Spectral Analysis and Optimal Synchronizability of Complex Networks 复杂网络谱分析及最优同步性能研究 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

Synchronization

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

 Spectra of network Laplacian matrices  Spectra and synchronizability of some typical complex networks  Networks with best synchronizabilityContentsContents

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

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

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

Recall: Eigenvalues of synchronizing if or if Case III: Sync region Case II: Sync region Case I: No sync bigger is better Case IV: Union of intervals Network Synchronization: Network Synchronization: Criteria

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, ) A Brief History

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

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

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

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

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

Upper Bounds for Largest Eigenvalues - average degree of all neighbors of node u - 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) R. Merris, A note on Laplacian graph eigenvalues, Linear Algebra and its Applications, 285 (1998) O. Rojo, R. Sojo, H. Rojo, An always nontrivial upper bound for Laplacian graph eigenvalues, Linear Algebra and its Applications, 312 (2000) for any graph

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

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 Spectra of Typical Complex Networks Piet Van Mieghem, Graph Spectra for Complex Networks (2011)

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

Rectangular Random Networks 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 E. Estrada and G. Chen (2015)

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) Upper bound Lemma 1: Diameter D = diagonal length / r  Lemma 2: Based on a result of Alon-Milman (1985) 

Spectral Role in Network Synchronization

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

Statistics Determines Synchronizability? Answer - Not necessarily Example: 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 …… Graph Z. S. Duan, G. R. Chen and L. Huang, Phys. Rev. E, 2007

But they have different synchronizabilities:Eigenvalues of are and Eigenratio satisfies: G1G1 G2G2

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 Topology Determines Synchronizability ?

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

Answer:  = maximum Observation: Homogeneous + Symmetrical Topology

 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)ProblemProblem

Nishikawa et al., Phys. Rev. Lett. 91, (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, (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 graphsProgressProgress

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

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 Non-Convex Optimization

Optimal 3-Regular Networks

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

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 …… ???

Thank You ! Homogeneity Symmetry

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[13] J.Gomez-Gardenes, Y.Moreno, A.Arenas, Paths to synchronization on complex networks, PRL 98 (2007) [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) [15] T.G.Lewis, Network Science –Theory and Applications, Wiley 2009 [16] T.M.Fiedler, Algebraic Connectivity of Graphs, Czech Math J 23 (1973) [17] W.N.Anderson, T.D.Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra 18 (1985) [18] R.Merris, A note on Laplacian graph eigenvalues, Linear Algebra and its Applications 285 (1998) [19] O.Rojo, R.Sojo, H.Rojo, An always nontrivial upper bound for Laplacian graph eigenvalues, Linear Algebra and its Applications 312 (2000) [20] G.Chen, Z.Duan, Network synchronizability analysis: A graph-theoretic approach, CHAOS 18 (2008) [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

Acknowledgements 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