Optimizing Weighted Hodge Laplacian Flows on Simplicial Complexes

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.

💡 Research Summary

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The paper “Optimizing Weighted Hodge Laplacian Flows on Simplicial Complexes” extends the well‑established theory of graph Laplacian dynamics to the higher‑order setting of simplicial complexes by introducing a systematic framework for weighted Hodge Laplacian flows. The authors first define a weighted inner product on the space of k‑chains (C_k(K,\mathbb{R})) using a diagonal weight matrix (W_k) that assigns a positive scalar to each k‑simplex. With this inner product, the weighted boundary operator (\partial_k) and its adjoint (\partial_k^\dagger) acquire matrix representations that differ from the unweighted case by pre‑ and post‑multiplication with the appropriate weight matrices. Consequently, the weighted down‑Laplacian and weighted up‑Laplacian are given by

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