Spectral analysis of complex networks and modeling methods by statistics

(1)网络的谱分析：作为组合中心极限定理，我们导出了卡氏积图的距离-k图的渐近谱分布。用q-变形Hermite多项式描述了加权超立方体的渐近谱分布。完全确定了小阶有向路的曼哈顿积的谱分布，并得到了它们在无限长范围内的渐近性质. (2)建模方法中动力学的统计和渐近行为：将量子概率方法应用于经典随机游动，重建转移概率和递归概率。本文给出了蜘蛛网上量子行走（Grover行走）的一个充分必要条件，使其具有局域性。

(1) Spectral analysis of networks: The asymptotic spectral distribution of the distance-k graph of the cartesian product graph is derived as a combinatorial central limit theorem. The asymptotic spectral distribution of weighted hypercubes are described in terms of q-deformed Hermite polynomials. The spectral distribution is completely determined for the Manhattan product of directed paths of small orders, and thier asymptotic behaviors are obtained in the infinite length.(2) Statistical and asymptotic behaviors of dynamics for the modeling methods: A quantum probabilistic method is applied to classical random walks for reconstructing transition probability and recurrence probability. A necessary and sufficient condition is obtained for a quantum walk (Grover walk) on a spidernet in order to exhibit the localization.