Several Mechanisms Of The Emergence Of Cooperation In Complex Systems

Complex system studies concentrate on the integrated macroscopic phenomena of self-organization and emergence caused by microscopic interactions. In the last two decades one major progress in the study of complex systems is on their coarse-grained description-the complex networks. Thus networks are ubiquitous in nature and hu-man society. Most typical complex networks include the Internet, World Wide Web, social networks, scientific collaboration networks, power grids, protein-protein inter-action networks, genetic regulatory networks, etc. Unlike traditional graph theory, the studies on complex networks emphasize on their large scale statistical properties, in which statistical physicists have made significant contributions. People have found that complex networks in real lives obey several universal properties such as short distance, power-law degree distributions and high cluster coefficients. They have built many mathematical models according to those properties, among which well known are the small world networks and scale free networks. Opun sufficient investigations on the net-work models one still needs to recover the dynamical freedoms attached to the nodes, as the ultimate goals of complex network studies are trying to understand how network structures affect the dynamic processes and conversely how dynamic processes drive the evolution of the network structures.Bio-systems and Social economy are typical complex systems. The units that com-prise these systems are no longer traditional physical particles, but are various compli-cated organisms, so that the interactions among the agents don’t appear as the four kind of fundamental forces and the evolution of these systems are not directly governed by classical or quantum laws. One challenging topic is how to study the society and bio-systems in a mathematical and quantitative approach. In fact, one existing framework on the evolution of bio-systems and social economical systems is the evolutionary game theory. Recently evolutionary game studies have turned from well-mixed populations to structured population and have closely connected with researches on complex networks, which has been called evolutionary games on graphs or spatial evolutionary games. So far the most successful application of mathematics in many body systems is in statisti-cal mechanics, while spacial game models resemble lattice statistical models in several ways, thus the later could serve as a guid for the former. Indeed, in recent years statisti-cal physicists have made a large amount of contributions in these problems. Important concepts in statistical mechanics such as phase transition, criticality and universality have all been adopted. Many famous methods in statistical mechanics such as the mean field approximation and master equation method have been extended to these problems as well. Conversely, cross-disciplinary investigations about these systems have raised many new challenges and opportunities to statistical mechanics.We have systematically investigated the mechanisms of the emergence of coopera-tion in several evolutionary game models on networks. Our chief works are as follows:1. Social division of work and the emergence of cooperation:one prerequisite of cooperation is division of work. Through division of work and specialization indi-viduals could make full use of their special talents and increase the productivity. The phenomena of division of work is ubiquitous in animals as well as human societies. It is division of work that makes cooperation meaningful. We divided the whole population into two different work types and investigated its impact on cooperation. We assume that two individuals have higher productivity if their types are different, that is their payoff matrix are amplified by a multiplicative factor; while the payoff matrix remains unchanged, if their are of a same type. Simulation results show that division into dif-ferent types together with the spacial lattice structure can induce several social ranks, which is beneficial for cooperation. Besides, we found that moderate values for both the type fraction parameter and multiplicative factor are more favorable for cooperation.2. Life processes in evolutionary game models:most evolutionary dynamics in-clude merely birth-death process, though the whole life process including birth, growth, aging, disease, death and so on are all very important for the description of bio-systems. Based on previous investigations about different interaction and strategy update time scales, we proposed that the time scale factor should be correlated with individuals’ fitness, so that the description and investigation on individuals’life processes become more complete. Simulation results show that the correlation between time scale and individuals’fitness can further increase the cooperation level. Besides, moderate time scale values lead to the highest cooperation level.3. Impact of intermediate strategies:spatial game models resemble some lattice statistical models, such as the Ising model. In statistical mechanics, there is another model——the q state Potts model which is an important generalization of Ising model, and in which two spin values is generalized to q values. Inspired by that, we general-ized the usual cooperation-defection binary strategy spatial game to a q-strategy model, where fractional valued mixed strategies are allowed. We interpret the fractional strate-gies as intermediate strategies, namely, neither completely cooperate nor defect. We showed that intermediate strategies can sustain cooperation in a broader range of the temptation to defect parameter, because some intermediate strategies are optimal strate-gies in that they can on the one hand reduce the exploitation by the pure defectors and on the other hand keep a certain spatial reciprocity effect.