复杂网络传播动力学的数学分析

项目名称： 复杂网络传播动力学的数学分析

项目编号： No.11331009

项目类型： 重点项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 靳祯

作者单位： 山西大学

项目金额： 240万元

中文摘要： 网络传播动力学主要是用动力学的方法分析特定网络上传染性疾病或信息的扩散机制及演化规律，它既依赖于网络的结构动力学又依赖于疾病或信息的传播机制，该方面研究已受到广泛的关注。但现有研究存在的共同问题是数学理论分析和证明不足，其根本原因是网络结构的引入带来了系统的随机性与模型维数的增加。为此，本项目将综合利用概率统计、随机过程及图论等来刻画网络的度分布、相关系数、聚类系数、团簇系数、介数等结构参数；针对规则与随机、静态与动态及耦合等网络研究结构参数在传播动力系统中的表征，以此建立几类典型的网络传播动力学模型；利用稳定性、分支、谱及反应扩散方程等有关理论进行数学分析和证明。该研究拟给出结构参数在传播模型中的恰当表征，解决网络传播动力系统理论分析和证明中的关键技术，揭示不同网络结构对传播动力系统的本质影响，最终建立复杂网络传播动力学的基本理论框架，为网络上疾病或信息传播的应用研究提供可靠的理论依据。

中文关键词： 复杂网络;结构参数;动力系统;传播动力学模型;传播阈值

英文摘要： Transmission dynamics on complex networks is mainly to characterize spreading behaviours and evolutionary rules of infectious disease or information taking place on special networks. It depends on both the network structure dynamics and the mechanism of the disease or information spreading, and has received widespread attention. However, a common problem in current research is the lack of mathematical theory and analysis in studying networked transmission dynamics. The essential reason is that the inclusion of network structure results in the randomness of the system and the increased model dimension. To this end, by employing theories of probability and statistics, stochastic processes and graph theory, we will study structure parameters (including degree distribution, correlation coefficient, clustering coefficient, cluster/clique coefficient, betweenness and so on), and how these parameters are incorporated into spreading dynamical systems with regular and random networks, static complex networks, dynamical networks, and coupled networks. Then, we will build several typical spreading dynamical models and analyze their dynamical behaviours by using the stability theory, matrix theory, differential equations, bifurcation theory, spectral theory and the theory of reaction-diffusion equations. We aim to obtain the appropriate characterization of structure parameters in spreading dynamical system, solve the theoretical analysis in spreading dynamical systems and key technique in the proof, and reveal the fundamental influence of different network structures on transmission dynamics. In this proposal, the ultimate goal is to establish a comprehensive mathematical framework for studying the networked transmission dynamics, and provide a theoretical basis for the applications of the spreading of disease or information on networks.

英文关键词： Complex networks;Structure parameters;Dynamic systems;Transmission dynamics model ;Propagation threshold