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.

翻译：本文提出一种多层非均匀随机图模型（MIRG），其各层可由单边图与多边图共同构成。在单层情形下，已有研究证明非均匀随机图底层权重分布的规则变化会导致典型度分布的规则变化。我们将这一对应关系推广至多层情形，通过证明权重分布的多元规则变化将导致渐近度分布的多元规则变化。进一步地，在特定条件下，权重分布中存在的极值依赖结构将被渐近度分布继承。通过考虑渐近度分布，可将更广泛的Chung-Lu图与Norros-Reittu图纳入MIRG各层。此外，我们证明了当MIRG度序列的尾指数大于1时，Hill估计量具有相合性。仿真结果表明，在实际应用中可从观测到的MIRG中稳定检测到隐式规则变化。