# Convergence of the Non-Uniform Physarum Dynamics

## TL;DR

This paper proves convergence of a generalized Physarum dynamics to the optimal solution of a weighted basis pursuit problem, extending previous results from uniform cases to more general, non-uniform and functional dynamics.

## Contribution

It establishes convergence for non-uniform Physarum dynamics with general reactivity functions, broadening the understanding beyond uniform and specific shortest path cases.

## Key findings

- Convergence shown for non-uniform Physarum dynamics under general conditions.
- Extended convergence results to dynamics with multiplicative factors involving functions g_e.
- Demonstrated convergence for a broader class of problems beyond shortest path.

## Abstract

Let $c \in \mathbb{Z}^m_{> 0}$, $A \in \mathbb{Z}^{n\times m}$, and $b \in \mathbb{Z}^n$. We show under fairly general conditions that the non-uniform Physarum dynamics \[ \dot{x}_e = a_e(x,t) \left(|q_e| - x_e\right) \] converges to the optimum solution $x^*$ of the weighted basis pursuit problem minimize $c^T x$ subject to $A f = b$ and $|f| \le x$. Here, $f$ and $x$ are $m$-vectors of real variables, $q$ minimizes the energy $\sum_e (c_e/x_e) q_e^2$ subject to the constraints $A q = b$ and $\mathrm{supp}(q) \subseteq \mathrm{supp}(x)$, and $a_e(x,t) > 0$ is the reactivity of edge $e$ to the difference $|q_e| - x_e$ at time $t$ and in state $x$. Previously convergence was only shown for the uniform case $a_e(x,t) = 1$ for all $e$, $x$, and $t$. We also show convergence for the dynamics \[ \dot{x}_e = x_e \cdot \left( g_e \left(\frac{|q_e|}{x_e}\right) - 1\right),\] where $g_e$ is an increasing differentiable function with $g_e(1) = 1$. Previously convergence was only shown for the special case of the shortest path problem on a graph consisting of two nodes connected by parallel edges.