第八届全国复杂网络学术会议 Spectra of transition matrix for networks: Computation and applications 章 忠 志 复旦大学计算机科学技术学院 Email: zhangzz@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/~zhangzz/ Blog: http://group.sciencenet.cn/home.php?mod=space&uid=311410

Main contents Introduction to relevant matrices Our works 1 Definition of various matrixes Relevance of spectra for transition matrix to structure and dynamics Our works 2 Computation of spectra for transition matrix of diverse networks Applications to spanning trees and walks 2019/5/6

Definitions Adjacency matrix A Diagonal degree matrix D Laplacian matrix L=D-A Probability transition matrix Normalized adjacency matrix Normalized Laplacian matrix Fundamental matrix of trapping …… 2019/5/6

Isotropic random walks Transition matrix describes the jumping probability for random walks on graphs - Isotropic random walks 2019/5/6

Random walks on graphs - At any node, go to one of the neighbors of the node with equal probability. 2019/5/6

Random walks on graphs - At any node, go to one of the neighbors of the node with equal probability. 2019/5/6

Random walks on graphs - At any node, go to one of the neighbors of the node with equal probability. 2019/5/6

Random walks on graphs - At any node, go to one of the neighbors of the node with equal probability. 2019/5/6

Random walks on graphs - At any node, go to one of the neighbors of the node with equal probability. 2019/5/6

Transition matrix Q is often called normalized adjacency matrix for non-bipartite graphs are the corresponding mutually orthogonal eigenvectors of unit length. Stationary distribution 2019/5/6

Transition matrix First passage time Commute time Eigentime identity 2019/5/6

Transition matrix Mixing rate Mixing time Return-to-origin probability 2019/5/6

Normalized Laplacian matrix are the corresponding mutually orthogonal eigenvectors of unit length. 2019/5/6

Normalized Laplacian matrix 2019/5/6

Our works Computation of spectra for various networks T-fractals, Hanoi graphs Treelike and loopy scale-free networks Applications of spectra for transition matrix Enumeration of spanning trees Determination of eigentime and trapping time 2019/5/6

Spectra of normalized Laplacian matrix of T fractals We obtain all the eigenvalues and their multiplicities. . The reciprocal of the smallest eigenvalue is approximately equal to the mean trapping time. EPL, 2011, 96:40009 2019/5/6

Spectra of transition matrix for Hanoi graphs 2019/5/6

What is the minimum number of moves ? The Hanoi towers game   What is the minimum number of moves ? 2019/5/6

Spectra of Hanoi graphs and applications Structural properties Spectral prosperities We obtain all the eigenvalues and their corresponding degeneracies. We determine the exact number of spanning trees and derive an explicit formula of the eigentime identity. Journal of Physics A, 2012, 45:345101. 2019/5/6

Spectra of transition matrix for fractal scale-free trees EPL, 2012, 99:10007 2019/5/6

Optimal and suboptimal networks minimizing eigentime identity for random walks

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