1. 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
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2. 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

3. Main contents Introduction to relevant matrices Our work1
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
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4. Definitions Adjacency matrix A Diagonal degree matrix D
Laplacian matrix L=D-A
Probability transition matrix
Normalized Laplacian matrix
Fundamental matrix
Modular matrix ……
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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.
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6. 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,
SI model: the eigenvector corresponding to the largest eigenvalue is related to the spreading power of nodes in a network.
Complexus 3 , (2006)
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7. 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, (2008)
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8. Laplacian matrix Spanning trees Algebraic connectivity
provides a upper bound for diameter of a network.
SIAM Journal of discrete mathematics, 1994, 7(3):
Spanning trees
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9. Laplacian matrix
Effective resistance
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10. Laplacian matrix Random walks
is degree of node z, m is the number of edges.
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11. Laplacian matrix Relevance to other dynamics Quantum walks
Synchronization
Generalized Gaussian structures
Ultimatum game
••••••
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12. 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
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13. Transition probability matrix
First passage time
Commute time
Eigentime identity
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14. Transition probability matrix
Mixing rate
Mixing time
Return-to-origin probability
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15. Normalized Laplacian matrix
are the corresponding mutually orthogonal eigenvectors of unit length.
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16. Normalized Laplacian matrix
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17. 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
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18. Spectra of adjacency matrix for a family of deterministic recursive trees
Journal of Physics A, 2009, 42:
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19. Laplacian eigenvalues and eigenvectors of deterministic recursive trees
Physical Review E, 2009, 80:016104
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20. Spectra of adjacent matrix and Laplacian matrix of small-world networks
Completed
2019/4/5

21. 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
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22. 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)
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23. 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:
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24. Application of spectra to random walks
Vicsek fractals
Physical Review E, 2010, 81:
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25. 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
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26. Random Walks on dual Sierpinski gasket
European Physical Journal B, 2011, 82:91-96.
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27. Relation to the Hanoi Towers Game
What is the minimum number of moves ?
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28. The Hanoi Towers Graphs
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29. 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
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30. Spectra of transition matrix: fractal scale-free networks
Completed
2019/4/5

31. Thank You!