网络的谱性质与应用

东南大学苏州研究院学科进展类讲座 网络的谱性质与应用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

合作者 • 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

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

Definitions • Adjacency matrix A • Diagonal degree matrix D • Laplacian matrix L=D-A • Probability transition matrix • Normalized Laplacian matrix • Fundamental matrix • Modular matrix • ……

Example

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.

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)

Adjacency matrix • Weighted percolation on directed networks: If the probability of removing node i is , the network disintegrates if is such that thelargesteigenvalue of the matrix with entriesis less than 1, where A is the adjacency matrix ofthenetwork. PRL 100, 058701 (2008)

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

LaplacianMatrix:Electricalnetworks • Every edge – a resistor of 1 ohm. • Voltage difference of 1 volt between u and v. • R(u,v) – inverse of electrical current from u to v. v _ + u

Laplacian matrix • Effective resistance

LaplacianMatrix:Randomwalks - Kinds of random walks: Unbiased random walks, biased random walks, self-avoid walks, quantum walks, ect

Random Walks on Graphs • At any node, go to one of the neighbors of the node with equal probability. -

Primary Indicators Important parameters of random walks First-Passage TimeF(s,t): Expected number of steps to reach t starting at s Mean Commute timeC(s,t): Steps from s to t, and then go backC(t,s) = F(s,t) + F(t,s) Mean Return timeT(s,s): mean time for returning to node sfor the first time after having left it Cover time, survival problity, …… New J. Phys. 7, 26 (2005)

Laplacian matrix • Random walks is degree ofnode z, m is the number of edges.

Laplacian matrix • Quantum walks • Synchronization • Generalized Gaussian structures • Ultimatum game • •••••• • Relevance to other dynamics

1 1 1 1/2 1 1 1 1/2 Transition probability matrix:definition Adjacency matrix A Transition matrix P

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

Transition probability matrix • First passage time • Commute time • Eigentime identity

Transition probability matrix • Mixing rate • Mixing time • Return-to-origin probability

Normalized Laplacian matrix are the corresponding mutually orthogonal eigenvectors of unit length.

Normalized Laplacian matrix

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

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

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

Spectra of adjacent matrix and Laplacian matrix of small-world networks Completed

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

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)

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. Journal of Mathematical Physics (in press)

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

Random walkson 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

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

Relation to the Hanoi Towers Game What is the minimum number of moves ?

The Hanoi Towers Graphs

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

Spectra of transition matrix: fractal scale-free networks Completed

Thank You!

网络的谱性质与应用

网络的谱性质与应用

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