1. 1 Synchronization in large networks of coupled heterogeneous oscillators Edward Ott University of Maryland

2. 2 Cellular clocks in the brain. Pacemaker cells in the heart. Pedestrians on a bridge. Electric circuits. Laser arrays. Oscillating chemical reactions. Examples of synchronized oscillators

3. 3 Male fireflies flashing in unison

4. 4 Cellular clocks in the brain (day-night cycle). Yamaguchi et al., Science, vol.302, p.1408 (2003). Incoherent Coherent

5. 5 Synchrony in the brain

6. 6 Coupled phase oscillators  Change of variables Limit cycle in phase space Many such ‘phase oscillators’: Couple them: Assumption: Attraction to limit cycle attractor is ‘strong’. Kuramoto 1975: ; i=1,2,…,N »1 Global coupling

7. 7 Framework N oscillators described only by their phase . N is very large. g(  )  The oscillator frequencies are randomly chosen from a distribution g(  ) with a single local maximum. (We assume the mean frequency is zero) nn nn

8. 8 Kuramoto model: all-to-all coupling Assumes sinusoidal all-to-all coupling. Macroscopic coherence of the system is characterized by n = 1, 2, …., N k = (coupling constant) = “order parameter”

9. 9 Order parameter measures coherence

10. 10 Results for the Kuramoto model There is a transition to synchrony at a critical value of the coupling constant. r Incoherence Synchronization g(0) g(  ) 

11. 11 Example: Order parameter in the incoherent case k

12. 12 Example: Order parameter in the coherent case k

13. 13 Derivation of k c “The order parameter”

14. 14 N→∞N→∞ Introduce the distribution function F( ,t) [the fraction of oscillators with phases in the range (  +d  ) and freqs. in the range (  +d  ) ] Conservation of number of oscillators: 0

15. 15 Incoherent solution Q.Is it stable? A.Yes, for k k c. (uniform distribution in angle) Stability analysis perturbation 0 0

16. 16 Dispersion relation Look for a solution of the form Get

17. 17 Example: Lorentzian g(  ) 3 poles: Close integral in upper half plane: < > < > -i  +i  -i Re(  ) Im(  ) k r kckc  g(  ) 

18. 18 Models of coupled heterogeneous oscillators Kuramoto model All-to-all Network. Sine coupled phase oscillators Ott et al ‘02; Pikovsky et al ‘96 Baek et al ‘04; Topaj et al ‘02 All-to-all Network. More general dynamics. Ichinomiya, Phys. Rev. E ‘04; ‘05 Restrepo et al., Phys. Rev. E ‘05; Chaos 06 More general network. More general Network. More general dynamics. Restrepo et al., PRL’06; Physica D’06 Sine coupled phase oscillators

19. 19 In recent years it has been realized that many processes in nature can be described in terms of interaction of elements in networks. Nodes Links d n = Degree of a node = number of links Why networks? 1 6 3 4 2 5 d 2 = 4 d 6 = 2 n = 1,2,…,N

20. 20 Real world networks are often complex Map of the Internet http://www.caida.org/Papers/Nae/

21. 21 In order to study the effect of a network on the emergence of synchronization, we will maintain the phase dynamics, but will introduce a network in the problem. All-to-all Network. More general Network. What follows is based on Juan G. Restrepo, Ed Ott, Brian R. Hunt, Physical Review E, 71 036151. Juan G. Restrepo, Ed Ott, Brian R. Hunt, Chaos, 2006. Sine coupled phase oscillators

22. 22 color phase < A visual example

23. 23 Kuramoto model on a network The network is introduced by means of a matrix A m is not connected to n A nm = 0. PDF of frequencies symmetric about 0. The nonzero elements of A can have any positive or negative value and correspond to the interaction strength at each link.

24. 24 Global order parameter: where is the node degree: PROBLEM: Find r vrs. k Order Parameter Description Local order parameter for node n where = time average.

25. 25 Order Parameter Form of the Dynamics and yield where.

26. 26 Neglect of h n Assumption: The degree is large. There are many terms in the determining. is noise-like. Neglect h n compared to O(d n ) The effect of finite h n is treated in Restrepo, Ott and Hunt Phys. Rev. E (2005), Sec 6.

27. 27 Time Averaged Approximation Locked nodes: Putting this in the equation for, using and, gives the time averaged approximation. (Restrepo, Ott, Hunt, PRE 2005) ;

28. 28 Frequency distribution approximation Assuming r n is independent of  n we can average over the frequencies to get Knowledge of each individual frequency is not needed. 0 The onset of synchronization corresponds to r n  0 +. We obtain an eigenvalue equation! g(0) g(  ) 

29. 29 Mean field approximation Assuming r n = r d n, we get the mean field approximation ( Ichinomiya, 2004 ): Near the transition, it has the extra assumption that the eigenvector u of A with the largest eigenvalue satisfies u n  d n. Only knowledge of the degree and frequency distributions is required.

