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Transport in weighted networks: optimal path and superhighways

Transport in weighted networks: optimal path and superhighways. Shlomo Havlin Bar-Ilan University Israel. Collaborators: Z. Wu, Y. Chen, E. Lopez, S. Carmi, L.A. Braunstein, S. Buldyrev, H. E. Stanley. Wu, Braunstein, Havlin, Stanley, PRL (2006)

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Transport in weighted networks: optimal path and superhighways

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  1. Transport in weighted networks: optimal path and superhighways Shlomo Havlin Bar-Ilan University Israel Collaborators: Z. Wu, Y. Chen, E. Lopez, S. Carmi, L.A. Braunstein, S. Buldyrev, H. E. Stanley Wu, Braunstein, Havlin, Stanley, PRL (2006) Yiping, Lopez, Havlin, Stanley, PRL (2006) Braunstein, Buldyrev, Cohen, Havlin, Stanley, PRL (2003)

  2. What is the research question? • In complex network, different nodes or links have different importance in the transport process. • How to identify the “superhighways”, the subset of the most important links or nodes for transport? Also important for immunization. • Identifying the superhighways and increasing their capacity enables to improve transport significantly. Immunization them will reduce epidemics.

  3. 30 6 10 2 3 4 50 1 15 8 Weighted networks • Networks with weights, such as “cost”, “time”, “resistance”“bandwidth” etc. associated with links or nodes • Many real networks such as world-wide airport network (WAN), E Coli. metabolic network etc. are weighted networks. • Many dynamic processes are carried on weighted networks. Barrat, Vespiggnani et al PNAS (2004)

  4. 30 6 10 2 3 4 50 1 15 8 Minimum spanning tree (MST) • The tree which connects all nodes with minimum total weight. • Union of all “strong disorder” optimal paths between any two nodes. • The MST is the part of the network that most of the traffic goes through • MST -- widely used in optimal traffic flow, design and operation of communication networks. B A In strong disorder the weight of the path is determined by the largest weight along the path!

  5. Optimal path – strong disorder Random Graphs and Watts Strogatz Networks CONSTANT SLOPE - typical range of neighborhood without long range links - typical number of nodes with long range links Analytically and Numerically LARGE WORLD!! Compared to the diameter or average shortest path or weak disorder (small world) N – total number of nodes Braunstein, Buldyrev, Cohen, Havlin, Stanley, Phys. Rev. Lett. 91, 247901 (2003);

  6. 0 0 15 18 12 0 7 0 0 Centrality of MST: How to find the importance of nodes in transport? • Number of times a node (or link) is used by the set of all shortest paths between all pairs of nodes - betweenes centrality. • Measure the frequency of a node being used by traffic. For ER, scale free and many real world networks Newman., Phys. Rev. E (2001) D.-H. Kim, et al., Phys. Rev. E (2004) K.-I. Goh, et al., Phys. Rev. E (2005)

  7. Minimum spanning tree (MST) High centrality nodes

  8. Incipient percolation cluster (IIC) • IIC is defined as the largest component at percolation criticality. • For a random scale-free or Erdös-Rényi graph, to get the IIC, we remove the links in descending order of the weight, until • is < 2. At , the system is at criticality. Then the largest connected component of the remaining structure is the • IIC. • The IIC can be shown to be a subset of the MST . R. Cohen, et al., Phys. Rev. Lett. 85, 4626 (2000)

  9. MST and IIC MST Superhighways and Roads The IIC is a subset of the MST I I C Superhighways

  10. Superhighways (SHW) and Roads sters

  11. Mean Centrality in SHW and Roads

  12. The average fraction of pairs of nodes using the IIC

  13. How much of the IIC is used? Square lattice ER + 3nd largest cluster ER,+ 2nd largest cluster ER SF, λ= 4.5 SF, λ= 3.5 The IIC is only a ZERO fraction of the network of order N2/3 !!

  14. Distribution of Centrality in MST and IIC

  15. Theory for Centrality Distribution For IIC inside the MST: For the MST: Good agreement with simulations!

  16. Application: improve flow in the network Comparison between two strategies: sI: improving capacity of all IIC links--highways sII: improving the highest centrality links in MST (same number as sI). BOTH, SAME COST Assume: multiple sources and sinks: randomly choose n pairs of nodes as sources and other n nodes as sinks • We study two transport problems: • Current flow in random resistor networks, where each link of the network represents a resistor. (Total flow, F: total current or conductance) • Maximum flow problem from computer science, where each link of the network has an upper bound capacity. (Total flow, F: maximum possible flow into network) Result: sI is better

  17. Application: compare two strategiescurrent flow and maximum flow sI:improve the IIC links. sII: improve the high C links in MST. • Two types of transport • Current flow: improve the conductance • Maximum flow: improve the capacity n=50 n=250 F0: flow of original network. FsI : flow after using sI. FsII: flow after usingsII. n=500 N=2048, <k>=4

  18. Summary • MST can be partitioned into superhighways which carry most of the traffic and roads with less traffic. • We identify the superhighways as the largest percolation cluster at criticality -- IIC. • Increasing the capacity of the superhighways enables to improve transport significantly. The superhighways of order N2/3 -- a zero fraction of the the network!! Wu, Braunstein, Havlin, Stanley, PRL (2006)

