Abstract
An extremely popular model-based graph partitioning approach that is used for both biological and social networks is the so-called modularity optimization approach originally proposed by Newman and its variations. In this chapter, we review several combinatorial and algebraic methods that have been used in the literature to study the computational complexities of these optimization problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Agarwal, G., Kempe, D.: Modularity-maximizing graph communities via mathematical programming. Eur. Phys. J. B 66(3), 409–418 (2008)
Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002)
Albert, R., DasGupta, B., Dondi, R., Kachalo, S., Sontag, E., Zelikovsky, A., Westbrooks, K.: A novel method for signal transduction network inference from indirect experimental evidence. J. Comput. Biol. 14(7), 927–949 (2007)
Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. In: Proceedings of the 36th ACM Symposium on Theory of Computing, pp. 72–80. ACM, New York (2004)
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1–3), 89–113 (2004)
Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)
Brandes, U., Delling, D., Gaertler, M., Görke, R., Hoefer, M., Nikoloski, Z., Wagner, D.: On modularity clustering. IEEE Trans. Knowl. Data Eng. 20(2), 172–188 (2007)
Charikar, M., Wirth, A.: Maximizing quadratic programs: extending Grothendieck’s inequality. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 54–68 (2004)
Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Boston, pp. 524–533 (2003)
Chlebík, M., Chlebíková, J.: Complexity of approximating bounded variants of optimization problems. Theor. Comput. Sci. 354(3), 320–338 (2006)
Coleman, T., Saunderson, J., Wirth, A.: Local-search 2-approximation for 2-correlation-clustering. In: Proceedings of the 16th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, Springer Verlag, vol. 5193, pp. 308–319 (2008)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
Danon, L., Duch, J., Diaz-Guilera, A., Arenas, A.: Comparing community structure identification. J. Stat. Mech. P09008 2005(9)
DasGupta, B., Desai, D.: Complexity of Newman’s community finding approach for social networks. J. Comput. Syst. Sci. 79(1), 50–67 (2013)
DasGupta, B., Andres Enciso, G., Sontag, E., Zhang, Y.: Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. Biosystems 90(1), 161–178 (2007)
Flake, G.W., Lawrence, S.R., Giles, C.L., Coetzee, F.M.: Self-organization and identification of web communities. IEEE Comput. 35, 66–71 (2002)
Fortunato, S., Barthélemy, M.: Resolution limit in community detection. Proc. Natl. Acad. Sci. 104(1), 36–41 (2007)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Company, New York (1979)
Giot, L., Bader, J.S., Brouwer, C., Chaudhuri, A., Kuang, B., Li, Y., Hao, Y.L., Ooi, C.E., Godwin, B., Vitols, E., Vijayadamodar, G., Pochart, P., Machineni, H., Welsh, M., Kong, Y., Zerhusen, B., Malcolm, R., Varrone, Z., Collis, A., Minto, M., Burgess, S., McDaniel, L., Stimpson, E., Spriggs, F., Williams, J., Neurath, K., Ioime, N., Agee, M., Voss, E., Furtak, K., Renzulli, R., Aanensen, N., Carrolla, S., Bickelhaupt, E., Lazovatsky, Y., DaSilva, A., Zhong, J., Stanyon, C.A., Finley, R.L., White, K.P., Braverman, M., Jarvie, T., Gold, S., Leach, M., Knight, J., Shimkets, R.A., McKenna, M.P., Chant, J., Rothberg, J.M.: A protein interaction map of Drosophila melanogaster. Science 302(5651), 1727–1736 (2003)
Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Natl. Acad. Sci 99, 7821–7826 (2002)
Guimera, R., Sales-Pardo, M., Amaral, L.A.N.: Classes of complex networks defined by role-to-role connectivity profiles. Nat. Phys. 3, 63–69 (2007)
Hann, J., Kamber, M.: Data Mining: Concepts and Techniques. Morgan Kaufmann, San Francisco (2000)
Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.-L.: The large-scale organization of metabolic networks. Nature 407, 651–654 (2000)
Kannan, R., Tetali, P., Vempala, S.: Markov-chain algorithms for generating bipartite graphs and tournaments. Random Struct. Algorithms 14, 293–308 (1999)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)
