Detecting hyperbolic geometry in networks: Why triangles are not enough

In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as scale invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We, therefore, introduce a differerent statistic, weighted triangles, which weighs triangles based on their evidence for geometry. We show analytically, as well as on synthetic and real-world data, that weighted triangles are a powerful statistic to detect hyperbolic geometry in networks.

Robust subgraph counting with distribution-free random graph analysis

Subgraphs such as cliques, loops, and stars play a crucial role in real-world networks. Random graph models can provide estimates for how often certain subgraphs appear, which can be tested against real-world networks. These estimated subgraph counts, however, crucially depend on the assumed degree distribution. Fitting a degree distribution to network data is challenging, in particular, for scale-free networks with power-law degrees. Therefore, in this paper we develop robust subgraph counts that do not depend on the entire degree distribution but only on its mean and mean absolute deviation (MAD), summary statistics that are easy to obtain for most real-world networks. By solving an optimization problem, we provide tight (the sharpest possible) bounds for the subgraph counts, for all possible subgraphs, and for all networks with degree distributions that share the same mean and MAD. We identify the extremal random graph that attains these tight bounds as the graph with a specific three-point degree distribution. We leverage the bounds on the maximal subgraph counts to obtain robust scaling laws for how the number of subgraphs grows as a function of the network size. The scaling laws indicate that sparse power-law networks are not the most extreme networks in terms of subgraph counts but dense power-law networks are. The robust bounds are also shown to hold for several real-world data sets.