Expressivity of Geometric Inhomogeneous Random Graphs -- Metric and Non-Metric
Benjamin Dayan, Marc Kaufmann, Ulysse Schaller
TL;DR
This work extends the framework for evaluating generative graph models to Geometric Inhomogeneous Random Graphs (GIRGs), including non-metric distance variants, to better capture real-world network geometry. It introduces joint estimation techniques for the GIRG parameters $c$ and $\alpha$, along with estimating the power-law exponent $\tau$, and presents an $O(dn)$ sampling method for Minimum-Component-Distance GIRGs, plus a coupling approach to sample cube-topology GIRGs from torus-topology ones. Empirical evaluation on 104 Facebook networks shows GIRGs align well with geometric features such as closeness and betweenness centrality and local clustering, outperforming several baseline models in these aspects, though they struggle with high-variance statistics and obtain smaller diameters than observed in real networks. The paper provides practical tools for practitioners modeling networks with GIRGs and outlines future directions including more non-metric variants, assortativity, disconnected graphs, and broader model comparisons, highlighting the role of geometry in network expressivity.
Abstract
Recently there has been increased interest in fitting generative graph models to real-world networks. In particular, Bläsius et al. have proposed a framework for systematic evaluation of the expressivity of random graph models. We extend this framework to Geometric Inhomogeneous Random Graphs (GIRGs). This includes a family of graphs induced by non-metric distance functions which allow capturing more complex models of partial similarity between nodes as a basis of connection - as well as homogeneous and non-homogeneous feature spaces. As part of the extension, we develop schemes for estimating the multiplicative constant and the long-range parameter in the connection probability. Moreover, we devise an algorithm for sampling Minimum-Component-Distance GIRGs whose runtime is linear both in the number of vertices and in the dimension of the underlying geometric space. Our results provide evidence that GIRGs are more realistic candidates with respect to various graph features such as closeness centrality, betweenness centrality, local clustering coefficient, and graph effective diameter, while they face difficulties to replicate higher variance and more extreme values of graph statistics observed in real-world networks.