Why the Metric Backbone Preserves Community Structure
This addresses a theoretical gap for network analysis researchers, showing that a common sparsification technique does not harm community detection, which is incremental as it confirms prior empirical findings with formal proof.
The paper tackles the problem of whether the metric backbone, a graph sparsification method based on shortest paths, preserves community structure in networks, and proves theoretically and empirically that it does so robustly, with empirical comparisons showing it is an efficient sparsifier.
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.