Let c ∈ Z m >0 , A ∈ Z n×m , and b ∈ Z n . We show under fairly general conditions that the non-uniform Physarum dynamicsẋ e = a e (x,t) (|q e | − x e ) converges to the optimum solution x * of the weighted basis pursuit problem minimize c T x subject to A f = b and | f | ≤ x. Here, f and x are m-vectors of real variables, q minimizes the energy ∑ e (c e /x e )q 2 e subject to the constraints Aq = b and supp(q) ⊆ 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 dynamicṡ 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.
This was generalized to the non-uniform dynamics (6) in REF .
Convergence of the Non-Uniform Physarum Dynamics
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