1. 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.cnzhangzz@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/~zhangzz/ http://homepage.fudan.edu.cn/~zhangzz/ Blog: http://group.sciencenet.cn/home.php?mod=space&uid=311410 第七届全国复杂网络学术会议

2. 复旦大学 Collaborators 2016-7-7  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

3. 复旦大学 2016-7-7 Main contents Introduction to relevant matrices 1 Our work 2 3/43 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

4. 复旦大学 Definitions 4/45 2016-7-7 Adjacency matrix A Diagonal degree matrix D Laplacian matrix L=D-A Probability transition matrix Normalized Laplacian matrix Fundamental matrix Modular matrix ……

5. 复旦大学 Adjacency matrix 5/45 2016-7-7  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.  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

6. 复旦大学 Adjacency matrix 6/45 2016-7-7  SIS model: the largest eigenvalue defines an epidemic threshold  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) 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.

7. 复旦大学 7/45 2016-7-7 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)  Weighted percolation on directed networks: Adjacency matrix

8. 复旦大学 Laplacian matrix 8/45 2016-7-7  Algebraic connectivity provides a upper bound for diameter of a network. SIAM Journal of discrete mathematics, 1994, 7(3): 443-457.  Spanning trees

9. 复旦大学 2016-7-7 9/43  Effective resistance Laplacian matrix

10. 复旦大学 2016-7-7 is degree of node z, m is the number of edges. 10/43  Random walks Laplacian matrix

11. 复旦大学 2016-7-7 Quantum walks Synchronization Generalized Gaussian structures Ultimatum game 11/43 Laplacian matrix  Relevance to other dynamics

12. 复旦大学 Transition probability matrix 12/45 2016-7-7 for non-bipartite graphs are the corresponding mutually orthogonal eigenvectors of unit length. Q is often called normalized adjacency matrix  Stationary distribution

13. 复旦大学 13/45 2016-7-7 Transition probability matrix  Commute time  First passage time  Eigentime identity

14. 复旦大学 14/45 2016-7-7  Mixing rate Transition probability matrix  Mixing time  Return-to-origin probability

15. 复旦大学 Normalized Laplacian matrix 15/45 2016-7-7 are the corresponding mutually orthogonal eigenvectors of unit length.

16. 复旦大学 16/45 2016-7-7 Normalized Laplacian matrix

17. 复旦大学 Our work 2016-7-7  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

18. 复旦大学 2016-7-7 Spectra of adjacency matrix for a family of deterministic recursive trees Journal of Physics A, 2009, 42: 165103.

19. 复旦大学 2016-7-7 Laplacian eigenvalues and eigenvectors of deterministic recursive trees Physical Review E, 2009, 80:016104

20. 复旦大学 2016-7-7 Spectra of adjacent matrix and Laplacian matrix of small-world networks Completed

21. 复旦大学 2016-7-7 Using Laplacian spectra to determine the number of spanning trees in Farey graph Farey sequence of order n denoted by

22. 复旦大学 2016-7-7 Spanning trees in Farey graph Two nodes and are linked to each other if they satisfy Physica A (in revision) Theoretical Computer Science, 2011, 412:865–875

23. 复旦大学 2016-7-7 Spanning trees in scale-free networks EPL, 2010, 90:68002. A counterintuitive conclusion that a network with more spanning trees may be relatively unreliable. Physical Review E, 2011, 83:016116. Fractality can significantly increase the number of spanning trees in fractal scale-free networks. Journal of Mathematical Physics (in press)

24. 复旦大学 2016-7-7 Application of spectra to random walks Physical Review E, 2010, 81:031118. 24/43 Vicsek fractals

25. 复旦大学 2016-7-7 Random walks on T fractals Physical Review E, 2010, 82:031140 25/43 is obtained from the relationship between characteristic polynomials at different generations. Our method can void the computation of eigenvalues.

26. 复旦大学 2016-7-7 Random Walks on dual Sierpinski gasket 26/43 European Physical Journal B, 2011, 82:91-96.

27. 复旦大学 Relation to the Hanoi Towers Game What is the minimum number of moves ? 2016-7-7 27/43

28. 复旦大学 The Hanoi Towers Graphs 2016-7-7 28/43

29. 复旦大学 2016-7-7 Spectra of transition matrix: T-fractal EPL, 2011, in press 29/43 We obtain all the eigenvalues and their multiplicities. The reciprocal of the smallest eigenvalue is approximately equal to the mean trapping time We obtain all the eigenvalues and their multiplicities. The reciprocal of the smallest eigenvalue is approximately equal to the mean trapping time

30. 复旦大学 2016-7-7 Spectra of transition matrix: fractal scale-free networks Completed 30/43

31. Thank You!