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: [email protected]
Homepage: http://homepage.fudan.edu.cn/~zhangzz/
Blog: http://group.sciencenet.cn/home.php?mod=space&uid=311410
Main contents
1
Introduction to relevant matrices
 Definition of various matrixes
 Relevance of spectra for transition matrix
to structure and dynamics
2
Our works
 Computation of spectra for transition
matrix of diverse networks
 Applications to spanning trees and walks
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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
……
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Transition matrix describes the jumping
probability for random walks on graphs
-
Isotropic random walks
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Random walks on graphs
-
At any node, go to one of the neighbors
of the node with equal probability.
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Random walks on graphs
-
At any node, go to one of the neighbors
of the node with equal probability.
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Random walks on graphs
-
At any node, go to one of the neighbors
of the node with equal probability.
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Random walks on graphs
-
At any node, go to one of the neighbors
of the node with equal probability.
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Random walks on graphs
-
At any node, go to one of the neighbors
of the node with equal probability.
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Transition matrix
Q is often called normalized adjacency matrix
for non-bipartite graphs
are the corresponding mutually orthogonal eigenvectors of unit length.
 Stationary distribution
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Transition matrix
 First passage time
 Commute time
 Eigentime identity
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Transition matrix
 Mixing rate
 Mixing time
 Return-to-origin probability
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Normalized Laplacian matrix
are the corresponding mutually orthogonal eigenvectors of unit length.
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Normalized Laplacian matrix
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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
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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
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Spectra of transition matrix for Hanoi graphs
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The Hanoi towers game
What is the
minimum
number of
moves
?
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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.
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Spectra of transition matrix for fractal
scale-free trees
EPL, 2012, 99:10007
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Optimal and suboptimal networks minimizing
eigentime identity for random walks
Thank You!