多自主体系统集体行为若干关键问题研究

Famously, the physicist Stephen Hawking has predicted that the 21st Century will be "the century of complexity". The investigation of multi-agent systems is an important entry point for the study of complex systems. The complex systems have become a hot issue of research in various fields, however, its theoretical basis has not been established till now. One of the main reasons may be that the process of how the interactions of agents on the micro-level affect the structure and function of the total system on the macro-level has not been well understood, and there still exist many key problems have not been solved. This project will study the following problems: (i) the synchronization property of some bio-cluster models in noisy environments, such as Vicsek model and Buhl model; (ii) the estimation of the critical conditions for synchronization in some bio-cluster models without noise; (iii) the convergence proofs of the inhomogeneous DW model and HK model, which are the open problems on opinion dynamics. These problems, which are closely related with the processes of synchronization and aggregation, are the key problems in multi-agent systems. Though a number of simulations have existed, the theoretic analysis of these problems is still lack, which has seriously hindered the development of the theory of complex systems. Combined with the control theory and percolation theory naturally, this project will investigate some new methods to solve these problems. Also, through this project, the interdisciplinary of multi-agent systems, control theory and percolation theory may be promoted, and some new methods may be provided for the study of other problems in complex systems.

复杂性科学被称为21世纪的科学。以多自主体系统为重要切入点的复杂系统研究已经成为热点和前沿方向，但一般性理论尚未建立，其主要原因是对微观层次上个体的行为及相互作用如何影响宏观层次上系统的结构和功能这一过程理解还不够深入，尚存在大量关键问题没有解决。本项目拟对噪声环境下生物集群模型（如Vicsek模型、Buhl模型等）的同步行为、无噪声生物集群模型的同步临界条件估计、以及舆论动力学中异质DW模型和HK模型收敛性公开问题展开深入研究。这些问题涉及同步或聚集等集体行为产生过程的本质，属于多自主体系统研究的核心问题，目前存在大量的仿真结果，但理论研究方面几乎还是空白。这极大妨碍了对复杂系统的深入了解和多自主体理论的进一步发展。本项目拟将控制论、渗流理论等学科相关工具与上述问题自然结合，积极探索新方法解决这些问题，并可能促进多自主体系统理论、控制论和渗流理论等学科的交叉融合，为复杂系统研究提供新方法

多个体系统由相互作用的多个个体构成，在自然界和社会经济领域无处不在。具有局部相互作用的微观个体如何导致宏观系统的整体行为，是系统学研究最基本的科学问题之一。本项目研究了该问题并原创性地提出了定性与定量分析方法：我们针对一类著名的自驱动粒子模型（Vicsek模型），首次引入渗流理论研究了局部规则导致系统整体行为的动态过程，给出了线性化Vicsek模型、局部规则的Cucker-Smale模型、以及k最近邻模型一些不含闭环系统连通性假设的精确定量结果；我们还将原始Vicsek模型分析转化为控制问题，首次对它作了严格理论分析，并对一些非齐次SPP模型包括Leader-follower模型也得出一些分析结果；解决了A.Jadbabaie等人在论文中所提噪声如何影响连通性问题；针对很多文献所关心的“鲁棒同步”问题给出了一个明确答案。此外，我们还研究了一类基本的舆论动力学模型，证明了它的收敛速度为一个负指数函数。最后，我们证明了随机几何图或离散渗流大连通分支的阶或连通分支数目的期望以负指数的速度趋于一个多项式，并且根据中心极限定理对这些随机变量的渐近大小作了精细刻画。该结果极大改进了H. Kesten（美国国家科学院院士）和M. Penrose等人结果。