Evolution of a Modified Binomial Random Graph by Agglomeration
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In the classical Erdős–Rényi random graph G(n, p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n, p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erdős-Rényi random graph G(n, p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n, p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n, p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real-world applications.
KeywordsErdős–Rényi model Random graph Inhomogeneous random graph Connectedness Phase transition
Mathematics Subject Classification05C80 60C05
We thank Serguei Popov for suggesting us the construction of the inhomogeneous random graph model studied in the paper. We also thank Luiz Renato Fontes for fruitful discussions during the early stages of this work. The first two authors were financially supported by DFG KA 2748/3-1 and the Austrian Science Fund (FWF): P26826, and the last one by FAPESP 2013/03898-8, 2015/03868-7 and CNPq 479313/2012-1. The second and the third author also thank, respectively, ICMC - Universidade de São Paulo and Universitá di Torino, for their hospitality. The authors are grateful to the anonymous reviewers for their interesting comments and suggestions.
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