On the R\'enyi index of random graphs
Abstract
Networks (graphs) permeate scientific fields such as biology, social science, economics, etc. Empirical studies have shown that real-world networks are often heterogeneous, that is, the degrees of nodes do not concentrate on a number. Recently, the R\'enyi index was tentatively used to measure network heterogeneity. However, the validity of the R\'enyi index in network settings is not theoretically justified. In this paper, we study this problem. We derive the limit of the R\'enyi index of a heterogeneous Erd\"os-R\'enyi random graph and a power-law random graph, as well as the convergence rates. Our results show that the Erd\"os-R\'enyi random graph has asymptotic R\'enyi index zero and the power-law random graph (highly heterogeneous) has asymptotic R\'enyi index one. In addition, the limit of the R\'enyi index increases as the graph gets more heterogeneous. These results theoretically justify the R\'enyi index is a reasonable statistical measure of network heterogeneity. We also evaluate the finite-sample performance of the R\'enyi index by simulation.
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