Asymptotic Modularity of Some Graph Classes

Modularity has been introduced as a quality measure for graph partitioning. It has received considerable attention in several disciplines, especially complex systems. In order to better understand this measure from a graph theoretical point of view, we study the modularity of a variety of graph classes. We first consider simple graph classes such as tori and hypercubes. We show that these regular graph families have asymptotic modularity 1 (that is the maximum possible). We extend this result to the general class of unit ball graphs of bounded growth metrics. Our most striking result concerns trees with bounded degree which also appear to have asymptotic modularity 1. This last result can be extended to graphs with constant average degree and to some power-law graphs.

[1]  Satu Elisa Schaeffer,et al.  Graph Clustering , 2017, Encyclopedia of Machine Learning and Data Mining.

[2]  R. Guimerà,et al.  Modularity from fluctuations in random graphs and complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[5]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Ulrik Brandes,et al.  On Modularity Clustering , 2008, IEEE Transactions on Knowledge and Data Engineering.

[7]  Charu C. Aggarwal,et al.  Graph Clustering , 2010, Encyclopedia of Machine Learning and Data Mining.

[8]  Bhaskar DasGupta,et al.  On the complexity of Newman's community finding approach for biological and social networks , 2011, J. Comput. Syst. Sci..

[9]  A. Arenas,et al.  Community detection in complex networks using extremal optimization. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  J. Bascompte,et al.  The modularity of pollination networks , 2007, Proceedings of the National Academy of Sciences.

[11]  M E J Newman,et al.  Modularity and community structure in networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[12]  R. Guimerà,et al.  Functional cartography of complex metabolic networks , 2005, Nature.

[13]  Jean-Loup Guillaume,et al.  Fast unfolding of communities in large networks , 2008, 0803.0476.

[14]  U. Alon,et al.  Spontaneous evolution of modularity and network motifs. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[15]  David R. Karger,et al.  Finding nearest neighbors in growth-restricted metrics , 2002, STOC '02.