Evolutionary Dynamics on Degree-Heterogeneous Graphs

Abstract

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process dynamics). The fixation probability for one species to take over a population of N individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree k, the fixation probability is proportional to k for voter model dynamics and to 1/k for invasion process dynamics.

Nonequivalence of updating rules in evolutionary games under high mutation rates

“…If the population structure is homogenous, populations where this assumption is true show equal fixation probabilities for all three of the evolutionary processes introduced above, as has been shown by Antal et al [29]. In generic graphs, on the other hand, this equivalence among BP, DP, and LD is lost [29].…”

“…Note that μ n λ n = f B (n) f A (n) , just as for the birth-death process, and therefore the fixation probability for the DP is indeed the same as for the BP [29]. This is not, however, true for the fixation times in the prisoner's dilemma example with payoff matrix Q as in Eq.…”

“…It is the equality of the fixation probabilities of BP, DP, and LD which motivates Antal et al [29] to refer to these three updating rules as equivalent for homogenous graphs. We remark that this is a very strong term for this specific property.…”

“…In generic graphs, on the other hand, this equivalence among BP, DP, and LD is lost [29]. This result entailed investigations into how different updating rules lead to different evolutionary dynamics on various graph structures [30,31].…”

Evolutionary prisoner׳s dilemma game on graphs and social networks with external constraint

“…A number of different updating mechanisms can be used to determine the evolving state of the graph, specifying how the composition of the population changes under natural selection. A considerable body of research has been devoted to this subject (Abramson and Kuperman, 2001;Santos and Pacheco, 2005;Antal et al, 2006;Ohtsuki et al, 2006;Ohtsuki and Nowak, 2006a,b;Pacheco et al, 2006;Santos et al, 2006a,b;Szabó and Fáth, 2007;Taylor et al, 2007;Fu and Wang, 2008;Roca et al, 2009b;Tarnita et al, 2009a,b;Perc and Szolnoki, 2010;Fehl et al, 2011;Allen et al, 2012;Cavaliere et al, 2012). Remarkably, Ohtsuki et al (2006) derived a simple rule as a good approximation for general graphs according to which natural selection favors cooperation if the benefit of the altruistic act, b, divided by its cost, c, exceeds the average number of neighbors, k (i.e., b=c 4 k implies cooperation).…”