Global mean-first-passage time of random walks on Vicsek fractals Wubin Fudan University

Introduction

Fractals Fractals are an important concept characterizing the features of real systems We can obtain explicit closed-form solutions on fractals

Sierpinski gasket

T-fractal

Eigenvalues Laplacian matrix Eigenvalues  

Discrete Random walks Assuming the time is discrete The walker jumps uniformly from its current location to one of its neighbors

Global mean-first-passage time First-passage time(FPT) The expected time to hit a target node for the first time for a walker starting from a starting node Global mean-first-passage time(GMFPT) FPT averaged over all pairs of nodes

Vicsek fractals        

Vicsek fractals  

GMFPT General method Numerical result Heavy demands on time and computational resources Suit for small networks

GMFPT Using eigenvalues Obtain the relation between GMFPT and network order directly

Comparison The relations between GMFPT and g The filled symbols are the numerical results the empty symbols correspond to the exact values

Comparison  

Bound in Trees The upper bound can be reached when the tree is a linear chain The lower bound can be reached when the tree is a star graph

Conclusions Using the connection between the FPTs and the Laplacian eigenvalues for general graphs, we have computed the GMFPT and obtained explicit solution GMFPT grows approximately as a power-law function of N The upper and lower bound for GMFPT can be achieved in linear chains and star graphs

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