The prisoners dilemma on a stochastic non-growth network evolution model

We investigate the evolution of cooperation on a non - growth network model with death/birth dynamics. Nodes reproduce under selection for higher payoffs in a prisoners dilemma game played between network neighbours. The mean field characteristics of the model are explored and an attempt is made to understand the size dependent behaviour of the model in terms of fluctuations in the strategy densities. We also briefly comment on the role of strategy mutation in regulating the strategy densties.

💡 Research Summary

The paper introduces a novel framework for studying the evolution of cooperation on a non‑growth network in which the number of nodes remains constant while a death‑birth (die‑birth) process continuously rewires the topology. At each discrete time step a randomly chosen node is removed together with all of its incident edges. A new node then enters the system by copying the strategy of an existing “parent” node; the probability of being chosen as a parent is proportional to the cumulative payoff that node earned by playing the Prisoner’s Dilemma (PD) with its neighbours. This selective replication implements evolutionary pressure favouring higher‑payoff strategies. In addition, with a small mutation probability μ the newborn may adopt the opposite strategy (C↔D), providing a source of stochastic novelty.

The underlying game is the classic two‑player PD with payoff ordering T > R > P > S. Each edge represents a pairwise interaction; after each round every node computes the average payoff obtained from all its neighbours and uses this value as its fitness for the replication step. The network is initially an Erdős‑Rényi random graph of size N and average degree ⟨k⟩, but its structure evolves as edges are deleted with the death of a node and recreated when the newborn connects to the parent’s neighbours.

A mean‑field (MF) analysis is performed by defining ρ(t) as the fraction of cooperators in the population. Assuming an infinite system, the authors derive a deterministic differential equation of logistic form:

 dρ/dt = ρ(1 − ρ)