Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions

TL;DR

This work addresses the computation of graph $p$-Laplacian eigenpairs for $p>2$ by reformulating the nonlinear problem as a constrained linear weighted Laplacian eigenproblem with edge and node densities $\mu=|\nabla f|^{p-2}$ and $\nu=|f|^{p-2}$. It then develops a family of differentiable energy functions $\mathcal{E}_{p,k}$ whose saddle points correspond to $p$-Laplacian eigenpairs, with the Morse index matching a linear indexing from the associated weighted Laplacian. The authors derive gradient-flow algorithms that solve a sequence of linear eigenproblems, enabling computation of higher eigenpairs without prior knowledge of lower ones, and prove a unique characterization for the first eigenpair via a dedicated energy $\mathcal{L}$. Numerical experiments on a unit-square graph illustrate the approach and highlight convergence behavior, including cases with eigenvalue multiplicity. Overall, the framework advances nonlinear spectral theory on graphs and provides practical, gradient-based tools for clustering and partitioning using the $p$-Laplacian spectrum.

Abstract

We address the problem of computing the graph $p$-Laplacian eigenpairs for $p\in (2,\infty)$. We propose a reformulation of the graph $p$-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between $p$-Laplacian eigenpairs and linear eigenpair of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any $p$-Laplacian eigenpair that matches the Morse index of the $p$-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the $k$-th energy function correspond to $p$-Laplacian eigenpairs having index equal to $k$. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first $p$-Laplacian eigenpair. Finally we develop novel gradient-based numerical methods suited to compute $p$-Laplacian eigenpairs for any $p\in(2,\infty)$ and present some experiments.

Figures (2)

• Figure 1: A graph with non-simple first eigenvalue. Assume $\nu_u=1\; \forall \, u\in V\setminus B$, then the graph is symmetric and the first eigenfunction of $\Delta_p$, $f$, is unique and necessarily agrees with the symmetry of the graph. This means that $\nabla f(3,4)=0$ and thus the density $\mu=|\nabla f|^{p-2}$ of eq. \ref{['linear_weighted_eigenpairs']} is zero on the edge $(3,4)$, splitting $\mathcal{G}$ in two connected components. As a result, $\lambda$ is not simple and $\mathcal{E}_{p,1}$ is not differentiable.
• Figure 2: First nine eigenfunctions as calculated by the proposed method for $p=3$ displayed in sequential order from $k=1$ to $k=9$ (left to right, top to bottom). For each $k$, the top panel shows the nodal values of the eigenfunctions, while the bottom panel reports the behavior of the log residual defined in eq. \ref{['residual']} as a function of time steps (iterations) $n$.

Theorems & Definitions (30)

• Definition 2.1: Connected graph
• Definition 2.2: $p$-Laplace operator
• Theorem 2.3: from Hua
• Definition 3.1
• Lemma 3.2
• Definition 3.3: Morse Index
• Proposition 3.4
• Proof 1
• Theorem 4.1
• Theorem 4.2