1. 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
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2. Main contents Introduction to relevant matrices Our works1
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
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3. 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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4. Isotropic random walks
Transition matrix describes the jumping probability for random walks on graphs -
Isotropic random walks
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5. Random walks on graphs -
At any node, go to one of the neighbors of the node with equal probability.
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6. Random walks on graphs -
At any node, go to one of the neighbors of the node with equal probability.
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7. Random walks on graphs -
At any node, go to one of the neighbors of the node with equal probability.
2019/5/6

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

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

10. 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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11. Transition matrix First passage time Commute time Eigentime identity
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12. Transition matrix Mixing rate Mixing time Return-to-origin probability
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13. Normalized Laplacian matrix
are the corresponding mutually orthogonal eigenvectors of unit length.
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14. Normalized Laplacian matrix
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15. 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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16. 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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17. Spectra of transition matrix for Hanoi graphs
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18. What is the minimum number of moves ?
The Hanoi towers game
What is the minimum number of moves ?
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19. 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:
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20. Spectra of transition matrix for fractal scale-free trees
EPL, 2012, 99:10007
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21. Optimal and suboptimal networks minimizing eigentime identity for random walks

22. Thank You!