Can one see the shape of a network?

TL;DR

This paper introduces a novel approach to network analysis by using edge-based properties and global characteristics like curvature and Euler characteristic, providing insights into the network's shape and structure.

Contribution

It develops a method to analyze networks through Ricci curvature and global invariants, offering a new perspective beyond local vertex properties.

Findings

Curvature flow can simplify networks to their essential structure.

Euler characteristic relates to the asymptotic behavior of curvature flow.

Global network shape can be inferred from curvature and topological invariants.

Abstract

Traditionally, network analysis is based on local properties of vertices, like their degree or clustering, and their statistical behavior across the network in question. This paper develops an approach which is different in two respects. We investigate edge-based properties, and we define global characteristics of networks directly. The latter will provide our affirmative answer to the question raised in the title. More concretely, we start with Forman's notion of the Ricci curvature of a graph, or more generally, a polyhedral complex. This will allow us to pass from a graph as representing a network to a polyhedral complex for instance by filling in triangles into connected triples of edges and to investigate the resulting effect on the curvature. This is insightful for two reasons: First, we can define a curvature flow in order to asymptotically simplify a network and reduce it to its essentials. Second, using a construction of Bloch, which yields a discrete Gauss-Bonnet theorem, we have the Euler characteristic of a network as a global characteristic. These two aspects beautifully merge in the sense that the asymptotic properties of the curvature flow are indicated by that Euler characteristic.