Topological estimation of the latent geometry of a complex network

TL;DR

This paper introduces novel topological methods based on persistent homology to estimate the hidden geometric structure of complex networks, overcoming challenges posed by long-range links that obscure latent geometry signatures.

Contribution

It develops a set of topological analysis techniques, including modified persistent homology and latent geometry mapping, to accurately infer the hidden geometry of complex networks.

Findings

Successfully applied to synthetic networks

Validated on empirical networks

Revealed topological properties despite long-range links

Abstract

Most real-world networks are embedded in latent geometries. If a node in a network is found in the vicinity of another node in the latent geometry, the two nodes have a disproportionately high probability of being connected by a link. The latent geometry of a complex network is a central topic of research in network science, which has an expansive range of practical applications such as efficient navigation, missing link prediction, and brain mapping. Despite the important role of topology in the structures and functions of complex systems, little to no study has been conducted to develop a method to estimate the general unknown latent geometry of complex networks. Topological data analysis, which has attracted extensive attention in the research community owing to its convincing performance, can be directly implemented into complex networks; however, even a small fraction (0.1%) of long-range links can completely erase the topological signature of the latent geometry. Inspired by the fact that long-range links in a network have disproportionately high loads, we develop a set of methods that can analyze the latent geometry of a complex network: the modified persistent homology diagram and the map of the latent geometry. These methods successfully reveal the topological properties of the synthetic and empirical networks used to validate the proposed methods.