a Mathematical Biosciences Institute, Ohio State University , Columbus , Ohio , USA.
b Department of Statistics, Colorado State University , Fort Collins , Colorado , USA.
J Biol Dyn. 2019;13(sup1):74-98. doi: 10.1080/17513758.2018.1508762. Epub 2018 Aug 16.
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.
我们研究了这样一个问题，即在给定的社交网站网络中，生物物种的种群将如何分布，以使它们通过连接的网站进行的社交接触最大化（或最小化）。这个问题在建模社交（或独居）物种的行为方面有应用，例如人类社会中社交群体的发展和偏远栖息地中独居动物的传播。我们表明，这个问题可以被表述为一个进化博弈，博弈的平衡状态对应于选择居住地点的策略，每个地点都有一定的概率，或者等效地，对应于种群在这些地点上的分布。博弈有一个对称的收益矩阵，因此可以通过求解相应的二次规划来分析：博弈的平衡策略是二次规划的 KKT 点，它可能是局部最大值、局部最小值或鞍点，但只有当它是严格的局部最大值时，它才是进化稳定的。一般来说，为了最大化社交接触，物种倾向于在网络站点上扩散，这些站点具有密集的连接，例如完整的子网或换句话说，网络团块。我们表明，在平衡状态下，种群可能分布在网络团块上，也可能不分布，但平衡状态的稳定性确实取决于所选子网的结构。特别是，我们表明，除非团块“连接”到相同或更大大小的另一个团块，否则种群在最大网络团块上的分布是进化稳定的，在这种情况下，种群可能能够切换或扩展到相邻的团块以增加或至少保持其总接触量。然而，种群在非团块子网上的分布在任何情况下都是进化不稳定的，或者充其量是弱进化稳定的，因为种群可以随时离开当前的分布，而不会减少其总接触量。我们的结论是，在最大网络团块上扩散的策略不仅是平衡策略，而且在进化上比非团块子网更稳定，因此从理论上再次证实了社交物种在社交网络中加入社交团块的进化优势。
