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Zhenhua Wu

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

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  1. 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: ShlomoHavlinBar-Ilan University VittoriaColizzaTurin, 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)

  2. 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

  3. 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).

  4. 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?

  5. 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)

  6. 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

  7. 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

  8. 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.

  9. Part 2. Specific questions Will the  change the critical fraction qcof a network? Will the  change the universality class of a network?

  10. 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.

  11. 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)

  12. 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!

  13. 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

  14. 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

  15. Part 3. Numerical Results Numerical results Result: indicates c = 3 for  > 0

  16. Part 3. Numerical Results Strong finite size effect successive slope  In our simulation, we can only reach 106 << 1014

  17. 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)

  18. 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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