1. Transport and Percolation in Complex Networks Guanliang Li Advisor: H. Eugene Stanley Collaborators: Shlomo Havlin, Lidia A. Braunstein, Sergey V. Buldyrev and José S. Andrade Jr.

2. Outline Part I: Towards design principles for optimal transport networks G. Li, S. D. S. Reis, A. A. Moreira, S. Havlin, H. E. Stanley and J. S. Andrade, Jr., PRL 104, 018701 (2010); PRE (submitted) Motivation: e.g. Improving the transport of New York subway system Questions: How to add the new lines? How to design optimal transport network? Part II: Percolation on spatially constrained networks D. Li, G. Li, K. Kosmidis, H. E. Stanley, A. Bunde and S. Havlin, EPL 93, 68004 (2011) Motivation: Understanding the structure and robustness of spatially constrained networks Questions: What are the percolation properties? Such as thresholds... What are the dimensions in percolation? How are spatial constraints reflected in percolation properties of networks?

3. Grid network L ~ L D. J. Watts and S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998). How to add the long-range links? Shortest path length AB =6 A B small-world network ~log L Randomly add some long-range links: AB =3 Part I: Towards design principles for optimal transport networks G. Li, S. D. S. Reis, A. A. Moreira, S. Havlin, H. E. Stanley and J. S. Andrade, Jr., PRL 104, 018701 (2010); PRE (submitted)

4. Kleinberg model of social interactions J. Kleinberg, Navigation in a Small World, Nature 406, 845 (2000). Rich in short-range connectionsA few long-range connections Part I How to add the long-range links?

5. Kleinberg model continued P(r ij ) ~ r ij - where 0, and r ij is lattice distance between node i and j Q: Which gives minimal average shortest distance ? Part I Steps to create the network: Randomly select a node i Generate r ij from P(r ij ), e.g. r ij = 2 Randomly select node j from those 8 nodes on dashed box Connect i and j

6. Optimal in Kleinbergs model Minimal occurs at =0 d is the dimension of the lattice Without considering the cost of links Part I

7. Considering the cost of links Each link has a cost length r (e.g. airlines, subway) Have budget to add long-range links (i.e. total cost is usually system size N ) Trade-off between the number N l and length of added long-range links From P(r) ~ r - : =0, r is large, N l = / r is small large, r is small, N l is large Q: Which gives minimal with cost constraint? Part I

8. With cost constraint is the shortest path length from each node to the central node Part I Minimal occurs at =3

9. vs. L (Same data on log-log plots) For 3, ~ L For =3, ~ (ln L) Part I With cost constraint Conclusion:

10. Different lattice dimensions Optimal occurs at =d+1 (Total cost ~ N) when N Part I With cost constraint

11. Conclusion, part I For regular lattices, d=1, 2 and 3, optimal transport occurs at =d+1 More work can be found in thesis (chapter 3): 1.Extended to fractals, optimal transport occurs at = d f +1 2.Analytical approach Part I Empirical evidence 1. Brain network: L. K. Gallos, H. A. Makse and M. Sigman, PNAS, 109, 2825 (2011). d f =2.1±0.1, link length distribution obeys P(r) ~ r - with =3.1±0.1 2. Airport network: G. Bianconi, P. Pin and M. Marsili, PNAS, 106, 11433 (2009). d=2, distance distribution obeys P(r) ~ r - with =3.0±0.2

12. Part II: Percolation on spatially constrained networks P(r) ~ r - controls the strength of spatial constraint =0, no spatial constraint ER network large, strong spatial constraint Regular lattice Questions: What are the percolation properties? Such as the critical thresholds, etc. What are the fractal dimensions of the embedded network in percolation? D. Li, G. Li, K. Kosmidis, H. E. Stanley, A. Bunde and S. Havlin, EPL 93, 68004 (2011)

13. The embedded network Start from an empty lattice Add long-range connections with P(r) ~ r - until k ~ 4 Two special cases: =0 Erdős-Rényi(ER) network large Regular 2D lattice Part II

14. Percolation process Remove a fraction q of nodes, a giant component (red) exists Start randomly removing nodes Increase q, giant component breaks into small clusters when q exceeds a threshold q c, with p c =1-q c nodes remained Part II small clustersgiant component exists pcpc p

15. Giant component in percolation =1.5 =2.5 =3.5 =4.5 Part II

16. Result 1: Critical threshold p c 0.33 1.52.02.53.03.54.05 pcpc 0.25 0.270.330.410.490.57 ER (Mean field) p c =1/ k =0.25 Intermediate regimeLattice p c 0.59 Part II

17. Result 2: Size of giant component: M M ~ N in percolation : critical exponent 1.52.02.53.03.54.05 0.67 0.700.760.870.930.94 ER (Mean field) =2/3 Intermediate regimeLattice =0.95 Part II =0.76

18. Result 3: Dimensions Examples: ER: d f, d e, but =2/3 2D lattice: d f =1.89, d e =2, =0.95 So we compare with d f /d e Percolation ( p=p c ): M ~ R d f Embedded network ( p=1 ): N ~ R d e M ~ N d f / d e = d f /d e M ~ N in percolation Part II

19. compare with d f /d e In percolationEmbedded network 1.52.02.53.03.54.05 dfdf 3.922.121.921.891.87 dede 5.652.762.182.001.99 d f /d e 0.690.760.880.94 0.700.760.870.930.94 Part II N

20. Conclusion, part II: Three regimes d, ER (Mean Field) > 2d, Regular lattice Intermediate regime, depending on Part II Transport properties still show three regimes. K. Kosmidis, S. Havlin and A. Bunde, EPL 82, 48005 (2008)

21. Summary For cost constrained networks, optimal transport occurs at =d+1 (regular lattices) or d f +1 (fractals) The structure of spatial constrained networks shows three regimes: 1. d, ER (Mean Field) 2.d < 2d, Intermediate regimes, percolation properties depend on 3. > 2d, Regular lattice