Phase transitions in a coevolving snowdrift game with costly rewiring

We propose and study a dissatisfied adaptive snowdrift game with a payoff parameter $r$ that incorporates a cost for rewiring a connection. An agent, facing adverse local environment, may switch action without a cost or rewire an existing link with a cost $a$ so as to attain a better competing environment. Detailed numerical simulations reveal nontrivial and nonmonotonic dependence of the frequency of cooperation and the densities of different types of links on $a$ and $r$. A theory that treats the cooperative and noncooperative agents separately and accounts for spatial correlation up to neighboring agents is formulated. The theory gives results that are in good agreement with simulations. The frequency of cooperation $f_C$ is enhanced (suppressed) at high rewiring cost relative to that at low rewiring cost when $r$ is small (large). For a given value of $r$, there exists a critical value of the rewiring cost below which the system evolves into a phase of frozen dynamics with isolated noncooperative agents segregated from a cluster of cooperative agents, and above which the system evolves into a connected population of mixed actions with continual dynamics. The phase boundary on the $a\text{−}r$ phase space that separates the two phases with distinct structural, population and dynamical properties is mapped out. The phase diagram reveals that, as a general feature, for small $r$ (small $a$), the disconnected and segregated phase can survive over a wider range of $a\left(r\right)$.

Figure 1

Schematic illustration of the coevolving snowdrift game with rewiring cost. Open (closed) circle represents an agent of C action (D action). (a) Possible processes of C-action agent. (1) When a C-action agent meets a D-action neighbor, he may rewire the link to another agent with a probability ${\mathcal{P}}_{CR}$. Otherwise, he may switch or keep his action, with the probabilities shown. (2) When a C-action agent meets another C-action neighbor, he is satisfied. (b) Possible processes of D-action agent. (1) When a D-action agent meets another D-action neighbor, he may rewire the link to another agent with a probability ${\mathcal{P}}_{DR}$. Otherwise, he may switch or keep his action, with the probabilities shown. (2) When a D-action agent meets a C-action neighbor, he is satisfied.

Figure 2

The frequency of cooperation $f_C$ as a function of the rewiring cost $a$, for different values of the cost-to-benefit ratio. Results obtained by simulations (symbols) and by the theory given by Eqs. (9)–(11) (curves) are shown for comparison.

Figure 3

The link densities $l_{CC}$, $l_{CD}$, and $l_{DD}$ as a function of the rewiring cost $a$, for different values of the SG cost-to-benefit ratio. Results obtained by simulations (symbols) and by the theory given by Eqs. (9)–(11) (curves) are shown for comparison.

Figure 4

Phase diagram on the $a\text{−}r$ plane. The phase boundary as obtained by simulations (symbols) and by our theory (curve) are shown. The phase boundary separates a static phase characterized by a network that is disconnected, segregated in action, and frozen in dynamics (see inset) from an active phase characterized by a network that is connected, mixed in action and nonstop coevolving dynamics (see inset). Open (closed) circles represent C-action (D-action) agents, and the lines represent connections.

Figure 5

The frequency of cooperation $f_C$ as a function of the SG cost-to-benefit ratio $r$ for $a=0.1$, $0.3$, and $T=1+r$.