报告人： 林 苑指导老师：章忠志 副教授复旦大学2010.7.30

Introduction about random walks Concepts Applications

Our works Fixed-trap problem Multi-trap problem Hamiltonian walks Self-avoid walks

Introduction about random walks Concepts Applications

Our works Fixed-trap problem Multi-trap problem Hamiltonian walks Self-avoid walks

At any node, go to one of the neighbors of the node with equal probability.

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At any node, go to one of the neighbors of the node with equal probability.

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• At any node, go to one of the neighbors of the node with equal probability.

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• At any node, go to one of the neighbors of the node with equal probability.

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• At any node, go to one of the neighbors of the node with equal probability.

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• At any node, go to one of the neighbors of the node with equal probability.

Random walks can be depicted accurately by Markov Chain.

Markov Chain Laplacian matrix

Generating Function

Mean transit time Tij

Tij ≠ Tji

Mean return time Tii

Mean commute time Cij

Cij =Tij+Tji

PageRank of Google Cited time Semantic categorization Recommendatory System

One major issue: How closed are two nodes? Distance between nodes

Classical methods Shortest Path Length Numbers of Paths

Based on Random Walk (or diffusion) Mean transit time, Mean commute time

The latter methods should be better, however… Calculate inverse of matrix for O(|V|)

times. Need more efficient way to calculate.

Imagine there are traps (or absorbers) on several certain vertices.

Imagine there are traps (or absorbers) on several certain vertices.

We are interested the time of absorption.

For simplicity, we first consider the problem that only a single trap.

Trapping in scale-free networks with hierarchical organization of modularity, Zhang Zhongzhi, Lin Yuan, et al.Physical Review E, 2009, 80: 051120.

Scale-free topology Modular organization

For a large number of real networks, these two features coexist: Protein interaction network Metabolic networks The World Wide Web Some social networks … …

Lead to the rising research on some outstanding issues in the field of complex networks such as exploring the generation mechanisms for scale-free behavior, detecting and characterizing modular structure.

The two features are closely related to other structural properties such as average path length and clustering coefficient.

Understand how the dynamical processes are influenced by the underlying topological structure.

Trapping issue relevant to a variety of contexts.

We denote by Hg the network model after g iterations.

For g=1, The network consists of a central

node, called the hub node, And M-1 peripheral (external) nodes.

All these M nodes are fully connected to each other.

We denote by Hg the network model after g iterations.

For g>1, Hg can be obtained from Hg-1 by adding M-1

replicas of Hg-1 with their external nodes being linked to the hub of original Hg-1 unit.

The new hub is the hub of original Hg-1 unit.

The new external nodes are composed of all the peripheral nodes of M-1 copies of Hg-1.

Xi

First-passage time (FPT)

Markov chain

gN

vv

i

ivi tXF

gd

atXF

1

)1()(

)(

Define a generating function

0

)()(t

tii ztXFzF

Define a generating function

(Ng-1)-dimensional vector

W is a matrix with order (Ng-1)*(Ng-1) with entry wij=aij/di(g)

0

)()(t

tii ztXFzF

)()( zFzWzF

0)()(')( zFWzFzWI

Setting z=1,

0)()(')( zFWzFzWI

eWIFWWIF 11 )()1()()1('

Setting z=1,

(I-W)-1

Fundamental matrix of the Markov chain representing the unbiased random walk

0)()(')( zFWzFzWI

eWIFWWIF 11 )()1()()1('

For large g, inverting matrix is prohibitively time and memory consuming, making it intractable to obtain MFPT through direct calculation. Time Complexity : O(N3) Space Complexity : O(N2)

Hence, an alternative method of computing MFPT becomes necessary.

1

1

1

1

1, )1()()(

1)1(

)(

2

)()(

g

m

t

igm

pg

pp

tg itPiQ

gKtP

gK

M

gKtP

1

1

1

11, )1()(

)(

)1(

)(

)1()(

g

m

t

igm

h

m

th

g

g itQiPgK

M

gK

MtQ

1

10

)()()(

)()(

2

)()()x(

g

m

gmp

g

ppt

tgg xPxQ

gK

xxPx

gK

M

gK

xxtPP

0

1

1

)()1()(

)(

)(

)1()()x(

t

g

m

mm

h

g

h

gt

gg xPMxgK

xQx

gK

MxtQQ

g gN

i

N

iiii gTgTgT

2

1

2

)()1()(

2

11

1 )1()1()1(1

||||)(

g

m

Pgm

Hmmmm

mgPg

Pgi TNTNTNMT

MTgT

)(2)4105()1

)(414166(1

)1( 223233

MMgMMMM

MMMM

M

MMT g

g

g

g

The larger the value of M, the more efficient the trapping process.

The MFPT increases as a power-law function of the number of nodes with the exponent much less than 1.

MMggNT ln/)1ln(1)(~

The above obtained scaling of MFPT with order of the hierarchical scale-free networks is quite different from other media. Regular lattices Fractals (Sierpinski, T-fractal…) Pseudofractal (Koch, Apollonian)

More Efficient The trap is fixed on hub. The modularity.

[1] Zhang Zhongzhi, Lin Yuan, et al. Trapping in scale free networks with hierarchical organization of modularity, Physical Review E, 2009, 80: 051120.

[2] Zhang Zhongzhi, Lin Yuan, et al. Mean first-passage time for random walks on the T-graph, New Journal of Physics, 2009, 11: 103043.

[3] Zhang Zhongzhi, Lin Yuan, et al. Average distance in a hierarchical scale-free network: an exact solution. Journal of Statistical Mechanics: Theory and Experiment, 2009, P10022.

[4] Lin Yuan, Zhang Zhongzhi. Exactly determining mean first-passage time on a class of treelike regular fractals, Physical Review E, (under review).

[5] Zhang Zhongzhi, Lin Yuan. Random walks in modular scale-free networks with multiple traps, Physical Review E, (in revision).

[6] Zhang Zhongzhi, Lin Yuan. Impact of trap position on the efficiency of trapping in a class of dendritic scale-free networks, Journal of Chemical Physics, (under review).

[7] Zhang Zhongzhi, Lin Yuan. Scaling behavior of mean first-passage time for trapping on a class of scale-free trees, European Physical Journal B, (under review).

Thank You