Dynamic on Networks
II. Consensus
dynamics
Ernesto Estrada
Department of Mathematics & Statistics,
University of Strathclyde
Glasgow G1 1XH, UK
www.estradalab.org
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2
3
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5
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i
i
j
opinion of j at time t
j
opinion of i at time t


ui t    u j t   ui  j  ,
i  1,, n
j ~i
ui 0  zi ,
zi  
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u t   LG ut ,
u0  u0
 1 if uv  E ,

Luv   ku if u  v,
 0 otherwise.

A
B
D
C
E
A
B
C
D
E
A
2
1 1
0
0
B
1
4
C
1 1
2
0
0
D
0
1
0
1
0
E
0
1
0
0
1
1 1 1
9
LKA
2

0
K  0

0
0

0 0 0 0

4 0 0 0
0 2 0 0

0 0 1 0
0 0 0 1 
0

1
A  1

0
0

1 1 0 0

0 1 1 1
1 0 0 0

1 0 0 0
1 0 0 0 
10
   j  1 

 n 
j  1,, n
 2j 

 n 
j  1,, n
Pn
 j L   2  2 cos
Cn
 j L   2  2 cos
Sn
SpL  0 1n  2 n
Kn
SpL  0 nn 1




11
n
The consensus set A   is the subspace span{1},
that is

A  u  n | ui  u j , i, j

u k  1   I   L  u k 
0    1/ kmax
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L  UΛU T
 1 1  2 1

 1 2  2 2
U



  n   n 
2
 1
  n 1 

  n 2

 

  n n 
 1

0
Λ


0

0
2

0
0

 0
  

  n 

0  1  2    n
13
e
tL
e 
 t UΛU T
  Ue tΛ UT
 e t111  e t2  2 2    e tn  n n
T
T
T
ut   e tLu0






ut   et1 1 u0 1  et2 2 u0 2    etn n u0 n
T
T
T
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Theorem: Let G be a connected network. The undirected
consensus model converges to the consensus set with a
rate of convergence that is dictated by  2 .
Proof:
1  0 and  j  0, j  1 . Thus


1T u 0
ut    u 0 1 
1 as t  
n
T
1
Hence ut   A as t  .
As  2 is the smallest positive eigenvalue of the graph Laplacian, it dictates
the slowest mode of convergence in the previous equation.
15
As the states of the nodes evolve towards the consensus set,
one has
d T
T

1 ut   1T  Lu t   ut  L1  0
dt
Then
1T ut    ui t 
i
is a constant of motion for the consensus dynamics.
Furthermore, the state trajectory generated by the
consensus model converges to the projection of its initial
state, in the Euclidean norm, onto the consensus space,
since
1T u 0
1T u 0
arg min u  u 0  T 1 
1
uA
11
n
16
The trajectory of the consensus model retains the centroid
of the node’s states as its constant of motion.
u0
1
1T u  u0   0
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Proposition: A necessary and sufficient condition for the
consensus model to converge to the consensus subspace
from an arbitrary initial condition is that the network is
connected, i.e., it contains a spanning tree.
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2
3
4
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5
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1
2
3
4
5
6
 1 1 0 0 0 0 
  1 3  1 0 0  1
 0 1 3 1 1 0 
L
0 0 1 2 1 0 
 0 0  1  1 3  1
 0 1 0 0 1 2 


leaders
 3  1

Ll  

1
2


 1 1 0 0 


 1 3 1 0 
Lf 
0  1 3  1


 0 0 1 2 


0 0


 0  1
L fl  
1 0 


 1 0 


22
u f 
L f
u    0

 l
L fl  u f  0
  u



0   ul   I 
u f  L f u f  L fl ul
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Theorem: If the network G is connected then L f is positive
definite.
Proof:
Since,
L is positive semidefinite. If G is connected N L  span1.

