Recent developments in the study of transport in random networks

1. Recent developments in the study of transport in random networks Shai CarmiBar-Ilan UniversityHavlin group Minerva meetingEilat, March 2009

2. Networks • Why do we care about networks? • Networks appear everywhere: • Communication (Internet, p2p,…) • Transportation (roads, airlines,…). • Social sciences (social networks, business relations,…) • Life sciences (gene regulation, food webs,…)

3. Transport in networks • Networks are commonly used as a platform for transport of:* Information (communication and social networks)* Passengers and commodities (transportation networks)* Current (electric circuits)* Diseases (social networks) • Quantities of interest:* Time to reach target * Maximal capacity * Number of links crossed * Congestion and avalanches* Load at each node * Diffusion coefficients

4. Network models • Most naïve model: a regular lattice.* Only good for purely spatial, local, interactions. • Erdos-Renyi (ER) network model: fully random.* Fixed number of nodes N, each link exists with probability p.* Narrow degree distribution: where k is node degree. • Scale-free (SF) networks: emergence of hubs.* Broad degree distribution: * Nodes with extremely high degree exist (hubs).* Other ingredients possible, e.g., growth, correlations.* Found to describe most real-world systems.

5. Transport models: outline • Random walk with priorities. • Random walk with trapping. • Maximum flow.

6. Transport models • Random walk with priorities. • Random walk with trapping. • Maximum flow.

7. Motivation • Some communication networks use random walkto search for other computers or spread information. • Some data packets have higher priority than others. • How does priority policy affect the diffusion in the network?

8. A B Model definition • Two species of particles, A and B with densities ρA and ρB. • A is high priority, B is low priority. • Symmetric random walk (nearest neighbors). • Protocols • B can move only after all the A’s in its site have already moved. • If motion is impossible, choose again. Site protocol: A site is randomly chosen and sends a particle. Particle protocol: A particle is randomly chosen and jumps out.

9. Solution in lattices • Write a Markov chain for the number of particles in a site. • Solve for the stationary probabilities. • Derive analytically the fraction of empty sites in both protocols. • Diffusion is normal: <R2>=Dt. • Apply the site protocol selection rule and find D for each species. • In the particle protocol define r as the fraction of free B's to total B’s.independent of ρB and approaches for large densities. • Derive diffusion coefficients similarly.

10. Solution in networks • In the particle protocol: or, the probability of a site to be empty decreases exponentially with its degree. • In scale-free networks, A’s move freely, and tend to aggregate at the hubs. • Therefore, B’s at the hubs have very low probability to escape. • Since the B’s themselves are attracted to the hubs, they eventually become trapped and their motion is arrested. • Time for a B to leave a site of degree k . • Waiting time distribution → sub-diffusion. Lattice, ER Average waiting time for B particles. Real Internet Distribution of waiting times for B particles. SF SF,ER

11. Transport models • Random walk with priorities. • Random walk with trapping. • Maximum flow.

12. Motivation • Consider again a random walk process in a network. • In a communication or a social network, a message can disappear; for example, due to failure. • How long will the message survive before being trapped?

13. Model definition • Particles initially evenly distributed over the network. • Symmetric random walk (nearest neighbors). • m of the nodes are absorbing. • Whenever the particle reaches the trap it is absorbed. • What is the survival probability ρ(t)?

14. A simple theory • Denote the total number of links entering the traps by km. • The total number of links is N<k>. • Thus, the probability per unit time of a particle to enter the trap is approximately proportional to km /N<k>. • , and the problem is reduced to evaluating km for different topologies. • In ER networks:- Approximation is good when . - Explicit dependence on both m and N. • For dense enough SF networks:-kmin is the minimum degree (one trap).

15. Results • The average time before trapping T usually scales as N. • In SF networks when one of the hubs is a trap -Only for infinite γ SF and ER networks are equivalent. • SF networks become less vulnerable as links are added. • For ER networks A=1-1/<k>. • Conclusion:A simple mean-field approach is usually useful to solve trapping problem in networks, and leads to interesting observations. Theory- linesSimulation- symbols ER

16. Transport models • Random walk with priorities. • Random walk with trapping. • Maximum flow.

17. Motivation • Users in communication networks (e.g., peer-to-peer) wish to exchange files by sending them through the network links. • How many users can exchange files without interfering with each other? • What is the maximum capacity of the network for a given number of users?

18. Model definition • Assume the network contains n sources and n sinks. • Consider three types of transport:* Maximum flow (= #of parallel paths)* Electric current* Multi-commodity flow • Non directed,non weighted(unit capacities/resistances). nSinks nSources S1 T2 Multi-commodity flow Regular flow Rest of network S2 T1

19. Theory for small n • For a single source/sink pair with degrees k1 and k2, F≈min(k1,k2). • For small n, replace k1 by the total number of links leaving the sources, and similarly for k2. • The distribution of flows is:* For ER networks:* For SF networks: • Flow per user increases with n up tothe optimal number of usersabove which the approximation is invalid. Small n theory Simulations

20. Theory for large n F1 • A different approach is needed:Find the total flow by conditioning on the number of paths of a given length. • F = F1 + F2 + F3 + … • I ≈ F1/1 + F2/2 + F3/3 + … • For direct linkage, <F1>=n2p. Implicit sum formulas for <F2>, <F3>. Sources Sinks F2 F3 ER networks Theory- linesSimulation- symbols

21. Multi-commodity flow • Flow from a source is directed towards specific sink. Thus the contribution of the different source/sink pairs can be separated. • Result for ER network: where kn is the effective degree of the network when n pairs communicate. • The network will saturate at the percolation threshold, when and thus . • In SF networks the absence of percolation threshold leads to increased capacity. Small n approximation

22. Summary • Transport in networks is important and interesting. • We introduced models inspired by problems in real networks. • We used probability theory and computer simulations to obtain analytical an numerical solutions. • Deep relations between network structure and dynamics were uncovered. For example:* Halting of low priority particles in highly connected nodes.* Effect of failure in hubs on particle survival probability.* Optimal number of users in flow network.* Influence of inter-node distance on electrical current.* Interplay between percolation theory and maximal network flow.

23. Collaborators • My advisor: Prof. Shlomo Havlin, Bar Ilan Univ., Israel. • Other collaborators: Prof. Daniel ben-Avraham (Clarkson Univ., NY, USA)Prof. Panos Argyrakis (Aristotle Univ., Thessaloniki, Greece)Prof. H. Eugene Stanley (Boston Univ., MA, USA). Shlomo Dani Panos Gene

24. Thank you for your attention! See also: • M. Maragakis, S. Carmi, D. ben-Avraham, S. Havlin, and P. Argyrakis. "Priority diffusion model in lattices and complex networks". Phys. Rev. E (RC) 77, 020103 (2008). • S. Carmi, Z. Wu, S. Havlin, and H. E. Stanley. "Transport in networks with multiple sources and sinks". Europhys. Lett. 84, 28005 (2008) . • A. Kittas, S. Carmi, S. Havlin, and P. Argyrakis. "Trapping in complex networks“. Europhys. Lett. 84, 40008 (2008).