Convergence rate of $\ell^p$-relaxation on a graph to a $p$-harmonic function with given boundary values

TL;DR

This paper studies the convergence rate of $ ext{l}^p$-relaxation dynamics on graphs to $p$-harmonic functions, establishing bounds on the approximation time that are optimal and extend previous work to include boundary conditions.

Contribution

It provides the first bounds on the mean approximation time for $ ext{l}^p$-relaxation with boundary values on arbitrary graphs, showing these bounds are tight and extending prior results.

Findings

Mean approximation time is at most polynomial in the number of vertices.

Bounds are tight and match the known lower bounds for specific graph classes.

Results extend to graphs with given average degree, with sharper bounds.

Abstract

We analyze the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Let $V = I \bigcup B$, where the set of interior vertices $I \ne \emptyset$ is disjoint from the set of boundary vertices $B \neq \emptyset$. Given $p > 1$ and an initial opinion profile $f_0: V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v_t \in I$ is selected, and the opinion there is updated to the value $f_{t}(v_t)$ that minimizes the sum $\sum_{w \sim v_t} \lvert f_t(v_t)-f_{t-1}(w) \rvert^p$ over neighbours $w$ of $v_t$. The case $p=2$ yields linear averaging dynamics, but for all $p \ne 2$ the dynamics are nonlinear. It is well known that almost surely, $f_t$ converges to the $p$-harmonic extension $h$ of $f_0 \vert_{B}$. Denote the number of steps needed to obtain $\lVert f_t - h \rVert_{\infty} \le \epsilon$ by $\tau_p(\epsilon).$ Recently, Amir, Nazarov, and Peres~\cite{noboundarycase} analyzed the same dynamics without boundary. For individual graphs, adding boundary values can slow down the convergence considerably; indeed, when $p = 2$ the approximation time is controlled by the hitting time of the boundary by random walk, and hitting times can be much larger than mixing times, which control the convergence when $B=\emptyset$. Nevertheless, we show that for all graphs with $n$ vertices, the mean approximation time $\E[\tau_p(\epsilon)]$ is at most $n^{\beta_p}$ (up to logarithmic factors in $\frac{n}{\epsilon}$ for $p \in [2, \infty)$, and polynomial factors in $\epsilon^{-1}$ for $p \in (1, 2)$), where $\beta_p=\max\big(\frac{2p}{p-1},3\big)$. This matches the definition of $\beta_p$ given in \cite{noboundarycase} and answers Question 6.2 in that paper. The exponent $\beta_p$ is optimal in both settings. We also prove sharp bounds for $n$-vertex graphs with given average degree, that are technically more challenging.