Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure

Range expansion with mutation and selection: dynamical phase transition in a two-species Eden model

The colonization of unoccupied territory by invading species, known as range expansion, is a spatially heterogeneous non-equilibrium growth process. We introduce a two-species Eden growth model to analyze the interplay between uni-directional (irreversible) mutations and selection at the expanding front. While the evolutionary dynamics leads to coalescence of both wild-type and mutant clusters, the non-homogeneous advance of the colony results in a rough front. We show that roughening and domain dynamics are strongly coupled, resulting in qualitatively altered bulk and front properties. For beneficial mutations the front is quickly taken over by mutants and growth proceeds Eden-like. In contrast, if mutants grow slower than wild-types, there is an antagonism between selection pressure against mutants and growth by the merging of mutant domains with an ensuing absorbing state phase transition to an all-mutant front. We find that surface roughening has a marked effect on the critical properties of the absorbing state phase transition. While reference models, which keep the expanding front flat, exhibit directed percolation critical behavior, the exponents of the two-species Eden model strongly deviate from it. In turn, the mutation-selection process induces an increased surface roughness with exponents distinct from that of the classical Eden model

The pace of evolution across fitness valleys

How fast does a population evolve from one fitness peak to another? We study the dynamics of evolving, asexually reproducing populations in which a certain number of mutations jointly confer a fitness advantage. We consider the time until a population has evolved from one fitness peak to another one with a higher fitness. The order of mutations can either be fixed or random. If the order of mutations is fixed, then the population follows a metaphorical ridge, a single path. If the order of mutations is arbitrary, then there are many ways to evolve to the higher fitness state. We address the time required for fixation in such scenarios and study how it is affected by the order of mutations, the population size, the fitness values and the mutation rate

Phase transition in a spatial Lotka-Volterra model

Spatial evolution is investigated in a simulated system of nine competing and mutating bacterium strains, which mimics the biochemical war among bacteria capable of producing two different bacteriocins (toxins) at most. Random sequential dynamics on a square lattice is governed by very symmetrical transition rules for neighborhood invasion of sensitive strains by killers, killers by resistants, and resistants by by sensitives. The community of the nine possible toxicity/resistance types undergoes a critical phase transition as the uniform transmutation rates between the types decreases below a critical value $P_c$ above which all the nine types of strain coexist with equal frequencies. Passing the critical mutation rate from above, the system collapses into one of the three topologically identical states, each consisting of three strain types. Of the three final states each accrues with equal probability and all three maintain themselves in a self-organizing polydomain structure via cyclic invasions. Our Monte Carlo simulations support that this symmetry breaking transition belongs to the universality class of the three-state Potts model.Comment: 4 page

Evolution of new regulatory functions on biophysically realistic fitness landscapes

Regulatory networks consist of interacting molecules with a high degree of mutual chemical specificity. How can these molecules evolve when their function depends on maintenance of interactions with cognate partners and simultaneous avoidance of deleterious "crosstalk" with non-cognate molecules? Although physical models of molecular interactions provide a framework in which co-evolution of network components can be analyzed, most theoretical studies have focused on the evolution of individual alleles, neglecting the network. In contrast, we study the elementary step in the evolution of gene regulatory networks: duplication of a transcription factor followed by selection for TFs to specialize their inputs as well as the regulation of their downstream genes. We show how to coarse grain the complete, biophysically realistic genotype-phenotype map for this process into macroscopic functional outcomes and quantify the probability of attaining each. We determine which evolutionary and biophysical parameters bias evolutionary trajectories towards fast emergence of new functions and show that this can be greatly facilitated by the availability of "promiscuity-promoting" mutations that affect TF specificity