Properties and applications of spectra for networks 第七届全国复杂网络学术会议 Properties and applications of spectra for networks 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

Collaborators Prof. Chen Guanrong(陈关荣), CityU of Hongkong Prof. Comellas Francesc, Universitat Politecnica de Catalunya, Barcelona, Spain Qi Yi(齐轶), Master student (graduated) Wu Shunqi(伍顺琪), Master student Wu Bin(吴斌), Master student Lin Yuan(林苑), Undergraduate student 2019/4/5

Main contents Introduction to relevant matrices Our work 1 Spectral properties of various matrixes and their relevance to structure and dynamics Our work 2 Computation of spectra for different matrixes and their applications to network structure and random walks 2019/4/5

Definitions Adjacency matrix A Diagonal degree matrix D Laplacian matrix L=D-A Probability transition matrix Normalized Laplacian matrix Fundamental matrix Modular matrix …… 2019/4/5

Adjacency matrix The diameter of a connected graph G is less than the number of distinct eigenvalues of the adjacency matrix of G. Electronic Journal of Linear Algebra, 2005, 14:12-31 From the greatest eigenvalue (often called spectrum radius), one can provide a lower bound for diameter of a network. J. Combin. Theory Ser. B 91 (1) (2004) 143–146. 2019/4/5

Adjacency matrix SIS model: the largest eigenvalue defines an epidemic threshold ACM Trans. Inf. Syst. Secur. 10 ,13 (2008) Spectrum radius plays a central role in determining critical couplings for the onset of coherent behavior. Phys. Rev. E 71, 036151 2005 . SI model: the eigenvector corresponding to the largest eigenvalue is related to the spreading power of nodes in a network. Complexus 3 , 131-146 (2006) 2019/4/5

Adjacency matrix Weighted percolation on directed networks: If the probability of removing node i is , the network disintegrates if is such that the largest eigenvalue of the matrix with entries is less than 1, where A is the adjacency matrix of the network. PRL 100, 058701 (2008) 2019/4/5

Laplacian matrix Spanning trees Algebraic connectivity provides a upper bound for diameter of a network. SIAM Journal of discrete mathematics, 1994, 7(3): 443-457. Spanning trees 2019/4/5

Laplacian matrix Effective resistance 2019/4/5

Laplacian matrix Random walks is degree of node z, m is the number of edges. 2019/4/5

Laplacian matrix Relevance to other dynamics Quantum walks Synchronization Generalized Gaussian structures Ultimatum game •••••• 2019/4/5

Transition probability matrix Q is often called normalized adjacency matrix for non-bipartite graphs are the corresponding mutually orthogonal eigenvectors of unit length. Stationary distribution 2019/4/5

Transition probability matrix First passage time Commute time Eigentime identity 2019/4/5

Transition probability matrix Mixing rate Mixing time Return-to-origin probability 2019/4/5

Normalized Laplacian matrix are the corresponding mutually orthogonal eigenvectors of unit length. 2019/4/5

Normalized Laplacian matrix 2019/4/5

Our work Calculating spectra of adjacent and Laplacian matrices for particular networks Applying Laplacian spectra to enumerate spanning trees Using Laplacian spectra to determine mean first-passage time Spectra of transition matrix for some networks and their applications 2019/4/5

Spectra of adjacency matrix for a family of deterministic recursive trees Journal of Physics A, 2009, 42: 165103. 2019/4/5

Laplacian eigenvalues and eigenvectors of deterministic recursive trees Physical Review E, 2009, 80:016104 2019/4/5

Spectra of adjacent matrix and Laplacian matrix of small-world networks Completed 2019/4/5

Farey sequence of order n denoted by Using Laplacian spectra to determine the number of spanning trees in Farey graph Farey sequence of order n denoted by 2019/4/5

Spanning trees in Farey graph Theoretical Computer Science, 2011, 412:865–875 Two nodes and are linked to each other if they satisfy Physica A (in revision) 2019/4/5

Spanning trees in scale-free networks A counterintuitive conclusion that a network with more spanning trees may be relatively unreliable. Fractality can significantly increase the number of spanning trees in fractal scale-free networks. EPL, 2010, 90:68002. Physical Review E, 2011, 83:016116. 2019/4/5

Application of spectra to random walks Vicsek fractals Physical Review E, 2010, 81:031118. 2019/4/5

Random walks on T fractals is obtained from the relationship between characteristic polynomials at different generations. Our method can void the computation of eigenvalues. Physical Review E, 2010, 82:031140 2019/4/5

Random Walks on dual Sierpinski gasket European Physical Journal B, 2011, 82:91-96. 2019/4/5

Relation to the Hanoi Towers Game   What is the minimum number of moves ? 2019/4/5

The Hanoi Towers Graphs 2019/4/5

Spectra of transition matrix: T-fractal We obtain all the eigenvalues and their multiplicities. The reciprocal of the smallest eigenvalue is approximately equal to the mean trapping time EPL, 2011, in press 2019/4/5

Spectra of transition matrix: fractal scale-free networks Completed 2019/4/5

Thank You!