On the Rényi index of random graphs

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\'{e}nyi index was tentatively used to measure network heterogeneity. However, the validity of the R\'{e}nyi index in network settings is not theoretically justified. In this paper, we study this problem. We derive the limit of the R\'{e}nyi index of a heterogeneous Erd\"{o}s-R\'{e}nyi random graph and a power-law random graph, as well as the convergence rates. Our results show that the Erd\"{o}s-R\'{e}nyi random graph has asymptotic R\'{e}nyi index zero and the power-law random graph (highly heterogeneous) has asymptotic R\'{e}nyi index one. In addition, the limit of the R\'{e}nyi index increases as the graph gets more heterogeneous. These results theoretically justify the R\'{e}nyi index is a reasonable statistical measure of network heterogeneity. We also evaluate the finite-sample performance of the R\'{e}nyi index by simulation.