Abstract
This paper discusses a complete and efficient algorithm for enumerating densely-connected k-Plexes. A k-Plex is a kind of pseudo-clique which imposes a Disconnection Upper Bound (DUB) by the parameter k for each constituent vertex. However, since the parameter is usually fixed not depending on sizes of our targeted pseudo-cliques, we often have k-Plexes not densely-connected. In order to overcome this drawback, we introduce another constraint using a parameter j designating Connection Lower Bound (CLB). Based on CLB, we can additionally enjoy a monotonic j-core operation and design an efficient depth-first algorithm which can exclude hopeless vertex sets which cannot be extended to their supersets satisfying both DUB and CLB. Our experimental results show it can work well as a useful tool for detecting densely-connected pseudo cliques in large networks including one with over 800, 000 vertices.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Berlowitz, D., Cohen, S., Kimelfeld, B.: Efficient enumeration of maximal \(k\)-Plexes. In: Proceedings of the 2015 ACM SIGMOD Conference, pp. 431–444 (2015)
Luce, D.R.: Connectivity and generalized cliques in sociometric group structure. Psychometrika 15(2), 169–190 (1950)
Alba, R.D.: A graph-theoretic definition of a sociometric clique. J. Math. Soc. 3(1), 113–126 (1973)
Mokken, R.: Cliques, clubs and clans, quality & quantity. Int. J. Meth. 13(2), 161–173 (1979)
Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393(6684), 440–442 (1998)
Seidman, S.B., Foster, B.L.: A graph-theoretic generalization of the clique concept. J. Math. Soc. 6(1), 139–154 (1978)
Abello, J., Resende, M.G.C., Sudarsky, S.: Massive quasi-clique detection. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 598–612. Springer, Heidelberg (2002)
Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)
Newman, M.E.J.: Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74(3), 036104 (2006)
Scott, J.P., Carrington, P.J. (eds.): The SAGE Handbook of Social Network Analysis. Sage Publications, London (2011)
Pattillo, J., Youssef, N., Butenko, S.: Clique relaxation models in social network analysis. In: Thai, M.T., Pardalos, P.M. (eds.) Handbook of Optimization in Complex Networks: Communication and Social Networks. Springer Optimization and Its Applications, vol. 58, pp. 143–162. Springer, New York (2012)
Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5(3), 269–287 (1983)
Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments. Theor. Comput. Sci. 363(1), 28–42 (2006)
Eppstein, D., Strash, D.: Listing all maximal cliques in large sparse real-world graphs. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 364–375. Springer, Heidelberg (2011)
Wu, B., Pei, X.: A parallel algorithm for enumerating all the maximal k-Plexes. In: Washio, T., Zhou, Z.-H., Huang, J.Z., Hu, X., Li, J., Xie, C., He, J., Zou, D., Li, K.-C., Freire, M.M. (eds.) PAKDD 2007. LNCS (LNAI), vol. 4819, pp. 476–483. Springer, Heidelberg (2007)
Okubo, Y., Haraguchi, M., Tomita, E.: Structural change pattern mining based on constrained maximal k-Plex search. In: Ganascia, J.-G., Lenca, P., Petit, J.-M. (eds.) DS 2012. LNCS, vol. 7569, pp. 284–298. Springer, Heidelberg (2012)
Rymon, R.: Search through systematic set enumeration. In: Proceedings of International Conference on Principles of Knowledge Representation Reasoning - KR 1992, pp. 539–550 (1992)
Slater, N., Itzchack, R., Louzoun, Y.: Mid size cliques are more common in real world networks than triangles. Netw. Sci. 2(3), 387–402 (2014)
Batagelj, V. and Zaversnik, M.: An \(O(m)\) algorithm for cores decomposition of networks. In: CoRR 2003, cs.DS/0310049 OpenURL
Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998)
Leskovec, J., Krevl, A.: SNAP datasets: Stanford large network dataset collection (2014). http://snap.stanford.edu/data
Acknowledgments
The authors of [1] have kindly provided us their program codes of MaxKplexEnum. We would like to sincerely appreciate their kindness.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Zhai, H., Haraguchi, M., Okubo, Y., Tomita, E. (2016). A Fast and Complete Enumeration of Pseudo-Cliques for Large Graphs. In: Bailey, J., Khan, L., Washio, T., Dobbie, G., Huang, J., Wang, R. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2016. Lecture Notes in Computer Science(), vol 9651. Springer, Cham. https://doi.org/10.1007/978-3-319-31753-3_34
Download citation
DOI: https://doi.org/10.1007/978-3-319-31753-3_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31752-6
Online ISBN: 978-3-319-31753-3
eBook Packages: Computer ScienceComputer Science (R0)