Random walks on complex networks Zhongzhi ZHANG (章忠志) School of Computer Science, Fudan University Email: zhangzz@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/~zhangzz/

Contents Introduction to random walks on graphs Applications of random walks Trapping on graphs: Introduction and Review Trapping on complex networks Future work 2018/9/9

Random Walks on Graphs Kinds of random walks: - Kinds of random walks: Unbiased random walks, biased random walks, self-avoid walks, quantum walks, ect 2018/9/9

Random Walks on Graphs - At any node, go to one of the neighbors of the node with equal probability. 2018/9/9

Random Walks on Graphs - At any node, go to one of the neighbors of the node with equal probability. 2018/9/9

Random Walks on Graphs - At any node, go to one of the neighbors of the node with equal probability. 2018/9/9

Random Walks on Graphs - At any node, go to one of the neighbors of the node with equal probability. 2018/9/9

Random Walks on Graphs - At any node, go to one of the neighbors of the node with equal probability. 2018/9/9

Important parameters of random walks Primary Indicators First-Passage Time F(s,t): Expected number of steps to reach t starting at s Mean Commute time C(s,t): Steps from s to t, and then go back C(t,s) = F(s,t) + F(t,s) Mean Return time T(s,s): mean time for returning to node s for the first time after having left it Cover time, survival problity, …… New J. Phys. 7, 26 (2005) 2018/9/9

FPT for general graphs Pseudoinverse of the Laplacian matrix N: Node number is the Laplacian matrix defined as 2018/9/9

FPT for complete graph Study random walks on simple structure is of exceptional importance. 2018/9/9

FPT for a linear graph 2018/9/9

Applications of random walks Community detection Recommendation systems Electrical networks Spanning trees Information Retrieval Natural Language Processing Machine Learning Graph partitioning In economics: random walk hypothesis PageRank algorithm 2018/9/9

Application to Community detection World Wide Web Protein interaction networks Metabolic networks Social networks Biological networks Food Webs Properties of community may be quite different from the average property of network. More links “inside” than “outside” 2018/9/9

Application to recommendation systems Help for setting up efficient recommendation systems IEEE Trans. Knowl. Data Eng. 19, 355 (2007) 2018/9/9

Connections with electrical networks 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 C(u,v) = F(s,t) + F(t,s) =2mR(u,v), R(u,v)= C(u,v)/ (2m) dz is degree of z, m is the number of edges 2018/9/9

Formulas for effective resistance 2018/9/9

Random walks and spanning trees EPL (Europhysics Letters), 2010, 90:68002 2018/9/9

Trapping problem: Random walks on graphs with an immobile trap Trapping time for node i denoted by Average trapping time Research goal: obtain the dependence of average trapping time on the system size 2018/9/9

Previous work Trapping on Sierpinski gasket Physical Review E, 2002, 65: 021105 2018/9/9

Previous work: Trapping on T-fractal Physical Review E, 2008, 77: 011128 2018/9/9

Previous work: Trapping on regular lattices Journal of Mathematical Physics, 1969, 10: 753 2018/9/9

Our work Random walks on scale-free networks A pseudofractal scale-free web Apollonian networks Modular scale-free networks Koch networks A fractal scale-free network Scale-free networks with the same degree sequences 2018/9/9

Main contributions Unveil the effect of trap location on ATT Methods for finding Mean first-passage time (MFPT) Backward equations Generating functions Laplacian spectra Electrical networks Uncover the impacts of structures on MFPT Scale-free behavior Modular structure Fractal structure Unveil the effect of trap location on ATT Relate random walks to Hanoi Towers Game 2018/9/9

Walks on pseudofractal scale-free web Physical Review E, 2009, 79: 021127. Contributions：(1)New method (2)New findings 2018/9/9

Walks on Apollonian network The most efficient structure for diffusion EPL, 2009, 86: 10006. 2018/9/9

Walks on modular scale-free networks Generating function method Physical Review E, 2009, 80: 051120. 2018/9/9

ATT on uncorrelated scale-free networks In uncorrelated random SF webs, Physical Review E, 2009, 79: 061113. 2018/9/9

Walks on the Koch network Physical Review E, 2009, 79: 061113. 2018/9/9

Impact of degree heterogeneity on MFPT Journal of Physics A, 2010, 43: 395102 Chaos, 2010, 20: 043112 2018/9/9

Effect of fractality on MFPT Definition of fractal networks Protein interaction network Box-covering method WWW Metabolic network Mean mass (number of nodes) within a box: Hollywood film actors network Last year Song et al. discovered the fractal scaling in scale-free networks. They found that the number of boxes to cover a given network scales in a power law with the box size ell. The mean number of nodes within a box scales in a power law. This power-law behavior seems to be contradictory to the small-world feature. The number of nodes with a distance ell from a given vertex increases exponentially with ell. The box size is defined as the maximum distance within a box. Song, Havlin, and Makse, Nature (2005). 2018/9/9

Walks on a fractal scale-free network EPL (Europhysics Letters), 2009, 88: 10001. 2018/9/9

Walks on scale-free networks with identical degree sequences Physical Review E, 2009, 79: 031110. 2018/9/9

Walks on scale-free networks with identical degree sequences Advantages： (1) without crossing edges (2) always connected. Physical Review E, 2009, 80: 061111 2018/9/9

Influence of trap position on MFPT on various networks 2018/9/9

Significant impact of trap position on MFPT in non-fractal scale-free trees Journal of Mathematical Physics, 2009, 50: 033514. Journal of Physics A, 2011, 44: 075102 2018/9/9

Global mean first-passage time Significant impact of trap position on MFPT in scale-free Koch networks Mean receiving time Mean sending time Global mean first-passage time European Physical Journal B, 2011, (in press). 2018/9/9

No essential impact of trap position on MFPT in fractal scale-free trees Journal of Physics A, 2011, 44: 075102 2018/9/9

No qualitative effect of trap location on MFPT in extended T-graphs Physical Review E, 2008, 77: 011128 Physical Review E, 2010, 82: 031140 New Journal of Physics, 2009, 11: 103043 2018/9/9

Random Walks on Vicsek fractals Physical Review E, 2010, 81:031118. 2018/9/9

Random Walks on dual Sierpinski gasket 2018/9/9

Relation to the Hanoi Towers Game   What is the minimum number of moves ? 2018/9/9

The Hanoi Towers Graphs 2018/9/9

Future work Walks with multiple traps 1 Quantum walks on networks 2 Self-avoid walks 3 Biased walks, e.g. walks on weighted nets 4 2018/9/9

Thank You!