Topology Design for Optimal Network Coherence

TL;DR

This paper introduces a scalable greedy algorithm for designing network topologies that optimize coherence, leveraging submodularity and efficient Laplacian pseudoinverse updates to handle large networks effectively.

Contribution

It demonstrates that network coherence is submodular and develops a fast, scalable greedy algorithm with rank-one updates for optimal network design.

Findings

The greedy algorithm achieves near-optimal solutions.

The method scales to large networks beyond previous heuristics.

Fast rank-one updates significantly improve computational efficiency.

Abstract

We consider a network topology design problem in which an initial undirected graph underlying the network is given and the objective is to select a set of edges to add to the graph to optimize the coherence of the resulting network. We show that network coherence is a submodular function of the network topology. As a consequence, a simple greedy algorithm is guaranteed to produce near optimal edge set selections. We also show that fast rank one updates of the Laplacian pseudoinverse using generalizations of the Sherman-Morrison formula and an accelerated variant of the greedy algorithm can speed up the algorithm by several orders of magnitude in practice. These allow our algorithms to scale to network sizes far beyond those that can be handled by convex relaxation heuristics.