Spectral Analysis and Optimal Synchronizability of Complex Networks

1. Spectral Analysis and Optimal Synchronizability of Complex Networks 复杂网络谱分析及最优同步性能研究 Guanrong (Ron) Chen City Univesity of Hong Kong

2. Synchronization

3. Synchrony can be essential 2009  “Clock synchronization is a critical component in the operation of wireless sensor networks, as it provides a common time frame to different nodes.” IEEE Signal Processing Magazine (2012)

5. Contents • Spectra of network Laplacian matrices • Spectra and synchronizability of some typical complex networks • Networks with best synchronizability

6. A General Dynamical Network Model A linearly and diffusively coupled network: f (.) – Lipschiz Coupling strength c > 0 A = [aij] – Adjacency matrix H– Coupling matrix function If there is a connection between node iand node j(j ≠ i), then aij= aji= 1 ; otherwise, aij= aji= 0 and aii= 0, i= 1, … , N Laplacian matrix L = D – A whereD = diag{ d1 , … , dN} For connected networks:

7. Network Synchronization Complete state synchronization: Linearized at equilibrium s: Only is important f (.) Lipschitz, or assume:

8. Network Synchronization:Criteria Master stability equation: (L.M. Pecora and T. Carroll, 1998) A (conservative) sufficient condition: Maximum Lyapunov exponent whichis a function of is called a synchronization region Synchronizing:if or if

9. Network Synchronization:Criteria Recall: Eigenvalues of orif synchronizingif Case III: Sync region Case II: Sync region Case IV: Union of intervals Case I: No sync bigger is better bigger is better

10. A Brief History Synchronizability characterized by Laplacian eigenvalues: 1. unbounded region(X.F. Wang and G.R. Chen, 2002) 2. bounded region(M. Banahona and L.M. Pecora, 2002) 3. union of several disconnected regions (A. Stefanski, P. Perlikowski, and T. Kapitaniak, 2007) (Z.S. Duan, C. Liu, G.R. Chen, and L. Huang, 2007 - 2009)

11. Spectra of Networks Some theoretical results Relation with network topology Role in network synchronizability Spectrum

12. Theoretical Bounds of Laplacian Eigenvalues - maximum,minimum, average degree D– Diameter of the graph • Graph Theory Textbooks • F.M. Atay, T. Biyikoglu, J. Jost, Network synchronization: Spectral versus statistical properties, Physica D, 224 (2006) 35–41.

13. Theoretical Bounds of Laplacian Eigenvalues (both in increasing order) Distribution: (interlacing property) For any node degree there exists a such that C. J. Zhan, G. Chen and L. F. Yeung, Physica A (2010)

14. Lemma(Hoffman-Wielanelt, 1953) For matrices B – C = A: Frobenius Norm  For Laplacian L = D – A:  Cauchy inequality

15. Difference between Eigenvalue and Degree Sequences ER random-graph networks N=2500, average of 2000 runs BA scale-free networks N=2500, average of 2000 runs Above: relative variances below: relative errors C. J. Zhan, G. Chen and L. F. Yeung, Physica A (2010)

16. Upper Bounds for Largest Eigenvalues - for any graph - average degree of all neighbors of nodeu - total number of common neighbors of u and v W.N. Anderson, T.D. Morley, Eigenvalues of the Laplucian of a graph, Linear and Multilinear Algebra, 18 (1985) 141-145. R. Merris, A note on Laplacian graph eigenvalues, Linear Algebra and its Applications, 285 (1998) 33-35. O. Rojo, R. Sojo, H. Rojo, An always nontrivial upper bound for Laplacian graph eigenvalues, Linear Algebra and its Applications, 312 (2000) 155-159.

