Optimal network geometry detection for weak geometry

TL;DR

This paper explores the limits of detecting underlying geometric structures in networks with weak geometric signals, proposing a mixed-integer linear programming approach to identify effective distinguishing statistics.

Contribution

It introduces a novel mixed-integer linear programming method to detect weak geometric signals in networks, extending analysis to models with power-law degrees including hyperbolic random graphs.

Findings

Effective subgraph and degree-based statistics identified for weak geometry detection

Mixed-integer programming successfully distinguishes geometric from non-geometric networks even with weak signals

Method applicable to models with power-law degrees, including hyperbolic random graphs

Abstract

Network geometry, characterized by nodes with associated latent variables, is a fundamental feature of real-world networks. Still, when only the network edges are given, it may be difficult to assess whether the network contains an underlying source of geometry. This paper investigates the limits of geometry detection in networks in a wide class of models that contain geometry and power-law degrees, which include the popular hyperbolic random graph model. We specifically focus on the regime in which the geometric signal is weak, characterized by the inverse temperature $1/T<1$. We show that the dependencies between edges can be tackled through Mixed-Integer Linear Problems, which lift the non-linear nature of network analysis into an exponential space in which simple linear optimization techniques can be employed. This approach allows us to investigate which subgraph and degree-based statistic is most effective at detecting the presence of an underlying geometric space. Interestingly, we show that even when the geometric effect is extremely weak, our Mixed-Integer programming identifies a network statistic that efficiently distinguishes geometric and non-geometric networks.