, and S. Xu. Scalable large near-clique detection in large-scale networks via sampling. In SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 815-824, 2015. URL: http://dx.doi.org/10.1145/2783258.2783385.
  43. A. Montanari and A. Saberi. The spread of innovations in social networks. Proc. Natl. Acad. Sci. USA, 107(47):20196-20201, 2010. Google Scholar
  44. J. Moon and L. Moser. On cliques in graphs. Israel J. Math., 3(1):23-28, 1965. Google Scholar
  45. M. E. J. Newman. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA, 98(2):404-409, 2001. URL: http://dx.doi.org/10.1073/pnas.98.2.404.
  46. M. E. J. Newman. Properties of highly clustered networks. Physical Review E, 68(2):026121, 2003. URL: http://dx.doi.org/10.1103/PhysRevE.68.026121.
  47. M. E. J. Newman. Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E, 74(3):036104, 2006. URL: http://dx.doi.org/10.1103/PhysRevE.74.036104.
  48. V. Nikiforov. The number of cliques in graphs of given order and size. Trans. Amer. Math. Soc., 363(3):1599-1618, 2011. Google Scholar
  49. V. Raman and S. Saket. Short cycles make W-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles. Algorithmica, 52(2):203-225, 2008. Google Scholar
  50. A. A. Razborov. On the minimal density of triangles in graphs. Comb. Probab. Comput., 17(4):603-618, 2008. URL: http://dx.doi.org/10.1017/S0963548308009085.
  51. C. Reiher. The clique density theorem. Ann. of Math. (2), 184(3):683-707, 2016. URL: http://dx.doi.org/10.4007/annals.2016.184.3.1.
  52. R. A. Rossi, D. F. Gleich, and A. H. Gebremedhin. Parallel maximum clique algorithms with applications to network analysis. SIAM J. Sci. Comput., 37(5), 2015. URL: http://dx.doi.org/10.1137/14100018X.
  53. A. Sala, L. Cao, C. Wilson, R. Zablit, H. Zheng, and B. Y. Zhao. Measurement-calibrated graph models for social network experiments. In Proceedings of the World Wide Web Conference, pages 861-870. ACM, 2010. URL: http://dx.doi.org/10.1145/1772690.1772778.
  54. A. E. Sarıyüce, C. Seshadhri, A. Pınar, and Ü. V. Çatalyürek. Finding the hierarchy of dense subgraphs using nucleus decompositions. In Proceedings of the 24th International Conference on World Wide Web, WWW '15, pages 927-937, Republic and Canton of Geneva, Switzerland, 2015. International World Wide Web Conferences Steering Committee. URL: http://dl.acm.org/citation.cfm?id=2736277.2741640.
  55. D. Saxton and A. Thomason. Hypergraph containers. Invent. Math., 201(3):925-992, Sep 2015. URL: http://dx.doi.org/10.1007/s00222-014-0562-8.
  56. C. Seshadhri, A. Pinar, and T. G. Kolda. Fast triangle counting through wedge sampling. In Proceedings of the SIAM Conference on Data Mining, 2013. URL: http://arxiv.org/abs/1202.5230.
  57. E. Tomita and T. Kameda. An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Global Optim., 37(1):95-111, 2007. Google Scholar
  58. E. Tomita, A. Tanaka, and H. Takahashi. The worst-case time complexity for generating all maximal cliques and computational experiments. Theor. Comput. Sci., 363(1):28-42, 2006. URL: http://dx.doi.org/10.1016/j.tcs.2006.06.015.
  59. C. Tsourakakis. The k-clique densest subgraph problem. In Proceedings of the 24th International Conference on World Wide Web, WWW '15, pages 1122-1132, Republic and Canton of Geneva, Switzerland, 2015. International World Wide Web Conferences Steering Committee. URL: http://dl.acm.org/citation.cfm?id=2736277.2741098.
  60. C. Tsourakakis, F. Bonchi, A. Gionis, F. Gullo, and M. Tsiarli. Denser than the densest subgraph: Extracting optimal quasi-cliques with quality guarantees. In Proc. of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '13, 2013. Google Scholar
  61. S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa. A new algorithm for generating all the maximal independent sets. SIAM J. Comput., 6(3):505-517, 1977. Google Scholar
  62. J. Ugander, L. Backstrom, and J. Kleinberg. Subgraph frequencies: Mapping the empirical and extremal geography of large graph collections. In Proceedings of World Wide Web Conference, pages 1307-1318, 2013. Google Scholar
  63. J. Ugander, B. Karrer, L. Backstrom, and C. Marlow. The anatomy of the facebook social graph. arXiv preprint arXiv:1111.4503, 2011. Google Scholar
  64. D. Watts and S. Strogatz. Collective dynamics of `small-world' networks. Nature, 393:440-442, 1998. URL: http://dx.doi.org/10.1038/30918.
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