Optimizing Weighted Hodge Laplacian Flows on Simplicial Complexes

TL;DR

This work extends consensus dynamics to higher-order interactions by formulating weighted Hodge Laplacian flows on simplicial complexes. It develops a comprehensive framework: weighted Hodge Laplacians with up/down components, a weighted Hodge decomposition, and two semidefinite programs that optimally tune upper and lower simplex weights to improve spectral metrics such as $\mathrm{tr}(L_k^+)$ and $\lambda_{\min >0}(L_k)$. Theoretical results establish convexity of these objectives in the weights and provide practical SDP-based tools for globally optimal weight design, complemented by numerical experiments on Vietoris–Rips complexes showing substantial performance gains over uniform weights. These contributions enable principled, globally optimal weight design for higher-order network flows with potential impact on distributed optimization, control of multi-agent systems, and topological data analysis.

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.

Figures (4)

• Figure 1: A Vietoris Rips complex generated from 30 points sampled uniformly on the unit square with threshold 0.5. The complex contains 30 nodes, 151 edges, and 339 faces. Each face in the complex is shaded according to the optimal face weights computed with the semidefinite program \ref{['sdp-traceinv']} for the weighted Hodge Laplacian $L_1$.
• Figure 2: Optimal face weights for the 339 faces of the Vietoris Rips complex from Fig. 1. The optimal weights provide a 8.9% improvement of the Hodge Laplacian pseudoinverse compared to the unweighted Hodge Laplacian (with unity weights).
• Figure 3: Optimal face weights for the 339 faces of the Vietoris Rips complex from Fig. 1 computed with the semidefinite program \ref{['sdp-mineig']} to maximize the smallest non-zero eigenvalue of the weighted Hodge Laplacian $L_1$, with the same face ordering as in Fig. 2. The optimal weights provide a 228% increase of the smallest non-zero eigenvalue compared to the unweighted Hodge Laplacian (with unity weights).
• Figure 4: (top) Edge flows for the unweighted Hodge Laplacian; (bottom) edge flows for the weighted Hodge Laplacian with optimal weights computed with the semidefinite program \ref{['sdp-traceinv']} to maximize the smallest non-zero eigenvalue of the weighted Hodge Laplacian $L_1$, exhibiting much faster convergence.

Theorems & Definitions (14)

• Theorem 1: 5.11 in wu2018weighted
• Corollary 2
• proof
• Lemma 3: Prop. 2.6 in wu2018weighted
• Theorem 4: Weighted analogue of Prop. 1 in barbarossa2020topological
• proof
• Theorem 5
• Theorem 6: Weighted analog of Thm 6.2 in hirani2010least
• proof
• Corollary 7: Hodge Laplacian Flow Decomposition