Equilibrium Distributions of Populations of Biological Species on Networks of Social Sites.

We investigate the problem of how a population of biological species would distribute over a given network of social sites so that their social contacts through the connected sites can be maximized (or minimized). This problem has applications in modelling the behaviours of social (or solitary) species such as the development of social groups in human society and the spread of solitary animals in distant habitats. We show that this problem can be formulated as an evolutionary game, with the equilibrium state of the game corresponding to a strategy for choosing the residing sites, each with a certain probability, or equivalently, to a distribution of the population on these sites. The game has a symmetric payoff matrix, and can therefore be analyzed via the solution of a corresponding quadratic programme: An equilibrium strategy of the game is a KKT point of the quadratic programme, which may be a local maximizer, local minimizer, or saddle point, but it is evolutionarily stable if and only if it is a strict local maximizer. In general, with a goal to maximize the social contacts, the species tend to spread on network sites where there are dense connections such as a complete subnetwork or in other words, a network clique. We show that at equilibrium, the population may or may not distribute on a network clique, but the stability of the equilibrium state does depend on the structure of the selected subnetwork. In particular, we show that the distribution of the population on a maximal network clique is evolutionarily stable unless the clique is 'attached' to another clique of the same or larger size, when the population may be able to switch or expand to the neighbouring clique to increase or at least maintain its total amount of contacts. However, the distribution of the population on a non-clique subnetwork is always evolutionarily unstable or weakly evolutionarily stable at the very best, for the population can always move away from its current distribution without decreasing its total amount of contacts. We conclude that the strategies to spread on maximal network cliques are not only equilibrium strategies but also evolutionarily more stable than those on non-clique subnetworks, thus theoretically reaffirming the evolutionary advantages of joining social cliques in social networks for social species.