Why the Metric Backbone Preserves Community Structure

TL;DR

This paper demonstrates both theoretically and empirically that the metric backbone of a weighted graph preserves community structure despite removing many intra-community edges, acting as an effective sparsifier.

Contribution

It provides a formal proof of community structure preservation in the metric backbone and empirically compares it with other sparsification methods.

Findings

Metric backbone preserves community structure in real networks.

Theoretical proof of robustness of community structure after backbone extraction.

Empirical validation shows metric backbone as an efficient sparsifier.

Abstract

The metric backbone of a weighted graph is the union of all-pairs shortest paths. It is obtained by removing all edges $(u,v)$ that are not the shortest path between $u$ and $v$. In networks with well-separated communities, the metric backbone tends to preserve many inter-community edges, because these edges serve as bridges connecting two communities, but tends to delete many intra-community edges because the communities are dense. This suggests that the metric backbone would dilute or destroy the community structure of the network. However, this is not borne out by prior empirical work, which instead showed that the metric backbone of real networks preserves the community structure of the original network well. In this work, we analyze the metric backbone of a broad class of weighted random graphs with communities, and we formally prove the robustness of the community structure with respect to the deletion of all the edges that are not in the metric backbone. An empirical comparison of several graph sparsification techniques confirms our theoretical finding and shows that the metric backbone is an efficient sparsifier in the presence of communities.