Networking effects on evolutionary snowdrift game in networks with fixed degrees

We study the effects of spatial structures other than the degree distribution on the extent of the emergence of cooperation in an evolutionary snowdrift game. By swapping the links in three different types of regular lattices with a fixed degree k, we study how the frequency of cooperator fC changes as the clustering coefficient (CC), which signifies how the nearest neighbors of a vertex are connected, and the sharing coefficient (SC), which signifies how the next-nearest neighbors of a vertex are shared by the nearest neighbors, are varied. For small k, a non-vanishing CC tends to suppress fC. A non-vanishing SC also leads to a suppressed fC for the networks studied. As the degree increases, the sensitivity of fC to the network properties is found to become increasingly weak. The result is discussed within the context of the ranking patterns of average payoffs as k changes. An approximation for fC, which is based on the idea of a finite fully connected network and gives results in good agreement with numerical results, is derived in the limit of large k.