Optimizing Weighted Hodge Laplacian Flows on Simplicial Complexes

TL;DR

This paper develops a framework for optimizing weighted Hodge Laplacian flows on simplicial complexes, enabling efficient determination of weights that improve network flow metrics through convex optimization.

Contribution

It introduces a novel weighted Hodge Laplacian flow framework, proves joint convexity of key spectral functions, and formulates semidefinite programs for optimal weight selection.

Findings

Optimal weights significantly improve spectral metrics over uniform weights

Joint convexity allows efficient global optimization of weights

Numerical results confirm the effectiveness of the proposed approach

Abstract

Simplicial complexes are generalizations of graphs that describe higher-order network interactions among nodes in the graph. Network dynamics described by graph Laplacian flows have been widely studied in network science and control theory, and these can be generalized to simplicial complexes using Hodge Laplacians. We study weighted Hodge Laplacian flows on simplicial complexes. In particular, we develop a framework for weighted consensus dynamics based on weighted Hodge Laplacian flows and show some decomposition results for weighted Hodge Laplacians. We then show that two key spectral functions of the weighted Hodge Laplacians, the trace of the pseudoinverse and the smallest non-zero eigenvalue, are jointly convex in upper and lower simplex weights and can be formulated as semidefinite programs. Thus, globally optimal weights can be efficiently determined to optimize flows in terms of these functions. Numerical experiments demonstrate that optimal weights can substantially improve these metrics compared to uniform weights.