Percolation analysis on scale-free networks with correlated link weights
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Percolation analysis on scale-free networks with correlated link weights. Zhenhua Wu. Advisor: H. E. Stanley Boston University Co-advisor: Lidia A. Braunstein Universidad Nacional de Mar del Plata. Collaborators: Shlomo Havlin Bar- Ilan University Vittoria Colizza Turin, Italy

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Zhenhua wu

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)


General question

General question

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


Properties of real world networks

Part 1. Motivation:

Properties of real-world networks

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).


What part of network is more important for traffic

Part 1. Motivation:

What part of network is more important for traffic?

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?


Why is it important

Part 1. Motivation:

Why is it important?

  • 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)


Zhenhua wu

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


Effect of

Part 2. Model:

Effect of 

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


Percolation properties only depend on the sign of

Part 2. Model:

Percolation properties only depend on the sign of 

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.


Specific questions

Part 2.

Specific questions

Will the  change the critical fraction qcof a network?

Will the  change the universality class of a network?


Comparison of q c for different

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

Comparison of qc for different 

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.


Critical degree distribution exponent c

Part 2. c: previous result

Critical degree distribution exponent, c

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)


What is the c for 0

Part 2. c: question about c for  > 0

What is the 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!


How to find out p c 0 for 0

Part 2. percolation threshold pc:

How to find out pc= 0 for  > 0?

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


T c the critical weight at p c

Part 2. Tc:

Tc , the critical weight at pc

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


Numerical results

Part 3. Numerical Results

Numerical results

Result: indicates c = 3 for  > 0


Strong finite size effect

Part 3. Numerical Results

Strong finite size effect

successive slope 

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


Part 4 summary

Part 4. Summary

  • 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)


Acknowledgements

Acknowledgements

  • 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,


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