Optimal network geometry detection for weak geometry

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.