### What Is Most Influential to Enhance Network Reciprocity? Is Topology So Critically Influential on Network Reciprocity

As mentioned above, many studies have dealt with network reciprocity in the last decades. From the network topology viewpoint, previous works have assumed evolutionary games to be played not only on regular graphs where respective vertices have identical degrees (e.g., a ring or square lattice, which we refer to as regular networks), but also on complex networks where vertices have different degrees, which we call degree-heterogeneous networks.

Reviewing previous studies on network reciprocity, we notice that each study draws a different conclusion based on its particular assumptions relating to the various choices we have described, although several comprehensive review papers have been shown (Szabo and Fath 2007; Perc and Szolnoki 2010). Thus, we cannot generally answer quantitatively such questions as to whether network topology is more influential than the update rule or; whether the factorial effect of degree-heterogeneous vis-a-vis regular networks is more significant than that of synchronous vis-a-vis asynchronous updates.

### Strategy Update and Its Dynamics

Concerning the update rule, we assume, as mentioned before. Those four are commonly used in previous studies. In the first case of Fermi-PW, a focal player, I adopt a randomly chosen player jâ€™s strategy with a probability calculated by a Fermi function. Although we recognize that the noise effect influences the cooperation level to some extent (Vukov et al. 2006), the parameter Ï„ in the Fermi function is set to 0.2. The second level is Linear-PW, in which the strategy of a player j is chosen as in Fermi-PW, but a linear function gives the probability. A critical difference between Fermi-PW and Linear-PW is that the former might allow copying an opponentâ€™s strategy even if the opponentâ€™s approach has a far smaller payoff than that of the focal player.

### Mutation

We assume that an agent is infected with a mutation in the strategy update process where they take C or D randomly with probability Îµ. We consider two levels of the modifications; Îµ Â¼ {0, 0.01}.

### Experimental Setting

We simulate a series of perfect factorial experiments consisting of 8 Â¼ 1536 combination cases, reflecting the above-described choices: (a) the number of players has three levels, (b) the average degree has four levels, (c) the update rule has four levels, (d) the update dynamics has two levels, (e) the mutation has two levels, and (f) the network has eight levels, Each simulation runs as follows. Initially, an equal percentage of strategies are distributed randomly to the players allocated on different network vertices. Then, several generations are run until the frequency of cooperation arrives at a certain quasi-equilibrium.

### Which Main Effect Is Significant?

Examining the differences that arise in choosing which game area is evaluated, we found that all and D&R had similar tendencies in general, but PDCH appeared different. In PDCH, all factorial effects except for the number of players were observed to be stronger than those for All and D&R. In particular, the impact of the network structure is remarkably more robust than the case for all and D&R. This implies that one might overestimate the network reciprocity that results from network topology when using the boundary games (PDCH) to evaluate the cooperation level, as the results are not consistent with the network reciprocity obtained.

### Main Factorial Effects

The bold lines indicate level average values and distributions mean frequency of all combination cases scattering from entire defective situation to reasonable cooperation level in X-axis. For example, in the case of (a) the number of agents, Y-axis values of three bar lines are the average cooperation fractions of the respective experimental levels, which are based on 1536/3 Â¼ 512 combination cases. In the following text, we will restrict our discussion to the results for all unless otherwise noted because no significant differences among the results of All, D&R, and PDCH were found.

### Summary

We established a new model for the spatial prisonerâ€™s dilemma game. An agent selects a game partner and an adaptation reference independently using a stochastically skewed selection process based on the partnerâ€™s payoff among her neighbors. This model can emulate relationships in a whole human society because humans tend to select one or two favorable friend(s) from among the many potential acquaintances provided by a social network.