17. Lower Bounds for Smallest Eigenvalues e– edge connectivity (number of edges whose removal will disconnect the graph e.g., for a chain, e = 1 ; for a triangle,e = 2) v – node connectivity D – diameter

18. Spectra of Typical Complex Networks Regularly-connected networks Complete graph, ring, chain, star … Random-graph networks For every pair of nodes, connect them with a certain probability. Poisson node distribution Small-world networks Similar to random graph, but with short average distance and large clustering, with near-Poisson node distribution Scale-free networks Heterogeneous, with power-law degree distribution Piet Van Mieghem, Graph Spectra for Complex Networks (2011)

19. Regularly-Connected Networks • Complete graphs: • 2K-rings: K=1: N - even: K≠1: • Chains: • Stars:

20. Rectangular Random Networks E. Estrada and G. Chen (2015) RRN:N nodes are randomly uniformly and independently distributed in a unit rectangle (It can be generalized to higher-dimensional setting) Two nodes are connected by an edge if they are inside a ball of radius r > 0 Example: N =250 a = 1 r = 0.15

21. Theorem: For RRN, eigenvalues are bounded by Lower bound The worst case: all nodes are located on the diagonal and E. Estrada and G. Chen (2015)

22. Upper bound Lemma 1: Diameter D= diagonal length / r  Lemma 2: Based on a result of Alon-Milman (1985)  E. Estrada and G. Chen (2015)

23. Spectral Role in Network Synchronization

24. Topology affectsEigenvalues affects Sync S - any subgraph, with Implication:small local changesaffecteigenratio, hence network synchronizability, which however do not affect much the network statistical properties in general: degree distribution, clustering, average distance, … • Recall: (bigger is better) Statistical properties depend on H, yet depends on S

25. Statistics Determines Synchronizability? Answer - Not necessarily Example: Graph Graph • They have same structural characteristics: Graphs and have same degree sequence {3,3,3,3,3,3} • So, all nodes have degree 3 ; same average distance 7/5 ; and same node betweenness 2 ; same …… Z. S. Duan, G. R. Chen and L. Huang, Phys. Rev. E, 2007

26. But they have different synchronizabilities: Eigenvalues of are and Eigenvalues of are and Eigenratio satisfies: G2 G1

27. Topology Determines Synchronizability? Answer:Not necessarily • Lemma 1: For any given connected undirected graph G , all its nonzero eigenvalues will not decrease with the number of added edges; namely,by adding any edge e, one has • Note: Z. S. Duan, G. R. Chen and L. Huang, Phys. Rev. E, 2007

28. What Topology  Good Synchronizability? Example Given Laplacian: Q: How to replace 0 and -1 while keeping the connectivity (and all row-sums = 0), such that = maximum ?

29. Answer: = maximum Observation:Homogeneous + Symmetrical Topology

30. Problem • With the same numbers of node and edges, while keeping the connectivity,what kind of network has the best possible synchronizability? • Computationally, this is NP-hard • Objective:Find analytic solutions Such that “row-sum = 0” and (Laplacian) (connected)

31. Progress • Nishikawa et al., Phys. Rev. Lett. 91, 014101(2003), regular networks with uniform small (node or edge) betweenness [we found: edge betweenness is more important than node betweenness] • Donetti et al., Phys. Rev. Lett. 95, 188701(2005) Entangled networks have biggest eigenratios [we found: not necessarily] • Donetti et al., J. Stat. Mech.: Theory and Experiment 8, 1742(2006), algorithm based on algebraic graph theory; big spectral gap  big eigenratio [we found: the opposite] • Zhou et al., Eur. Phys. J. B. 60, 89(2007), algorithm based on smallest clustering coefficient • Hui, Ann Oper. Res., July (2009), algorithm based on entropy • Xuan et al., Physica A 388, 1257(2009), algorithm based on short average path length • Mishkovski et al., ISCAS, 681(2010), fast generating algorithm • Shi, Chen, Thong, Yan., IEEE CAS Magazine, 13, 1(2013), 3-regular optimal graphs

32. Our Approach • Homogeneity + Symmetry • Same node degree • Minimize average path length • Minimize path-sum • Maximize girth D. Shi, G. Chen, W. W. K. Thong and X. Yan, IEEE Circ. Syst. Magazine, (2013) 13(1) 66-75

33. Non-Convex Optimization Illustration: Grey：networks with same numbers of nodes and edges Green：degree-homogeneous networks Blue：networks with maximum girths Pink：possible optimal networks Red：near homogenous networks White：Optimal solution location

34. Optimal 3-Regular Networks

35. Optimal 3-Regular Networks

36. Summary Optimal network topology (in the sense of having best possible synchronizability)should have • Homogeneity • Symmetry • Shortest average path-length • Shortest path-sum • Longest girth • Anything else ?

