Document

Modularity of Erdös-Rényi Random Graphs

File

LIPIcs.AofA.2018.31.pdf
• Filesize: 496 kB
• 13 pages

Cite As

Colin McDiarmid and Fiona Skerman. Modularity of Erdös-Rényi Random Graphs. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 31:1-31:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.31

Abstract

For a given graph G, modularity gives a score to each vertex partition, with higher values taken to indicate that the partition better captures community structure in G. The modularity q^*(G) (where 0 <= q^*(G)<= 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity value. Given the prominence of modularity in community detection, it is an important graph parameter to understand mathematically. For the Erdös-Rényi random graph G_{n,p} with n vertices and edge-probability p, the likely modularity has three distinct phases. For np <= 1+o(1) the modularity is 1+o(1) with high probability (whp), and for np --> infty the modularity is o(1) whp. Between these regions the modularity is non-trivial: for constants 1 < c_0 <= c_1 there exists delta>0 such that when c_0 <= np <= c_1 we have delta<q^*(G)<1-delta whp. For this critical region, we show that whp q^*(G_{n,p}) has order (np)^{-1/2}, in accord with a conjecture by Reichardt and Bornholdt in 2006 (and disproving another conjecture from the physics literature).

Subject Classification

ACM Subject Classification
• Theory of computation → Random network models
Keywords
• Community detection
• Newman-Girvan Modularity
• random graphs

Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

References

1. William Aiello, Fan Chung, and Linyuan Lu. A random graph model for power law graphs. Experimental Mathematics, 10(1):53-66, 2001. URL: http://dx.doi.org/10.1080/10586458.2001.10504428.
2. Aaron F. Alexander-Bloch, Nitin Gogtay, David Meunier, Rasmus Birn, Liv Clasen, Francois Lalonde, Rhoshel Lenroot, Jay Giedd, and Edward T. Bullmore. Disrupted modularity and local connectivity of brain functional networks in childhood-onset schizophrenia. Frontiers in Systems Neuroscience, 4, 2010. URL: http://dx.doi.org/10.3389/fnsys.2010.00147.
3. James P. Bagrow. Communities and bottlenecks: Trees and treelike networks have high modularity. Physical Review E, 85(6):066118, 2012. URL: http://dx.doi.org/10.1103/PhysRevE.85.066118.
4. Ulrik Brandes, Daniel Delling, Marco Gaertler, Robert Görke, Martin Hoefer, Zoran Nikoloski, and Dorothea Wagner. On finding graph clusterings with maximum modularity. In Graph-Theoretic Concepts in Computer Science, pages 121-132. Springer, 2007. URL: http://dx.doi.org/10.1007/978-3-540-74839-7_12.
5. Ulrik Brandes, Daniel Delling, Marco Gaertler, Robert Gorke, Martin Hoefer, Zoran Nikoloski, and Dorothea Wagner. On modularity clustering. Knowledge and Data Engineering, IEEE Transactions on, 20(2):172-188, 2008. URL: http://dx.doi.org/10.1109/TKDE.2007.190689.
6. Fan Chung. Spectral graph theory, volume 92. American Mathematical Soc. Providence, RI, 1997. URL: http://dx.doi.org/10.1090/cbms/092.
7. Fan Chung, Linyuan Lu, and Van Vu. The spectra of random graphs with given expected degrees. Internet Mathematics, 1(3):257-275, 2003. URL: http://dx.doi.org/10.1073/pnas.0937490100.
8. Amin Coja-Oghlan. On the Laplacian Eigenvalues of G_n,p. Combinatorics, Probability and Computing, 16:923-946, 2007. URL: http://dx.doi.org/10.1017/S0963548307008693.
9. Fabien De Montgolfier, Mauricio Soto, and Laurent Viennot. Asymptotic modularity of some graph classes. In Algorithms and Computation, pages 435-444. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-25591-5_45.
10. P. Erdős and A. Rényi. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5:17-61, 1960.
11. Santo Fortunato. Community detection in graphs. Physics Reports, 486(3):75-174, 2010. URL: http://dx.doi.org/10.1016/j.physrep.2009.11.002.
12. Santo Fortunato and Marc Barthélemy. Resolution limit in community detection. Proceedings of the National Academy of Sciences, 104(1):36-41, 2007. URL: http://dx.doi.org/10.1073/pnas.0605965104.
13. Beate Franke and Patrick J. Wolfe. Network modularity in the presence of covariates. preprint arXiv:1603.01214, 2016.
14. Alan Frieze and Michał Karoński. Introduction to random graphs. Cambridge University Press, 2015. URL: http://dx.doi.org/10.1017/cbo9781316339831.
15. Roger Guimerà, Marta Sales-Pardo, and Luís A. Nunes Amaral. Modularity from fluctuations in random graphs and complex networks. Physical Review E, 70:025101, 2004. URL: http://dx.doi.org/10.1103/physreve.70.025101.
16. Ido Kanter and Haim Sompolinsky. Graph optimisation problems and the Potts glass. Journal of Physics A, 20(11):L673, 1987. URL: http://dx.doi.org/10.1088/0305-4470/20/11/001.
17. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010. URL: http://dx.doi.org/10.1103/physreve.82.036106.
18. Andrea Lancichinetti and Santo Fortunato. Limits of modularity maximization in community detection. Physical Review E, 84(6):066122, 2011. URL: http://dx.doi.org/10.1103/physreve.84.066122.
19. Malwina J. Luczak and Colin McDiarmid. Bisecting sparse random graphs. Random Structures &Algorithms, 18(1):31-38, 2001. URL: http://dx.doi.org/10.1002/1098-2418(200101)18:1<31::aid-rsa3>3.3.co;2-t.
20. Colin McDiarmid and Fiona Skerman. Modularity in random regular graphs and lattices. Electronic Notes in Discrete Mathematics, 43:431-437, 2013. URL: http://dx.doi.org/10.1016/j.endm.2013.07.063.
21. Colin McDiarmid and Fiona Skerman. Modularity of regular and treelike graphs. Journal of Complex Networks, 5, 2017. URL: http://dx.doi.org/10.1093/comnet/cnx046.
22. Colin McDiarmid and Fiona Skerman. Extreme values of modularity. In preparation, 2018.
23. Colin McDiarmid and Fiona Skerman. Modularity and edge-sampling. In preparation, 2018.
24. Colin McDiarmid and Fiona Skerman. Modularity of Erdős-Rényi random graphs. Manuscript, 2018.
25. Colin McDiarmid and Fiona Skerman. Modularity of very dense graphs. In preparation, 2018.
26. Mark E. J. Newman. Networks: An Introduction. Oxford University Press, 2010. URL: http://dx.doi.org/10.1093/acprof:oso/9780199206650.001.0001.
27. Mark E. J. Newman and Michelle Girvan. Finding and evaluating community structure in networks. Physical Review E, 69(2):026113, 2004. URL: http://dx.doi.org/10.1103/physreve.69.026113.
28. Mason A. Porter, Jukka-Pekka Onnela, and Peter J. Mucha. Communities in networks. Notices of the AMS, 56(9):1082-1097, 2009.
29. Liudmila Ostroumova Prokhorenkova, Paweł Prałat, and Andrei Raigorodskii. Modularity in several random graph models. Electronic Notes in Discrete Mathematics, 61:947-953, 2017. URL: http://dx.doi.org/10.1016/j.endm.2017.07.058.
30. Jörg Reichardt and Stefan Bornholdt. When are networks truly modular? Physica D: Nonlinear Phenomena, 224(1):20-26, 2006. URL: http://dx.doi.org/10.1016/j.physd.2006.09.009.
31. Fiona Skerman. Modularity of Networks. PhD thesis, University of Oxford, 2016.
32. Stojan Trajanovski, Huijuan Wang, and Piet Van Mieghem. Maximum modular graphs. The European Physical Journal B-Condensed Matter and Complex Systems, 85(7):1-14, 2012. URL: http://dx.doi.org/10.1140/epjb/e2012-20898-3.