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
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)
Contents
 Spectra of network Laplacian matrices
 Spectra and synchronizability of some
typical complex networks
 Networks with best synchronizability
A General Dynamical Network Model
A linearly and diffusively coupled network:
N
xi  f ( xi )  c aij H ( x j )
j 1
f (.) – Lipschiz
xi  R n
i  1,2,..., N
Coupling strength c > 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: 0  1  2    N
Network Synchronization
N
xi  f ( xi )  c aij H ( x j )
xi  R n
i  1,2,..., N
j 1
Complete state
synchronization:
lim || xi (t )  x j (t ) || 2  0,
t 
i, j  1,2,  , N
Linearized at equilibrium s :
N


xi  [ Df ( xi )] xi  s  c aij [ DH ]xi  s  xi
j 1


f (.) Lipschitz, or assume:
|| f ( s) ||  M
Only
c[aij ] or {ci }
is important
Network Synchronization: Criteria
N


xi  [ Df ( xi )]s  c aij [ DH ]s  xi
j 1


0  1  2    N
Master stability equation: (L.M. Pecora and T. Carroll, 1998)
yi  [[ Df ]s   [ DH ]s ] yi ,
 : c2
A (conservative) sufficient condition:
Maximum Lyapunov exponent Lmax  0 which is a function of
Lmax
S max is called a synchronization region
Synchronizing:
if
0  1    
or if
0   2    3  

Network Synchronization: Criteria
Recall: Eigenvalues of L  D  A
synchronizing if
Case I:
No sync
0  1  2    N
0  1  c2   or
if
0  2 
Case II:
Sync region
Case III:
Sync region
S1  (1 , )
S2  ( 2 ,  3 )
2 bigger is better
2
bigger is better
N
2
 3
N
Case IV:
Union of
intervals
A Brief History
Synchronizability characterized by Laplacian eigenvalues:
1. unbounded region (X.F. Wang and G.R. Chen, 2002)
2 ,
0  1  2    N
S1  (1 , )
2. bounded region (M. Banahona and L.M. Pecora, 2002)
2 / N , 0  1  2    N
S2  ( 2 ,  3 )
3. union of several disconnected regions
S m  (1 ,  2 )  ( 3 ,  4 )    ( m , )
(A. Stefanski, P. Perlikowski, and T. Kapitaniak, 2007)
(Z.S. Duan, C. Liu, G.R. Chen, and L. Huang, 2007 - 2009)
Spectra of Networks
Some theoretical results
Relation with network topology
Role in network synchronizability
Spectrum
Theoretical Bounds of Laplacian Eigenvalues
d max , d min , d avg
- maximum, minimum, average degree
D – Diameter of the graph
N
N
2 
d min 
d max  N  2d max
N 1
N 1
d min 2

d max  N
1
 2  ...  N  N
ND
d avg  N ( A)
 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  (d1 , d 2 ,..., d N )T
  1 , 2 ,..., N T
(both in increasing order)
|| d ||1
||   d ||2
N


|| d ||2
|| d ||2
|| d ||1
Distribution: (interlacing property)
For any node degree d i there exists a
*   j | j  1,2,..., N 
such that d i  d i  *  d i  d i
C. J. Zhan, G. Chen and L. F. Yeung, Physica A (2010)
  1 , 2 ,..., N T
|| d ||1
||   d ||2
N


|| d ||2
|| d ||2
|| d ||1
d  (d1 , d 2 ,..., d N )T
Lemma (Hoffman-Wielanelt, 1953)
For matrices B – C = A:
2
2
|

(
B
)


(
C
)
|

||
A
||
i1 i
i
F
n
Frobenius Norm
 For Laplacian L = D – A:
2
2
|

(
L
)

d
|

||
A
||
i1 i
i
F
n
|| A ||2F  i 1  j 1| aij |2  i 1 di
n
2

 ||   d ||2 

 
||
d
||
2


d
d
i
2
i

|| d ||1 
 di  N  N


 || d ||2   d 2 / N  d i || d ||1

  i
Cauchy inequality
n
n
Difference between Eigenvalue and Degree Sequences
1000  d1000  42
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)
Upper Bounds for Largest Eigenvalues
N  N
- for any graph
N  max{ du  dv |  edges (u, v) }
N  max{ d u  d u }
d u - average degree of all neighbors of node u
N  max{ du  dv  | Nu  Nv | |  nodes u  v in G }
Nu  N v
- 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.
Lower Bounds for Smallest Eigenvalues
2  2[cos( / N )  cos(2 / N )]  2 cos( / N )(1  cos( / N ))d max
2  2e(1  cos( / N ))  4e sin 2 ( / 2 N )
e  v  2  1/( ND)
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
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)
Regularly-Connected Networks
 Complete graphs: 0, 2  ...  N  N
 2K-rings:
i
4sin 2 , i  1, 2,..., N  1
K=1:
N
N
 i 
N - even: 0, 4 sin 2  , i  1,2,...,
K≠1:
 Chains:
 Stars:
2 / N  1
 1, 4,
2
N
K
2 (i  1)l
i  2 K  2 cos
, i  2,3,..., N
N
l 1
N
 1,. .. , N  1
2
 i 
0, 4 sin 2 
, i  1,2, .. , N  1
2
N