37. Open Problem Looking for optimal solutions: Given: Move which -1 can maximize ? Where to add -1 can maximize ? Where to delete -1 …………………… can maximize ? And so on …… ???

38. Thank You ! Homogeneity Symmetry

39. References [1] F.Chung, Spectral Graph Theory, AMS (1992, 1997) [2] T.Nishikawa, A.E.Motter, Y.-C.Lai, F.C.Hoppensteadt, Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize? PRL 91 (2003) 014101 [3] H.Hong, B.J.Kim, M.Y.Choi, H.Park, Factors that predict better synchronizability on complex networks, PRE 65 (2002) 067105 [4] M.diBernardo, F.Garofalo, F.Sorrentino, Effects of degree correlation on the synchronization of networks of oscillators, IJBC 17 (2007) 3499-3506 [5] M.Barahona, L.M.Pecora, Synchronization in small-world systems, PRL 89 (5) (2002)054101 [6] H.Hong, M.Y.Choi, B.J.Kim, Synchronization on small-world networks, PRE 69 (2004) 026139 [7] F.M. Atay, T.Biyikoglu, J.Jost, Network synchronization: Spectral versus statistical properties, Physica D 224 (2006) 35–41 [8] A.Arenas, A.Diaz-Guilera, C.J.Perez-Vicente, Synchronization reveals topological scales in complex networks, PRL 96(11) (2006 )114102 [9] A.Arenas, A.Diaz-Guilerab, C.J.Perez-Vicente, Synchronization processes in complex networks, Physica D 224 (2006) 27–34 [10] B.Mohar, Graph Laplacians, in: Topics in Algebraic Graph Theory, Cambridge University Press (2004) 113–136 [11] F.M.Atay, T.Biyikoglu, Graph operations and synchronization of complex networks, PRE 72 (2005) 016217 [12] T.Nishikawa, A.E.Motter, Maximum performance at minimum cost in network synchronization, Physica D 224 (2006) 77–89

40. [13] J.Gomez-Gardenes, Y.Moreno, A.Arenas, Paths to synchronization on complex networks, PRL 98 (2007) 034101 [14] C.J.Zhan, G.R.Chen, L.F.Yeung, On the distributions of Laplacian eigenvalues versus node degrees in complex networks, Physica A 389 (2010) 17791788 [15] T.G.Lewis, Network Science –Theory and Applications, Wiley 2009 [16] T.M.Fiedler, Algebraic Connectivity of Graphs, Czech Math J 23 (1973) 298-305 [17] W.N.Anderson, T.D.Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra 18 (1985) 141-145 [18] R.Merris, A note on Laplacian graph eigenvalues, Linear Algebra and its Applications 285 (1998) 33-35 [19] O.Rojo, R.Sojo, H.Rojo, An always nontrivial upper bound for Laplacian graph eigenvalues, Linear Algebra and its Applications 312 (2000) 155-159 [20] G.Chen, Z.Duan, Network synchronizability analysis: A graph-theoretic approach, CHAOS 18 (2008) 037102 [21] T.Nishikawa, A.E.Motter, Network synchronization landscape reveals compensatory structures, quantization, and the positive effects of negative interactions, PNAS 8 (2010)10342 [22] P.V.Mieghem, Graph Spectra for Complex Networks (2011) [23] D. Shi, G. Chen, W. W. K. Thong and X. Yan, Searching for optimal network topology with best possible synchronizability, IEEE Circ. Syst. Magazine, (2013) 13(1) 66-75

41. Acknowledgements Prof Ernesto Estrada, University of Strathclyde, UK Prof Xiaofan Wang, Shanghai Jiao Tong University Prof Zhi-sheng Duan, Peking University Prof Jun-an Lu, Wuhan University Prof Dinghua Shi, Shanghai University Dr Juan Chen, Wuhan University Dr Choujun Zhan, City University of Hong Kong Dr Wilson W K Thong, City University of Hong Kong Dr Xiaoyong Yan, Beijing Normal University