The difficulty in formulating analytical treatments in co-evolving networks is studied in light of the Vazquez-Egu ' iluz-San Miguel voter model (VM) and a modified VM (MVM) that introduces a random mutation of the opinion as a noise in the VM. The density of active links, which are links that connect the nodes of opposite opinions, is shown to be highly sensitive to both the degree k of a node and the active links n among the neighbors of a node. We test the validity in the formalism of analytical approaches and show explicitly that the assumptions behind the commonly used homogeneous pair approximation scheme in formulating a mean-field theory are the source of the theory's failure due to the strong correlations between k, n and n2. An improved approach that incorporates spatial correlation to the nearest-neighbors explicitly and a random approximation for the next-nearest neighbors is formulated for the VM and the MVM, and it gives better agreement with the simulation results. We introduce an empirical approach that quantifies the correlations more accurately and gives results in good agreement with the simulation results. The work clarifies why simply mean-field theory fails and sheds light on how to analyze the correlations in the dynamic equations that are often generated in co-evolving processes.

The role of punishments in promoting cooperation is an important issue. We incorporate costly punishments into the snowdrift game (SG) by introducing a third punishing (P) character, and study the effects. The punishers, who carry basically a cooperative (C) character, are willing to pay a cost α so as to punish a non-cooperative (D) opponent by β. Depending on the initial fractions of the characters, α, β, and the cost-to-benefit ratio r in the SG, the three-character system evolves into a steady state consisting either only of C and P characters or only of C and D characters, in a well-mixed population. The former situation represents an enhancement in cooperation relative to the SG, while the latter is similar to the SG. The dynamics in approaching these different steady states are found to be different. Analytically, the key features in the dynamics and the steady states observed in simulations are captured by a set of differential equations. The sensitivity to the initial distribution of characters is studied by depicting the flow in a phase portrait and analyzing the nature of fixed points. The analysis also shows the role of P-character agents in preventing a system from invasion by D-character agents. Starting from a population consisting only of C and P agents, a D-character agent intended to invade the system cannot survive when the initial fraction of P agents is greater than r/β. Our model, defined intentionally as a simulation algorithm, can be readily generalized to incorporate many interesting effects, such as those in a networked population.

The N -person evolutionary snowdrift game (NESG) is generalized to study the effects of the additional benefit to all agents in a competing group of size N resulting from an earlier completion of a task when more agents are willing to share the work. Following replicator dynamics, an equation that can be used to solve for the steady state frequency of cooperation x ∗ in a well-mixed population as a function of the parameters representing the cost-to-benefit ratio c / b , additional reward w / b , and N is derived. Cooperation is enhanced in general for w ≠ 0 and a stable state with all cooperative agents (AllC state) emerges for small groups N and small c / b . In contrast, such a harmonious AllC state does not exist in the original NESG for c ≠ 0 . The condition for the existence of an AllC state is estimated to be ( N − 1 ) c w and good agreement is found when compared with numerical solutions. The observed disappearance of an AllC state and the drop in x ∗ with N are discussed in terms of the change in the stability of the AllC state as N increases. Previous results in NESG are recovered as the w = 0 case of our model. Thus, the pleasure of enjoying the completed task earlier promotes cooperation and the effects are more pronounced when small-group collective interactions are considered.