In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. The new model, that couples an elliptic equation enforcing PP density balance with an ODE governing the dynamics of the flow of information along the PP body, is analyzed by recasting it into an infinite-dimensional dynamical system. We are able to show well-posedness of the proposed model for sufficiently small times under the hypothesis of Hölder continuous diffusion coefficients and essentially bounded forcing functions, which play the role of food sources. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior.A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P 1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.

We extend our previous work on a biologically inspired dynamic Monge-Kantorovich model [18] and propose it as an effective tool for the numerical solution of the L 1 -PDE based optimal transportation model. Starting from the conjecture that the dynamic model is timeasymptotically equivalent to the Monge-Kantorovich equations governing L 1 optimal transport, we experimentally analyze a simple yet effective numerical approach for the quantitative solution of these equations.The novel contributions in this paper are twofold. First, we introduce a new Lyapunov-candidate functional that better adheres to the dynamics of our proposed model. It is shown that the Lie derivative of the new Lyapunov-candidate functional is strictly negative and, more remarkably, the OT density is the unique minimizer for this new Lyapunov-candidate functional, providing further support to the conjecture of asymptotic equivalence of our dynamic model with the Monge-Kantorovich equations. Second, we describe and test different numerical approaches for the solution of our problem. The ordinary differential equation for the transport density is projected into a piecewise constant or linear finite dimensional space defined on a triangulation of the domain. The elliptic equation is discretized using a linear Galerkin finite element method defined on uniformly refined triangles. The ensuing nonlinear differential-algebraic equation is discretized by means of a first order Euler method (forward or backward) and a simple Picard iteration is used to resolve the nonlinearity. The use of two discretization levels is dictated by the need to avoid oscillations on the potential gradients that prevent convergence of the scheme.We study the experimental convergence rate of the proposed solution approaches and discuss limitations and advantages of these formulations. An extensive set of test cases, including problems that admit an explicit solution to the Monge-Kantorovich equations are appropriately designed to verify and test the expected numerical properties of the solution methods. Finally, a comparison with literature methods is performed and the ensuing transport maps are compared. The results show that optimal convergence toward the asymptotic equilibrium point is achieved for sufficiently regular forcing function, and that the proposed method is accurate, robust, and computationally efficient. 1 arXiv:1709.06765v2 [math.NA]

Routing optimization is a relevant problem in many contexts. Solving directly this type of optimization problem is often computationally intractable. Recent studies suggest that one can instead turn this problem into one of solving a dynamical system of equations, which can instead be solved efficiently using numerical methods. This results in enabling the acquisition of optimal network topologies from a variety of routing problems. However, the actual extraction of the solution in terms of a final network topology relies on numerical details which can prevent an accurate investigation of their topological properties. In fact, in this context, theoretical results are fully accessible only to an expert audience and ready-to-use implementations for non-experts are rarely available or insufficiently documented. In particular, in this framework, final graph acquisition is a challenging problem in-and-of-itself. Here we introduce a method to extract network topologies from dynamical equations related to routing optimization under various parameters’ settings. Our method is made of three steps: first, it extracts an optimal trajectory by solving a dynamical system, then it pre-extracts a network, and finally, it filters out potential redundancies. Remarkably, we propose a principled model to address the filtering in the last step, and give a quantitative interpretation in terms of a transport-related cost function. This principled filtering can be applied to more general problems such as network extraction from images, thus going beyond the scenarios envisioned in the first step. Overall, this novel algorithm allows practitioners to easily extract optimal network topologies by combining basic tools from numerical methods, optimization and network theory. Thus, we provide an alternative to manual graph extraction which allows a grounded extraction from a large variety of optimal topologies. The analysis of these may open up the possibility to gain new insights into the structure and function of optimal networks. We provide an open source implementation of the code online.

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