Phase Transition Phenomenon In Complex Network Based On Game Theory

A variety of systems in our life can be abstract ed as networks, such as SocialNetwork and Traffic System. We all live in a set of complex systems that can beabstracted as complex networks.The main idea of this article is the analogy between a complex network based onsymmetric coordination game and the Ising model, which descript a system ofinteracting particles. We use Agent-Based Modeling and Simulation method to find outthat if they had the same property: phase transition.We introduce three complex network models in the first part of this article. Thestudy of complex networks started with the classic graph theory which descripts a classof network with a certain structure. The study of networks reached the first milestonewhen Erdos and Renyi brought the uncertainty to networks in the1950s. Theuncertaintymadethenetworkscomplex.It madethe“smallworld”phenomenon,whichis the character of the real world networks, happen in the ER Network Model. But thenetworks in real world are not pure random. They have structures with rules and specificfeatures. To add rules to the evolution and growing of networks, the WS Small-WorldNetwork Model and BA Scale-Free Network Model were proposed in the1990s. Allthree complex network models can descript the real world networks in certain aspects,such as short average shortest path, high clustering coefficient and scale free.Then we introduce the Game Theory and symmetric coordination game model.Game theory is the study of strategic decision making. It began with games from streets.Children played games and gangsters gambled. The players guessed what others woulddo and calculated how to win more. John von Neumann and John Nash made greatcontribution to the modern game theory. Prisoner’s Dilemma and Minority Game aretwo classic game models. Symmetric coordination games are a class of games withmultiple pure strategy Nash equilibria in which players choose the same strategies.Then we introduce the Ising model and the phase transition phenomenon. The Ising model is simplest model to descript a system of interacting particles. It allows theidentification of phase transitions. By thinking of the phase transition phenomenon inIsing model, we assume that there may also be a phase transition phenomenon in acomplex network based on symmetric coordination game.In the last, we use Agent-Based Modeling and Simulation method to verificationthe hypotheses. We use three complex network models, each based on three kinds ofsymmetric coordination game. We find that phase transition is related to long_rangedcorrelation. Then we study how network structure influence the phase transition.