30. 30 Results For a large class of networks, there is still a transition to synchrony at a critical coupling strength k C. We have developed several approximations to the order parameter past the transition. The critical coupling strength is given by is the largest eigenvalue of the matrix A defining the network. NetworkAll-to-all: Kuramoto model g(0) g(  ) 

31. 31 Generation of networks to test our theory We consider network with degree dist., p(d). 1.We first specify the degree d n of each node. 2.We imagine d n spokes stick out of node n. 3.We randomly connect pairs of spokes, avoiding self and double connections. 3 3 2 1 1 3 4 1

32. 32 Example: scale-free networks Simulation Time averaged theory Frequency distribution approximation Mean field theory r 2 12 We prescribe a degree distribution of the form p(d)  d - , d  50, and N = 2000.

33. 33 Simulation Time averaged theory Frequency distribution approximation Mean field theory Example: scale-free networks r2r2 1 2

34. 34 Some implications Heterogeneity in the degree distribution:  For a given number of connections, a heterogeneous network tends to synchronize easier (smaller k C ). Green network Red network

35. 35 Some implications Randomness in the interaction strengths:  For a given average interaction strength, a network with higher randomness tends to synchronize more easily. Green network Red network 3 3 3 3 3 3 3 3 4 2 1 3 3 5 3 3

36. 36 Degree-degree correlations: Highly connected core Disperse hubs Some implications  For the same degree distribution, a highly connected core tends to synchronize more easily.

37. 37 Further work Kuramoto model (Kuramoto, 1984) All-to-all Network. Coupled phase oscillators (simple dynamics). Ott et al.,02; Pikovsky et al.96 Baek et al.,04; Topaj et al.01 All-to-all Network. More general dynamics. Ichinomiya, Phys. Rev. E ‘04 Restrepo et al., Phys. Rev E ‘04; Chaos‘06 More general network. Coupled phase oscillators. More general Network. More general dynamics. Restrepo et al. Physica D ‘06

38. 38 Networks with general node dynamics Restrepo, Hunt, Ott, PRL ‘06; Physica D ‘06 Uncoupled node dynamics: Could be periodic or chaotic. Kuramoto is a special case: Main result: Q: depends on the collection of node dynamical behaviors (not on network topology).  : Max. eigenvalue of A; depends on network topology (not on node dynamics).

39. 39

40. 40 “Approximating the largest eigenvalue of network adjacency matrices” Reference: Paper with the above title: Restrepo, Ott, Hunt, Phys. Rev. E 76, 056119 (2007). Markovian theory: P(d´ in,d´ out | d in,d out ) = prob. a random out-link from a node with (d in,d out ) connects to a node with (d´ in, d´ out ).  = assortivity coefficient high in-degree nodehigh out-degree node low in-degree node low out-degree node  > 1 : Assortitive  < 1 : Disassortitive

41. 41 Test of Markovian theory Randomly generated A’s with N=25,000, = 20. P(d)  d -2.5 Dashed line: expansion about ρ=1. Solid black line: actual eigenvalue. Squares: Markovian approximation. P(d in,d out )  (d in d out ) -2.5

42. 42 Summary For a large class of networks, there is a transition to synchrony at a critical coupling constant determined by the maximum eigenvalue of the adjacency matrix. A larger maximum eigenvalue of the adjacency matrix favors a lower threshold for synchronization. Heterogeneity in the degree distribution, randomness in the couplings, and positive degree correlations favors synchronization. Our papers can be obtained from: http://www.math.umd.edu/~juanga/umdsyncnets.htm

43. 43 We will focus on the synchronization of coupled heterogeneous oscillators. What is the effect of complex interaction structure on dynamical processes taking place in networks?

44. 44 Effect of the nodes with small degree So far we have been using the average value of r n. However, Finite degree d n Fluctuations What is the effect of the fluctuations?

45. 45 Perturbations from the incoherent state The incoherent state is given by We introduce perturbations and assume. Onset of synchronization

46. 46 Time fluctuations as noise The original equations are If the degree d n is large but finite and the system is incoherent, the coupling term can be approximated by a Gaussian random noise with diffusion coefficient D n. Smaller degree d n Larger diffusion coefficient D n

47. 47 Theoretical results We obtain the eigenvalue equation If D n = 0, letting we get Positive D n corresponds to a smaller growth rate and thus to a larger critical coupling strength.

48. 48 d = 20 d = 50 d = 20 d = 50 d = 100 We plot the order parameter r 2 found numerically and the growth rate s found from our theory for three networks in which all nodes have degree 100, 50 and 20. (N = 500) Comparison k/k c r2r2 s

49. 49 AN EXAMPLE 5000 Lorenz systems: Network: N=5000; References: Restrepo, Ott, Hunt, Physica D (2006) [Ott, So, Barreto, Antonsen, Physica D(2002)]