Zhenhua Wu

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

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Properties of real-world networks
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
THK-C
THK-P

• 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).
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What part of network is more important for
traffic?
Part 1. Motivation
Thickness of links Traffic, ex number of
passengers
Bad choice
Good choice
How to choose the most import links?
Choose the links with the highest traffic
Introduce rank-ordered percolation
We remove links in ascending order of weight, T

• qc critical q to break network

What is effect of weights correlation on
percolation?
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Why is it important?
Part 1. Motivation

• World-wide airport network is closely related to
  epidemic spreading such as the case of SARS1.
• 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)
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Part 2. Model
Weighted scale-free networks
Scale-free (SF)
Power-law distribution
k1
k10
Define the weight on each link
? controls correlation
Definitions xij Uniform distributed random
numbers 0 lt xij lt 1. ki Degree of node i. Tij
Weight, ex, traffic, number of passengers in
WAN. For WAN, ? 0.5
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Effect of ?
Part 2. Model
For
Ex
Ex
The sign of ? determines the nature of the hubs
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Percolation properties only depend on the sign of
?
Part 2. Model
In studies of percolation properties, what
matters is the rank of the links according their
weight.
For
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Specific questions
Part 2.

• Will the ? change the critical fraction qc of a
  network?
• Will the ? change the universality class of a
  network?

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Comparison of qc for different ?
Part 2. qc simulations on the model (number of
nodes N 8,192)
N8,192

• qc critical q to break network

S
N/2
q fraction of links removed with lowest weights
0
Scale-free networks with ? gt 0 have larger qc
than networks with ? ? 0.
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Critical degree distribution exponent, ?c
Part 2. ?c previous result
?c is the ? below which pc is zero and above
which pc is finite
pc ? 1-qc Fraction of links remained to connect
the whole network
Scale-free networks with ? 0
Perfectly connected
?c 3 for scale-free networks with ? 0
R. Cohen, K. Erez, D. ben-Avraham, S. Havlin,
Phys. Rev. Lett. 85, 4626 (2000)
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What is the ?c for ? lt 0?
Part 2. ?c question about ?c for ? gt 0
Numerical results for ? lt 0
Theoretical results ?c 3 for ? 0
Numerical results for ? gt 0
If ?c (? gt 0) ? ?c (? 0)
different universality classes!
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How to find out pc 0 for ? gt 0?
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
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Tc , the critical weight at pc
Part 2. Tc
The divergence of Tc tells us whether pc 0
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Numerical results
Part 3. Numerical Results
Result indicates ?c 3 for ? gt 0
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Strong finite size effect
Part 3. Numerical Results
In our simulation, we can only reach 106 ltlt 1014
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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 ki and kj , which is quantified by ?,
  changes the properties of networks.
• Scale-free networks with ? gt 0, such as the WAN,
  have larger qc than scale-free networks with ? ?
  0.
• Scale-free networks with ? gt 0 and ? 0 belong
  to the same university class (have the same ?c
  3)

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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,