  19. Applications: compare 2 strategiescurrent flow and maximum flow • Two transport problems: • Current flow in random resistor networks, where each link of the network represents a resistor. (Total flow, F: total current or conductance) • Maximum flow problem in computer science[4], where each link of the network has a capacity upper bound. (Total flow, F: maximum possible flow into network) resistance/capacity = eax, with a = 40 (strong disorder) Multiple sources and sinks: randomly choose n pairs of nodes as sources and other n nodes as sinks Two strategies to improve flow, F, of the network: sI: improving the IIC links. sII: improving the high C links in MST. [4]. Using the push-relabel algorithm by Goldberg. http://www.avglab.com/andrew/soft.html

  20. Universal behavior of optimal paths in weighted networks with general disorder Yiping Chen Advisor: H.E. Stanley Y. Chen, E. Lopez, S. Havlin and H.E. Stanley “Universal behavior of optimal paths in weighted networks with general disorder” PRL(submitted)

  21. Collaborators: Eduardo Lopez and Shlomo Havlin Scale Free – Optimal Path Theoretically + Numerically Numerically Strong Disorder LARGE WORLD!! SMALL WORLD!! Weak Disorder Diameter – shortest path Braunstein, Buldyrev, Cohen, Havlin, Stanley, Phys. Rev. Lett. 91, 247901 (2003); Cond-mat/0305051

  22. Motivation: Different disorders are introduced to mimic the individual properties of links or nodes (distance, airline capacity…).

  23. Weighted random networks and optimal path: Weights w are assigned to the links (or nodes) to mimic the individual properties of links (or nodes). Optimal Path: the path with lowest total weight. (If all weights the same, the shortest path is the optimal path) 7 4 3 source 11 20 5 destination 2

  24. Previous results: Most extensively studied weight distribution (Generated by an exponential function) small: Weak disorder : all the weights along the optimal path contribute to the total weight along the optimal path . L large: Strong disorder : is dominated by the highest weight along the path. Y. M. Strelniker et al., Phys. Rev. E 69, 065105(R) (2004)

  25. Unsolved problem: General weight distribution Needed to reflect the properties of real world. Ex: • exponential function----quantum tunnelling effect • power-law----diffusion in random media • lognormal----conductance of quantum dots • Gaussian----polymers

  26. Questions: 1.Do optimal paths for different weight distributions show similar behavior? 2. Is it possible to derive a way to predict whether the weighted network is in strong or weak disorder in case of general weight distribution? 3. Will strong disorder behavior show up for any distributions when distribution is broad?

  27. Theory: On lattice Suppose the weight follows distribution (Total cost) where We define 0: , cannot dominate the total cost (Weak limit) 1: , dominates the total cost (Strong limit) L Assume S can determine the strong or weak behavior. Using percolation theory: Percolation exponent Structural & distributional parameter

  28. General distributions studied in simulation • Power-law • Power-law with additional parameter • Lognormal • Gaussian

  29. Answer to questions 1 and 2: My simulation result on 2D-lattice -0.22 L the linear size of lattice Strong: Weak: the length of optimal path Y. Chen, E. Lopez, S. Havlin and H.E. Stanley “Universal behavior of optimal paths in weighted networks with general disorder” PRL(submitted)

  30. Erdős-Rényi (ER) Networks For each pair of nodes, they have probability p to be connected Definition: A set of N nodes p My simulations on ER network show the same agreement with theory.

  31. Answer to question 3: Distributions that are not expected to have strong disorder behavior • Gaussian • Exponential ( the percolation threshold, constant for certain network structure) A is independent of which describes the broadness of distribution. No matter how broad the distribution is, can not be large, and no strong disorder will show up.

  32. Summary of answers to 3 questions 1. Do optimal paths in different weight distributions show similar behavior? Yes 2. Is it possible to derive a way to predict whether the weighted network is in strong or weak disorder in case of general weight distribution? Yes 3. Will strong disorder behavior show up for any distributions when distribution is broad? No

  33. Theory: On lattice Suppose follows distribution where S goes small: and are comparable (Weak) S goes large: (Strong) Percolation applies

  34. Percolation Theory Percolation properties: Percolation threshold (0.5 for 2D square lattice) In finite lattice with linear size L: The first and second highest weighted bonds in optimal path will be close to and follow its deviation rule. Strong disorder and percolation behave in the similar way Thus

  35. The result comes from percolation theory From percolation theory Transfer back to original disorder distribution

  36. Test on known result Apply our theory on disorder distribution , we get percolation threshold percolation exponent (Constants for certain structure) In 2D square lattice To have same behavior by keeping fixed, we get constant Compatible with the reported results. (The crossover from strong to weak disorder occurs at )

  37. Scaling on ER network Percolation at criticality on Erdős-Rényi(ER) networks is equivalent to percolation on a lattice at the upper critical dimension . Virtual linear size (N = number of nodes) Percolation exponent in ER network ( is now depending on number of nodes in ER network)

  38. Simulation result on ER networks In ER network, the percolation exponent (N=number of nodes) log-linear log-log Strong: Weak: Strong: Weak: From early report: L.A. Braunstein et al. Phys. Rev. Lett. 91, 168701 (2003)

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