Research On Convergence Of Dynamic Network Based On Vicsek Model

In recent years, the collective behaviors of multi-agent system have attracted lots of attention of scientists in diverse disciplines. Dynamic network frame is an effective approach to investigate the problems of collective behavior in multi-agent system. Research on convergence of dynamic network, which is one key problem in the field of dynamic network, has become an important and challenging frontier issue.Dynamic network models, especially the Vicsek model provide theoretical fundament for the corporative control and consensus research through utilizing simple evolution rules to generate complex self-organization phenomena. Based on Vicsek model, on the one hand, the biological complexity and swarm intelligence generating process could be better understood; on the other hand, the whole system could be controlled to show some expected emergence behaviors: such as, cooperation control of UAV-Unmanned air vehicle, formation control of robot system, attitude alignment of satellite clusters, and congestion control of communication network, etc. Therefore, the study on Vicsek dynamic network model convergence becomes significant both theoretically and practically.Under the above background, in this paper, we investigate the dynamic network model establishment and convergence property. The main work is divided into two aspects as follows:(1) Based on Vicsek model, a new weighted model is presented to improve the convergence efficiency of dynamic network.Although a great wide concern has been attracted on Vicsek model and its modifications, most of these studies are performed on non-weighted network, i.e., at each time step, every agent updates its direction according to the average direction of agents' motion in its neighborhood of radius R. However, in reality, kinds of network's topological structures are inhomogeneous, i.e., the degree of some node is much larger than others, and these nodes affect the structure and the dynamic process of network a lot. With considering disparity of neighbor numbers during evolution process, we propose a weighted self-propelled agent system to improve convergence efficiency by utilizing the degree in complex network and the topological structure of the dynamic network, wherein the weight is determined by an exponent form of each agent's neighbor number. The direction of each agent is updated by the weighted average directions of its neighbor, instead of its neighbors' average direction. In terms of weight, two weighted models are presented:①in model I, the weight is in proportion to each agent's degree, i.e., the number of neighbors;②in model II, the weight is in proportion to the index of each agent's degree, and the index is greater than zero. Convergence time and degree of consensus are defined as convergence efficiency. The difference of convergence efficiency between weighted model and classical Vicsek model are compared in this paper. The results indicate that weighted model can enhance convergence efficiency of dynamic network, even with some embedded noise. All of these provide a new and powerful mechanism for the investigation of biological swarm and artificial intelligent system.(2) In this paper, after the modeling and simulation research, we provide a theoretical proof that convergence speed of weighted model is faster than classical non-weighted Vicsek model. There are two main problems in the current studies: first, all studies of convergence speed and consensus degree are focused on Vicsek model or its simplified model rather than weighted model; second, although many researchers propose some models to improve system convergence efficiency, few ones prove that theoretically. Olfati-Saber etc. obtained the following two conclusions:①the larger the absolute value of the second largest eigenvalue for Laplace matrix is, the faster convergence speed of system is;②algebraic connectivity of weighted directed graph can be reflected by its mirror graph. Based on Olfati-Saber's theory and with the aid of matrix and graph theory such as Gersgorin theorem, we transform Laplace matrix of weighted model into a symmetric matrix and found that the absolute value of the second largest eigenvalue for weighted model's Laplace matrix is larger than that of classical non-weighted model, therefore, the weighted model proposed in this paper was proved to be able to improve the convergence efficiency of dynamic network.