Noise-Induced Network Topologies.

We analyze transport on a graph with multiple constraints and where the weight of the edges connecting the nodes is a dynamical variable. The network dynamics results from the interplay between a nonlinear function of the flow, dissipation, and Gaussian, additive noise. For a given set of parameters and finite noise amplitudes, the network self-organizes into one of several metastable configurations, according to a probability distribution that depends on the noise amplitude α. At a finite value α, we find a resonantlike behavior for which one network topology is the most probable stationary state. This specific topology maximizes the robustness and transport efficiency, it is reached with the maximal convergence rate, and it is not found by the noiseless dynamics. We argue that this behavior is a manifestation of noise-induced resonances in network self-organization. Our findings show that stochastic dynamics can boost transport on a nonlinear network and, further, suggest a change of paradigm about the role of noise in optimization algorithms.

Physarum can compute shortest paths.

Physarum polycephalum is a slime mold that is apparently able to solve shortest path problems. A mathematical model has been proposed by Tero et al. (Journal of Theoretical Biology, 244, 2007, pp. 553-564) to describe the feedback mechanism used by the slime mold to adapt its tubular channels while foraging two food sources s(0) and s(1). We prove that, under this model, the mass of the mold will eventually converge to the shortest s(0)-s(1) path of the network that the mold lies on, independently of the structure of the network or of the initial mass distribution. This matches the experimental observations by Tero et al. and can be seen as an example of a "natural algorithm", that is, an algorithm developed by evolution over millions of years.

Interplay of periodic dynamics and noise: Insights from a simple adaptive system.

We study the dynamics of a simple adaptive system in the presence of noise and periodic damping. The system is composed by two paths connecting a source and a sink, and the dynamics is governed by equations that usually describe food search of the paradigmatic Physarum polycephalum. In this work we assume that the two paths undergo damping whose relative strength is periodically modulated in time, and we analyze the dynamics in the presence of stochastic forces simulating Gaussian noise. We identify different responses depending on the modulation frequency and on the noise amplitude. At frequencies smaller than the mean dissipation rate, the system tends to switch to the path which minimizes dissipation. Synchronous switching occurs at an optimal noise amplitude which depends on the modulation frequency. This behavior disappears at larger frequencies, where the dynamics can be described by the time-averaged equations. Here we find metastable patterns that exhibit the features of noise-induced resonances.