Karrer, B., Newman, M.E.J.: Random graph models for directed acyclic networks. Phys. Rev. E 80, 046110 (2009)
Kefeng, D., Ping, Z., Huisha, Z.: Graph separation of 4-regular graphs is NP-complete. J. Math. Study 32(2), 137–145 (1999)
Kelley, J.E., Jr.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960)
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, pp. 57–65 (2000)
Lee, T.I., Rinaldi, M.J., Robert, F., Odom, D.T., Bar-Joseph, Z., Gerber, G.K., Hannett, N.M., Harbison, C.T., Thompson, C.M., Simon, I., Zeitlinger, J., Jennings, E.G., Murray, H.L., Gordon, D.B., Ren, B., Wyrick, J.J., Tagne, J.-B., Volkert, T.L., Fraenkel, E., Gifford, D.K., Young, R.A.: Transcriptional regulatory networks in Saccharomyces cerevisiae. Science 298(5594), 799–804 (2002)
Leicht, E.A., Newman, M.E.J.: Community structure in directed networks. Phys. Rev. Lett. 100, 118703 (2008)
Li, S., Armstrong, C.M., Bertin, N., Ge, H., Milstein, S., Boxem, M., Vidalain, P.-O., Han, J.-D.J., Chesneau, A., Hao, T., Goldberg, D.S., Li, N., Martinez, M., Rual, J.-F., Lamesch, P., Xu, L., Tewari, M., Wong, S.L., Zhang, L.V., Berriz, G.F., Jacotot, L., Vaglio, P., Reboul, J., Hirozane-Kishikawa, T., Li, Q., Gabel, H.W., Elewa, A., Baumgartner, B., Rose, D.J., Yu, H., Bosak, S., Sequerra, R., Fraser, A., Mango, S.E., Saxton, W.M., Strome, S., van den Heuvel, S., Piano, F., Vandenhaute, J., Sardet, C., Gerstein, M., Doucette-Stamm, L., Gunsalus, K.C., Harper, J.W., Cusick, M.E., Roth, F.P., Hill, D.E., Vidal, M.: A map of the interactome network of the metazoan C. elegans. Science 303, 540–543 (2004)
Maayan, A., Jenkins, S.L., Neves, S., Hasseldine, A., Grace, E., Dubin-Thaler, B., Eungdamrong, N.J., Weng, G., Ram, P.T., Rice, J.J., Kershenbaum, A., Stolovitzky, G.A., Blitzer, R.D., Iyengar, R.: Formation of regulatory patterns during signal propagation in a Mammalian cellular network. Science 309(5737), 1078–1083 (2005)
Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256, (2003)
Newman, M.E.J.: Detecting community structure in networks. Eur. Phys. J. B 38, 321–330 (2004)
Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103, 8577–8582 (2006)
Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004)
Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64(2), 026118–026134 (2001)
Nesterov, Y.: Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9, 141–160 (1998)
Pothen, A., Simon, D.H., Liou, K.P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11, 430–452 (1990)
Raghavan, U.N., Albert, R., Kumara, S.: Near linear time algorithm to detect community structures in large-scale networks. Phys. Rev. E 76, 036106 (2007)
Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Barabási, A.-L.: Hierarchical organization of modularity in metabolic networks. Science 297(5586), 1551–1555 (2002)
Shen-Orr, S.S., Milo, R., Mangan, S., Alon, U.: Network motifs in the transcriptional regulation network of Escherichia coli. Nat. Genet. 31, 64–68 (2002)
Simon, H.D., Teng, S.H.: How good is recursive bisection. SIAM J. Sci. Comput. 18, 1436–1445 (1997)
Swamy, C.: Correlation clustering: maximizing agreements via semidefinite programming. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, pp. 526–527 (2004)
Trevisan, L.: Max cut and the smallest eigenvalue. In: Proceedings of the 41st ACM Symposium on Theory of Computing, New York, pp. 263–272 (2009)
Vazirani, V.: Approximation Algorithms. Springer, Berlin (2001)
Acknowledgments
The author was supported by NSF grant IIS-1160995.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
DasGupta, B. (2014). Computational Complexities of Optimization Problems Related to Model-Based Clustering of Networks. In: Rassias, T., Floudas, C., Butenko, S. (eds) Optimization in Science and Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0808-0_5
Download citation
DOI: https://doi.org/10.1007/978-1-4939-0808-0_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-0807-3
Online ISBN: 978-1-4939-0808-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)