Statistical mechanics of complex networks and interacting populations

Abstract

In complex dynamical systems, microscopic processes lead to rich macroscopic behavior. Such are found in a variety of disciplines including complex networks, biological populations, and games. These are studied here using analytical arguments, computer simulations, and analysis of empirical data. First, bipartite collaboration networks are studied both empirically and numerically. Such graphs consist of two vertex types, actors and ties, and edges are allowed only between vertices with different types. Empirical measurements reveal stretched exponential degree distributions, significant clustering, and assortative mixing together with sublinear preferential attachment. A new growth model for the graphs is introduced and compared to earlier ones and empirics. All considered models predict some properties correctly while failing on several others, emphasizing the need for more developed models. Next, systems of two interacting populations, hosts and parasitoids, are studied in various environments. The phase diagram is computed both analytically and numerically on a Bethe lattice. The most crucial feature is the lack of a tricritical point: There is no boundary separating phases with both populations alive and both extinct. On scalefree graphs, the parasitoids are shown to behave similarly than the infected nodes in a well-known epidemic model, the susceptible-infected-susceptible model. Consequently, the well-known absense of an epidemic threshold is directly applicable. On square lattices, a similar system with distance-dependent spreading leads to noisy irregular spirals and oscillations with a fluctuating amplitude. The latter is a consequence of a separation of two time scales related to the oscillations. The patterns can also be understood as vortices with topological signs, and measures for the patterning based on these are introduced. Also on an empirical metapopulation landscape serving as the habitat of a butterfly and its parasitoid, spirals form and noise-sustained oscillations are observed. Finally, pattern formation and extinction probabilities are considered in two-dimensional rock-paper-scissors games, also interpretable as models of three interacting populations. It is known that a four-state variant leads to spirals, and that there is a crossover to extinction when their wave length outgrows the system size. Here, it is shown that with three states spirals do not form, but a length scale leading to a similar crossover still exists. A small asymmetry in the reaction rates is shown not to alter the average wave length of the spirals and thus not to influence the crossover.