Consensus Problems In Complex Heterogeneous Networks

Consensus behaviors is ubiquitous in nature and human society, e.g. nocking of birds, swarming of fishes, colonies of bacteria, a consensus of opinions, and so on. Consensus problems have a number of applications in multiple mobile robot systems, cooperative control of unmanned aerial vehicles, and distributed sensor networks, etc. There is a large and growing literature concerned with consensus problems in various systems. For a multi-agent system, consensus is to design a proper distributed algorithm to make all agents’states reach an agreement. The algorithm needs to be local in the sense that each agent can perform local computations and communicate with its immediate neighbors. With the arising and development of complex networks, consensus problems in different networks have become a hot problem. This thesis focuses on consensus algorithms in opinion dynamics in complex networks, heterogeneous networked systems and evolutionary snowdrift games.The main contributions of this dissertation are summarized as follows:Investigation on consensus algorithms of opinion dynamics in complex het-erogeneous networks. A discrete-time model of opinion dynamics is pro-posed in this paper. The neighborhood relationship is decided by confi-dence/influence radius of each agent in this model. We investigate the influence of heterogeneity in confidence/influence distribution on the be-havior of the network. The simulations suggest that the heterogeneity of single confidence or influence networks can promote the opinions to achieve consensus. It is shown that the heterogeneous influence radius systems con-verge in fewer time steps and more often in finite time than the heteroge- neous confidence radius systems.We find that heterogeneity does not always promote consensus, and there is an optimal heterogeneity so that the rela-tive size of the largest consensus cluster reaches maximum in heterogeneous confidence and influence networks.Investigation on consensus algorithms of heterogeneous multi-agent system-s. We study the problem of swarm aggregations of heterogeneous multi-agent systems in a directed balance network topology. It is assumed that agents in the network are nonidentical and there is a leader in the multi-agent system. The coupling matrix plays an important role in the stability analysis of consensus algorithms. We show that the heterogeneous agents will gather with a certain error under some assumptions and condition-s. The stability properties have been proven by theoretical analysis and verified via numerical simulation. The stability of the heterogeneous multi-agent systems has been achieved based on matrix theory and the Lyapunov stability theorem. Numerical simulation is given to demonstrate the effec-tiveness of the theoretical result.Investigation on consensus algorithms of heterogeneous multi-agent systems with periodically intermittent control. We study a class of heterogeneous swarming systems with periodically intermittent control. It is assumed that agents in the network are nonidentical and potential functions are hetero-geneous. Each agent is assumed to obtain information from the leader and the neighbors only on a series of periodically time intervals. The dynamics of the swarm members are affected by inter-individual interactions and the environment. Some sufficient conditions are provided to guarantee expo-nential stability of the heterogeneous multi-agent system. Although the information from the leader and the neighbors is not continuous, all fol-lowers can track the leader in certain error range. A numerical example is shown to demonstrate the effectiveness of the theoretical result.Investigation on consensus algorithms of evolutionary snowdrift games. We study the stochastic stability of evolutionary snowdrift games. We identify stochastically stable equilibria for two-player and multi-player evolution-ary snowdrift games. For the two cases with the same values of cost and benefit of cooperation, we show that like the two-player case, under cer-tain conditions, there is a unique stochastically stable equilibrium in the multi-player case, at which, however, the proportion of cooperators can be higher than that of the two-player case. More importantly, the proportion of cooperators can be manipulated as the stochastically stable equilibrium is being shifted by changing the game parameters. Therefore, the results indicate a promising approach to control the proportion of cooperators in large populations that has not been reported before. Besides theoretical analysis, we demonstrate our results through numerical computations and simulations as well.