1. Transport in weighted networks: optimal path and superhighways Collaborators: Z. Wu, Y. Chen, E. Lopez, S. Carmi, L.A. Braunstein, S. Buldyrev, H. E. Stanley Shlomo Havlin Bar-Ilan University Israel 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. 10 3 8 6 2 4 1 15 50 30 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. Weighted networks Barrat, Vespiggnani et al PNAS (2004)

4. 10 3 8 6 2 4 1 15 50 30 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. Minimum spanning tree (MST) A B 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. 18 0 0 0 15 0 7 12 0 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. Newman., Phys. Rev. E (2001) D.-H. Kim, et al., Phys. Rev. E (2004) K.-I. Goh, et al., Phys. Rev. E (2005) Centrality of MST: How to find the importance of nodes in transport? For ER, scale free and many real world networks

7. High centrality nodes Minimum spanning tree (MST)

8. 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) Incipient percolation cluster (IIC)

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

10. sters Superhighways (SHW) and Roads

11. Mean Centrality in SHW and Roads

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

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

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. 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 Application: improve flow in the network 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 Assume: multiple sources and sinks: randomly choose n pairs of nodes as sources and other n nodes as sinks

17. sII: improve the high C links in MST. sI: improve the IIC links. Two types of transport Current flow: improve the conductance Maximum flow: improve the capacity F 0 : flow of original network. F sI : flow after using sI. F sII : flow after using sII. N=2048, =4 n=50 n=250 n=500 Application: compare two strategies current flow and maximum flow

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 N 2/3 -- a zero fraction of the the network!! Wu, Braunstein, Havlin, Stanley, PRL (2006)

19. Two strategies to improve flow, F, of the network: sI: improving the IIC links. sII: improving the high C links in MST. 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) Multiple sources and sinks: randomly choose n pairs of nodes as sources and other n nodes as sinks resistance/capacity = e ax, with a = 40 (strong disorder) [4]. Using the push-relabel algorithm by Goldberg. http://www.avglab.com/andrew/soft.html Applications: compare 2 strategies current flow and maximum flow

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

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) 4 20 7 11 3 5 2 source destination

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

25. 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 Unsolved problem: General weight distribution

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

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

29. My simulation result on 2D-lattice Strong:Weak: -0.22 L the linear size of lattice the length of optimal path Answer to questions 1 and 2: 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-R é nyi (ER) Networks Definition: A set of N nodes p For each pair of nodes, they have probability p to be connected My simulations on ER network show the same agreement with theory.

31. Distributions that are not expected to have strong disorder behavior Gaussian Exponential 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. ( the percolation threshold, constant for certain network structure) Answer to question 3:

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

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

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

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

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

38. Scaling on ER network Percolation at criticality on Erd ő s-R é nyi(ER) networks is equivalent to percolation on a lattice at the upper critical dimension. Virtual linear size Percolation exponent in ER network (N = number of nodes) ( is now depending on number of nodes in ER network)

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