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Zhenhua Wu

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Percolation analysis on scale-free networks with correlated link weights

Zhenhua Wu

Advisor:

H. E. StanleyBoston University

Co-advisor:

Lidia A. BraunsteinUniversidad Nacional de

Mar del Plata

Collaborators:

ShlomoHavlinBar-Ilan University

VittoriaColizzaTurin, Italy

Reuven CohenBar-Ilan University

12/4/2007

Z. Wu, L. A. Braunstein, V. Colizza, R. Cohen, S. Havlin, H. E. Stanley, Phys. Rev. E 74, 056104 (2006)

Do link weights affect the network properties?

Outline

- Motivation
- Modeling approach
- qc: definition and simulations in the model
- pc: percolation threshold (define c and Tc)

- Numerical results
- Summary

Part 1. Motivation:

Example: world-wide airport network (WAN)

Large cities (hubs) have many routs k(degree)

Link weight Tij is “# of passengers”

Link weight Tij depends on degree ki and kj of airports i and j

-=-2.01

THK-C

THK-P

0.5

Real-world networks:

Heterogeneous connectivity

Heterogeneous weights

Correlation between connectivity and weights

A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, PNAS, 101, 3747 (2004).

Part 1. Motivation:

Thickness of links: Traffic, ex: number of passengers

Bad choice

Good choice

How to choose the most import links?

qc: critical q to break network

Choose the links with the highest traffic

Introduce rank-ordered percolation

We remove links in ascending order of weight, T

S: number of nodes in the largest connected cluster

Infinite system

q: fraction of links removed with lowest weights

Finite system

N: number of nodes

qc 1-pc

What is effect of weights & correlation on percolation?

Part 1. Motivation:

- World-wide airport network is closely related to epidemic spreading such as the case of SARS[1].
- Help to develop more effective immunization strategies.
- Biological networks such as the E. coli metabolic networks also has the same correlation between weights and nodes degree.[2]

[1] V. Colizza et al., BMC Medicine 5,34 (2007)

[2] P. J. Macdonald et al., Europhys. Lett.72, 308 (2005)

Part 2. Model:

Weighted scale-free networks

Scale-free (SF)

Power-law distribution

k=1

k=10

“hub”

Define the weight on each link:

: maximum degree

: controls correlation

Definitions:

xij: Uniform distributed random numbers 0 < xij < 1.

ki: Degree of node i.

Tij: Weight, ex, traffic, number of passengers in WAN.

For WAN, = 0.5

Part 2. Model:

For

Ex:

0.17

6

Ex:

0.17

0.08

12

6

8

0.13

8

0.13

4

0.25

The sign of determines the nature of the hubs

Part 2. Model:

In studies of percolation properties,

what matters is the rank of the links according their weight.

For

3

6

36

12

1

144

3

6

36

64

8

2

8

2

64

4

4

16

Ranking of the links remains unchanged for different of the same sign.

Part 2.

Will the change the critical fraction qcof a network?

Will the change the universality class of a network?

Part 2. qc: simulations on the model (number of nodes N = 8,192)

N=8,192

qc: critical q to break network

S

N/2

q: fraction of links removed with lowest weights

0

~0.7

~0.45

~1.0

Scale-free networks with > 0 have larger qc than networks with 0.

Part 2. c: previous result

cis the below which pc is zero and above which pc is finite

pc 1-qc

Fraction of links remained to connect the whole network

1

Scale-free networks with = 0

Perfectly connected

0

2

3

c=3 for scale-free networks with = 0

R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Phys. Rev. Lett.85, 4626 (2000)

Part 2. c: question about c for > 0

Numerical results

for < 0

Theoretical results

c = 3 for = 0

Numerical results

for > 0

If c ( > 0) c ( = 0)

different universality classes!

Part 2. percolation threshold pc:

Difficulties:

- Limit of numerical precision hard to determine
pc = 0 numerically.

- Correlation hard to find analytical solution for pc.

Solution:

Analytical approach with numerical solution

Part 2. Tc:

Network of our model

Maximum degree: kmax

: the critical weight at which the system is at pc

Smallest weight

kill

diverge

critical

weight

pc

Kill all the links

Largest weight

The divergence of Tc tells us whether pc = 0

Part 3. Numerical Results

Result: indicates c = 3 for > 0

Part 3. Numerical Results

successive slope

In our simulation, we can only reach 106 << 1014

- For the first time, we proposed and analyzed a model that takes into account the correlation between weights and node degrees.
- The correlation between weight Tij and nodes degree kiand kj , which is quantified by, changes the properties of networks.
- Scale-free networks with > 0, such as the WAN, have larger qcthan scale-free networks with 0.
- Scale-free networks with > 0 and = 0 belong to the same university class (have the same c= 3)

- Advisor: H. E. Stanley
- Co-advisor: Lidia A. Braunstein
- Professor Shlomo Havlin
- E. López, S. Sreenivasan, Y. Chen, G. Li, M. Kitsak
- Special thanks: E. López , P. Ivanov,