Networks: Condensation and Coevolving Dynamics

1. Condensation in/of Networks Jae Dong Noh NSPCS08, 1-4 July, 2008, KIAS

2. Getting wired • Moving and Interacting • Being rewired

3. References • Random walks • Noh and Rieger, PRL92, 118701 (2004). • Noh and Kim, JKPS48, S202 (2006). • Zero-range processes • Noh, Shim, and Lee, PRL94, 198701 (2005). • Noh, PRE72, 056123 (2005). • Noh, JKPS50, 327 (2007). • Coevolving networks • Kim and Noh, PRL100, 118702 (2008). • Kim and Noh, in preparation (2008).

4. Networks

5. Basic Concepts • Network = {nodes} [ {links} • Adjacency matrix A • Degree of a node i : • Degree distribution • Scale-free networks :

6. Random Walks

7. 1/5 1/5 1/5 1/5 1/5 Definition • Random motions of a particle along links • Random spreading

8. Stationary State Property • Detailed balance : • Stationary state probability distribution

9. SF networks w/o loops SF networks with many loops Relaxation Dynamics • Return probability

10. Mean First Passage Time • MFPT

11. Zero Range Process

12. Model • Interacting particle system on networks • Each site may be occupied by multiple particles • Dynamics : At each node i , • A single particle jumps out of i at the rate ui (ni ), • and hops to a neighboring node j selected randomly with the probability Wji .

13. transport capacity particle interactions Jumping rate ui (n) Hopping probability Wji • depends only on the occupation number at the departing site. • may be different for different sites (quenched disorder) independent of the occupation numbers at the departing and arriving sites Model Note that [ZRP with M=1 particle] = [ single random walker] [ZRP with u(n) = n ] = [ M indep. random walkers]

14. Stationary State Property [M.R. Evans, Braz. J. Phys. 30, 42 (2000)] • Stationary state probability distribution : product state • PDF at node i : where e.g.,

15. Condensation in ZRP • Condensation : single (multiple) node(s) is (are) occupied by a macroscopic number of particles • Condition for the condensation in lattices • Quenched disorder (e.g., uimp. = <1, ui≠imp. = 1) • On-site attractive interaction : if the jumping rate function ui(n) = u(n) decays ‘faster’ than ~(1+2/n) e.g.,

16. ZRP on SF Networks • Scale-free networks • Jumping rate • (δ>1) : repulsion • (δ=1) : non-interacting • (δ<1) : attraction • Hopping probability : random walks

17. Condensation on SF Networks • Stationary state probability distribution • Mean occupation number

18. normal phase transition line condensed phase Phase Diagram Complete condensation

19. Coevolving Networks

20. Synaptic Plasticity • In neural networks • Bio-chemical signal transmission from neural to neural through synapses • Synaptic coupling strength may be enhanced (LTP) or suppressed (LTD) depending on synaptic activities • Network evolution

21. 2 3 1 2 4 5 3 4 Co-evolving Network Model • Weighted undirected network + diffusing particles • Particles dynamics : random diffusion • Weight dynamics [LTP] • Link dynamics [LTD]: With probability 1/we, each link e is removed and replaced by a new one

22. Dynamic Instability • Due to statistical fluctuations, a node ‘hub’ may have a higher degree than others • Particles tend to visit the ‘hub’ more frequently • Links attached to the ‘hub’ become more robust, hence the hub collects more links than other nodes • Positive feedback  dynamic instability toward the formation of hubs

23. dynamic instability [N=1000, <k>=4] linear growth sub-linear growth Numerical Data for kmax dynamic phase transition

24. Poissonian + Poissonian + Isolated hubs Poissonian + Fat-tailed   Degree Distribution high density low density

25. Analytic Theory • Separation of time scales • particle dynamics : short time scale • network dynamics : long time scale • Integrating out the degrees of freedom of particles • Effective network dynamics : Non-Markovian queueing (balls-in-boxes) process

26. 1 K i  2 queue Non-Markovian Queueing Process • node i $ queue (box) • edge $ packet (ball) • degree k $ queue size K

27. queue Non-Markovian Queueing Process • Weight of a ball  • A ball  leaves a queue with the probability

28. Outgoing Particle Flux ~ uZRP(K) • Upper bound for fout(K,)

29. queue is trapped at K=K1 for • instability time t =  • - queue grows linearly after t >  Dynamic Phase Transition

30. Phase Diagram ballistic growth of hub sub-linear growth of hub

31. 2 3 1 2 4 5 3 4 A Variant Model • Weighted undirected network + diffusing particles • Particles dynamics : random diffusion • Weight dynamics • Link dynamics : Rewiring with probability 1/we • Weight regularization :

32. Rate equations for K and w 1 K i potential candidate for the hub  2 A Simplified Theory

33. no hub hub no condensation condensation Flow Diagram

34. Numerical Data

35. Summary • Dynamical systems on networks • random walks • zero range process • Coevolving network models • Network heterogeneity $ Condensation