0, 2  ...  N 1  1 N  N
2 / N  1 / N
Rectangular Random Networks
E. Estrada and G. Chen (2015)
RRN: N nodes are randomly uniformly and independently
distributed in a unit rectangle [a, b]2  R 2 with a  b  1
(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
Theorem: For RRN, eigenvalues are bounded by
1
2 8(ar ) 2
2


log
2 N
2
4
( N  1) N
N a  1
Lower bound
The worst case: all nodes are located on the diagonal
2 
1
1

ND N ( N  1)
and
N  N
E. Estrada and G. Chen (2015)
1
2 8(ar ) 2
2


log
2 N
2
4
( N  1) N
N a  1
Upper bound
Lemma 1: Diameter D = diagonal length / r

 a4 1 
D

 ar 
Lemma 2: Based on a result of Alon-Milman (1985)
2
 2  8d max
log
2 N
2
D
E. Estrada and G. Chen (2015)
Spectral Role in Network Synchronization
Topology affects Eigenvalues affects Sync
2  2 S / S
• Recall: d min  2
d max  N
S  
a
iS jG  S
ij
S - any subgraph, with
0 < |S|  N/2
(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 2
depends on S
Statistics Determines Synchronizability?
Answer - Not necessarily
Example:
Graph
G1
Graph
G2
• They have same structural characteristics:
Graphs G1 and G2 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
G1
G2
But they have different synchronizabilities:
Eigenvalues of G1 are 0, 3, 3, 3, 3 and 6
Eigenvalues of G2 are 0, 2, 3, 3, 5 and 5
2 (G1 )  3  2 (G2 )  2
Eigenratio r  2 / N satisfies: r (G1 )  0.5  r (G2 )  0.4
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
i (G  e)  i (G),
i  1,2,.., N
• Note:
2
2
N
2
N
Z. S. Duan, G. R. Chen and L. Huang, Phys. Rev. E, 2007
What Topology  Good Synchronizability ?
Example
Given Laplacian:
2 0 0 0
 0 2 1 1

 0 1 3 0
L
 0 1 0 3
 1 0 1 1

 1 0 1 1
1 1
0 0 
1 1

1 1
4 1

1 4 
Q: How to replace 0 and -1 while keeping the connectivity
(and all row-sums = 0), such that 2 N = maximum ?
Answer:
 3 1 0 1 0 1
 1 3 1 0 1 0 


 0 1 3 1 0 1
*
L 


1
0

1
3

1
0


 0 1 0 1 3 1


 1 0 1 0 1 3 

2 N = maximum
Observation: Homogeneous + Symmetrical Topology
Problem
 With the same numbers of node and edges, while
keeping the connectivity, what kind of network has
the best possible synchronizability?
Max
LL*
2
N
L * - set of N  N Laplacian matrices
Such that “row-sum = 0” and
(Laplacian)
2  0
(connected)
 Computationally, this is NP-hard
 Objective: Find analytic solutions
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), 3regular optimal graphs
Our Approach
• Homogeneity + Symmetry
• Same node degree d1 
 dN
• Minimize average path length l1 
• Minimize path-sum
• Maximize girth
g1 
li   j  i lij
 gN
D. Shi, G. Chen, W. W. K. Thong and X. Yan,
IEEE Circ. Syst. Magazine, (2013) 13(1) 66-75
 lN
Non-Convex Optimization
Illustration:
White：Optimal
solution location
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
Optimal 3-Regular Networks
Optimal 3-Regular Networks
Summary
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 ?
Open Problem
Looking for optimal solutions:
 1
 d1  1 0
 1 d

0
2





Given: 0



0


 1 0
0 d N 
Move which -1
 1
 d1  1 0
 1 d
0 
2

0





0


 1 0
0 d N 
 1
 d1  1 0
 1 d
 1 0 
2

0





1

0


 1 0
0 d N 
can maximize
2
?
N
can maximize
2
?
N
Where to delete -1 …………………… can maximize
2
?
N
Where to add -1
And so on …… ???
Thank You !
Homogeneity
Symmetry
References
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13(1) 66-75
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