Global meanfirstpassage time of random walks on Vicsek

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

Introduction

Fractals l l 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 l Laplacian matrix l Eigenvalues

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

Global mean-first-passage time l First-passage time(FPT) ü l 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) ü FPT averaged over all pairs of nodes

Vicsek fractals ß

Vicsek fractals ß

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

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

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

Comparison ß

Bound in Trees l l 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 l l l 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

Thank you