Topological dynamics of small degree networks

Abstract

In self-organizing networks, structure and dynamics interact in a unique way. Local activity sustained on the network organizes in global dynamic patterns to which the network topology adapts. It is believed that simple organisms lacking neural structures may derive their competence for complex behavior from this type of mechanistic interaction. One such organism is the slime mold Physarum polycephalum, which has emerged as an experimentally accessible model system for self-organizing Transport networks over the recent years. Its remarkable features include the formation of a network structure optimized for transport, and various higher-level behaviors such as efficient foraging decisions and learning. Although significant progress has been made in uncovering these behaviors and the origins of dynamic processes taking place on the network, their relation via structural remodeling still lacks detailed understanding. Combining experimental approaches with modeling and simulation, this work focuses on reorganization of the P. polycephalum network after fragmentation, a process occurring in a percolation transition in an expanding system. Data analysis and modeling are based on a novel method for graph extraction from spatial networks, which regards nodes as extended objects and fixes the number of degree two nodes in a graph. Results suggest that the network formation process is separated into four functionally distinctive phases that are closely related to the topological state. Prominently, topological development is concluded once P. polycephalum reaches a steady state in which it continues to expand while maintaining a constant degree distribution, characterized by nodes of degrees one to four, with a negligible number of large degree nodes. Using this small degree property, analytical solutions to the random graph and configuration models of graph theory are derived. Comparison to experimental data reveals a shift of the percolation transition, which through modeling and Simulation is attributed to active growth processes in the slime mold. A model consisting of a deterministic rate equation and a stochastic master equation is devised based on the concept that the topological evolution can be decomposed into a sequence of elementary processes of four distinct classes, each representing one possible type of interaction between nodes. The model, characterized by a set of rate constants obtained from experimental data, describes the topological Dynamics with excellent accuracy. In a simplified setting, the influence of interaction types is analyzed, and via simulations it is found that system growth shifts the percolation transition according to a power law. Furthermore, the percolation critical exponents are determined, concluding that the simulated master equation model shares a universality class with mean-field percolation, whereas preliminary two-dimensional simulations, and by extension, Network formation in P. polycephalum, are characterized by the exponents for two-dimensional percolation.