Detecting hyperbolic geometry in networks: why triangles are not enough

TL;DR

This paper demonstrates that traditional triangle-based metrics are inadequate for detecting hyperbolic geometry in networks, and introduces a new weighted statistic that effectively reveals such geometric structures.

Contribution

The paper introduces a novel weighted triangle statistic that improves detection of hyperbolic geometry in networks beyond standard clustering measures.

Findings

Traditional triangle counts fail to detect hyperbolic geometry.

The new statistic effectively identifies hyperbolic structure in synthetic and real networks.

The approach enhances understanding of geometric properties in complex networks.

Abstract

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 self-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 novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks.