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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.