Convergence of the Non-Uniform Directed Physarum Model

The directed Physarum dynamics is known to solve positive linear programs: minimize $c^T x$ subject to $Ax = b$ and $x \ge 0$ for a positive cost vector $c$. The directed Physarum dynamics evolves a positive vector $x$ according to the dynamics $\dot{x} = q(x) - x$. Here $q(x)$ is the solution to $Af = b$ that minimizes the "energy" $\sum_i c_i f_i^2/x_i$. In this paper, we study the non-uniform directed dynamics $\dot{x} = D(q(x) - x)$, where $D$ is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with $D$ being the identity matrix), as it allows each component of $x$ to react with different speed to the differences between $q(x)$ and $x$. Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.