Talk

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Spectral Analysis and Optimal Synchronizability
of Complex Networks ????????????????
Guanrong (Ron) Chen City Univesity of Hong Kong
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Synchronization
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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)
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Contents

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

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A General Dynamical Network Model
A linearly and diffusively coupled network
f (.) Lipschiz Coupling strength c gt 0
A aij Adjacency matrix H Coupling
matrix function
If there is a connection between node i and node
j (j ? i), then aij aji 1 otherwise, aij
aji 0 and aii 0, i 1, , N
Laplacian matrix L D A where D diag d1
, , dN
For connected networks
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Network Synchronization
Complete state synchronization
Linearized at equilibrium s
Only is important
f (.) Lipschitz, or assume
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Network Synchronization Criteria
Master stability equation (L.M. Pecora and T.
Carroll, 1998)
A (conservative) sufficient condition
Maximum Lyapunov exponent which is
a function of
is called a synchronization region
Synchronizing if
or if
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Network Synchronization Criteria
Recall Eigenvalues of
or if
synchronizing if
Case III Sync region
Case II Sync region
Case IV Union of intervals
Case I No sync
bigger is better bigger is better
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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)

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Spectra of Networks

• Some theoretical results
• Relation with network topology
• Role in network synchronizability

Spectrum
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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) 3541.

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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)
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Lemma (Hoffman-Wielanelt, 1953) For
matrices B C A
Frobenius Norm
? For Laplacian L D A
?
Cauchy inequality
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Difference between Eigenvalue and Degree Sequences
BA scale-free networks N2500, average of 2000
runs

• ER random-graph networks
• N2500, average of 2000 runs

Above relative variances below relative errors
C. J. Zhan, G. Chen and L. F. Yeung, Physica A
(2010)
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Upper Bounds for Largest Eigenvalues
- for any graph
- 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) 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.
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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
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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)
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Regularly-Connected Networks

• Complete graphs
• 2K-rings
• K1
• N - even
• K?1
• Chains
• Stars

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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 gt 0
Example N 250 a 1 r 0.15
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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)
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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)
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Spectral Role in Network Synchronization
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Topology affects Eigenvalues affects Sync
S - any subgraph, with

• Recall (bigger is better)
  

• 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,

Statistical properties depend on H, yet depends
on S
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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
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G2
G1

• But they have different synchronizabilities
• Eigenvalues of are and
  Eigenvalues of are and
• Eigenratio satisfies

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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
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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 ?
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Answer
? maximum
Observation Homogeneous Symmetrical Topology
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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)
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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

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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
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Non-Convex Optimization

• Illustration
• Greynetworks with same numbers of nodes and
  edges
• Greendegree-homogeneous networks
• Bluenetworks with maximum girths
• Pinkpossible optimal networks
• Rednear homogenous networks

WhiteOptimal solution location
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Optimal 3-Regular Networks
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Optimal 3-Regular Networks
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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 ?

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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 ???
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Thank You !
Homogeneity
Symmetry
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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) 3541
• 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) 2734
• 10 B.Mohar, Graph Laplacians, in Topics in
  Algebraic Graph Theory, Cambridge University
  Press (2004) 113136
• 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) 7789

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• 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

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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
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