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，推导出笛卡尔乘积图的距离-k图的渐近谱分布。用q变形的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径的曼哈顿积的谱分布，并得到了它们在无限长度上的渐近行为。(2)动力学的统计和渐近行为的建模方法：将量子概率方法应用于经典的随机游动，以重建转移概率和重现概率。得到了蜘蛛网上量子游动(格罗弗游动)表现局域化的充要条件。

(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.