Emergence of Multivariate Extremes in Multilayer Inhomogeneous Random Graphs

TL;DR

This paper introduces a multilayer inhomogeneous random graph model (MIRG) that captures multivariate extremes and dependence structures, extending prior single-layer results to more complex multilayer networks with practical estimation insights.

Contribution

It extends the regular variation correspondence from single-layer to multilayer inhomogeneous random graphs and analyzes extremal dependence and estimation consistency within this framework.

Findings

Multivariate regular variation in weights implies similar variation in degrees.

Extremal dependence in weights influences degree distribution dependence.

Hill estimator is consistent for degree tail index > 1 in MIRGs.

Abstract

In this paper, we propose a multilayer inhomogeneous random graph model (MIRG), whose layers may consist of both single-edge and multi-edge graphs. In the single layer case, it has been shown that the regular variation of the weight distribution underlying the inhomogeneous random graph implies the regular variation of the typical degree distribution. We extend this correspondence to the multilayer case by showing that the multivariate regular variation of the weight distribution implies the multivariate regular variation of the asymptotic degree distribution. Furthermore, in certain circumstances, the extremal dependence structure present in the weight distribution will be adopted by the asymptotic degree distribution. By considering the asymptotic degree distribution, a wider class of Chung-Lu and Norros-Reittu graphs may be incorporated into the MIRG layers. Additionally, we prove consistency of the Hill estimator when applied to degrees of the MIRG that have a tail index greater than 1. Simulation results indicate that, in practice, hidden regular variation may be consistently detected from an observed MIRG.