We consider a problem of how a population of biological or social species spreads or evolves in a given social network -- a problem related to research in many areas of biological, medical, and social sciences. For example, a wide range of human diseases spread through networks of possible transmission routes; news or rumors circulate among linked social sites on the internet; immediate relatives or friends influence each other on opinions or beliefs; companies purchase and sell products through trade partnerships. All these activities, biological or social, can be considered as an evolutionary process of a population of biological or social species over a given social network, and be modeled as an evolutionary game called social network game.
A social network consists of a set of sites where species reside and a set of links through which species can migrate from one site to another. By choosing a certain way to distribute over the network, the species tend to maximize their total social contacts, so they can increase their survival rates as viruses spread over closely connected hosts (animals or humans), or maximize their accessibilities as news or rumors propagate in the internet, or maintain their influences as individuals socialize with their relatives or friends, or promote their economic welfares as companies choose to join trade partnerships, etc.
With a goal of maximizing the social payoff, the species tend to migrate to sites with the most possible connections among them. Such a group of sites corresponds to a subgraph in the network with a dense set of edges among its nodes. The best one would be a complete subgraph or in other words, a network clique. However, we have shown that at equilibrium, the population may or may not reside on a network clique, but the stability of the equilibrium state does depend on the structure of the resided subgraph (Wang, Zhou, Wu 2018).
In particular, we have shown that the strategy to reside on a maximal clique is always an optimal strategy, while the population in equilibrium may be distributed on a non-clique subgraph as well. We have shown that the equilibrium state of the population on a clique is evolutionarily stable unless the clique is “attached" to another clique of the same or larger size, while the population state on a non-clique subgraph at equilibrium is evolutionarily unstable, as the population tends to shift to a more stable state by shrinking its spread to a maximal clique contained in the non-clique subgraph. In this sense, the strategies to spread on network cliques are not only optimal but also more stable than those on non-clique subgraphs for the evolution of populations on social networks.