由Sierpinski垫片得到的局部自相似网络上的随机游走及谱分析

近来研究发现，真实中的网络大都具有自相似结构，从而揭示了网络的分形特征。在网络研究中，一个中心问题就是了解网络的结构特征及对网络功能和动力学性质的影响。随机游走以及拉普拉斯矩阵的谱都可以很好地刻画网络的结构和功能，具有重要研究价值。目前，对具有全局自相似结构的规则网络上结构分析已有大量研究，而对具有局部自相似结构的非规则网络的研究还未涉及。然而，真实中的网络往往大部分都是非规则的，因此，局部自相似网络上的结构及动力学分析显得尤为重要。基于此，本项目拟从Sierpinski垫片(SG)入手，通过SG上的竖切和横切两种方式下得到的残余网络为我们的研究对象，研究这些局部自相似网络上的陷阱问题和拉普拉斯矩阵的谱，分析切割方式对于网络结构和动力学性质的影响。通过本项目的研究对非规则网络上的拓扑结构和动力学研究提供新的思路和方法。

Recently, it is found that most real networks have self-similar structure, which reveals the fractal characteristics of networks. In the study of complex networks, a central problem is to understand the structural characteristics of the network and its influence on the function and dynamic properties of the network. Random walk and the spectrum of Laplace matrix can well describe the structure and function of the network, which has important research value. At present, there are a lot of research results on the structure analysis of regular networks with global self-similar structure, but there is no relevant research on irregular networks with local self-similar structure. However, most of the real networks are irregular, so it is very important to analyze the structure and dynamics of local self-similar networks. Based on this, this project plans to start with Sierpinski gasket (SG), and take the residual networks obtained by vertical and horizontal cutting on SG as our research object, study the trapping problem on these local self-similar networks and the spectrum of Laplace matrix, and analyze the influence of cutting mode on network structure and dynamic properties. It is hoped that the research of this project can provide some new ideas and methods for the research of topology and dynamics on irregular networks.