u Lfuf  u
T
f
T
f

u f 
0 Lf  
0
And
u

0  N L
T
f
Then
u
T
f

u f 
nf
0 L f    0, u f  
0
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Theorem: Given fixed leader opinions u l , the equilibrium
point under the leader-follower dynamics is
u f  Lf1L fl ul
which is globally asymptotically stable.
Proof:
Lf  0
. Thus Lf1 exists and
u f  Lf1L fl ul is well defined.
Hence, the equilibrium point is unique. Moreover, because
equilibrium point is globally asymptotically stable.
L f  0 , this
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leaders
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A  L f , B  L fl
u f  Au f  Bul
Proposition: This system is controllable if and only if none of
the eigenvectors of A is (simultaneously) orthogonal to
(all columns of ) B. Moreover, if A does not have
distinct eigenvalues, then the system is not controllable.
Corollary: The system with a single leader is controllable
if and only if none of the eigenvectors of A is orthogonal to 1.
27
Proposition: Suppose that the leader-follower system is
uncontrollable. Then there exists an eigenvector of L
that has a zero component on the index that corresponds to
the leader.
Corollary: Suppose that none of the eigenvectors of L has a
zero component. Then the leader-follower system
is controllable for any choice of the leader.
28



J u f t , ul    uTl Ru l  uTf Qu f dt
0
Assess the energy needed by the system to transfer
state u 0 to u t . R is positive definite and Q is positive
semidefinite.
ul t   R 1LTfl Pˆ u f t 
where P̂ is found by solving the matrix Riccati equation:
ˆ P
ˆAP
ˆ BR 1BT P
ˆ Q  0
AT P
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1
0.923 0.942
0.885 0.913
Correlation coefficient
0.8
0.6
0.4
0.2
0
-0.2
closeness
betweenness
degree
-0.4
-0.6
-0.8
-0.611
-0.697
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x1 t   0,
x2 t   w21x1 t   x2 t ,
1
w21
2
w32
3
w43
w34
w42
4
x3 t   w32 x2 t   x3 t   w34 x4 t   x3 t ,
x4 t   w42 x2 t   x4 t   w43 x3 t   x4 t .
0
 0

  w21 w21
x t   
0
 w32

 0
 w42

0
0
w32  w34
 w43


0

xt 

 w34

w42  w43 
0
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u t   LDut ,
u0  u0
1
w21
2
w32
3
w43
w34
w42
4
 0 w21 0

 0 0 w32
A
0 0
0

0 0 w
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
0
 0

  w21 w21
T
T
LD   Diag A 1  A  
0
 w32

 0
 w42



0 

w42 
w43 

0 
0
0
w32  w34
 w43


0

 w34 

w42  w43 
0
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Definition: A directed graph is a rooted out-branching if:
1. It does not contain a directed cycle and
2. It has a vertex vr (root) such that for every other vertex
v there is a directed path from vr to v.
vr
vr
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Proposition: A directed network contains a rooted
out-branching subgraph if and only if rank LD  n  1 .
In that case, N LD is spanned by the all-ones vector.
Theorem: Let D be a directed network on n nodes. Then,
The spectrum of LD  lies in the region
z  C | z  kˆ
in
 kˆin

where k̂in is the maximum (weighted) in-degree in D. That is,
for every directed network, the eigenvalues of LD  have
non-negative real parts.
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kˆin  2
0
12
eig L  3,1,1,0
3
eig L
kˆin  2
The eigenvalues of L(D) are contained in the Geršgorin
disk of radius k̂in centred at k̂in .
35
Proposition: Let L  PJ Λ P 1 be the Jordan decomposition
of the Laplacian for the directed network D. When D contains
a rooted out-branching, the nonsingular matrix P can be
chosen such that
0

0 
0


0 
 0 J  2  
J Λ    0
0

0 




 

0




0
J

n 

where the i i  2,, n have positive real parts, and J i 
is the Jordan block associated with eigenvalue  i .
36
Consequently,
1 0  0


0 0  0
lim e tJ  Λ    0 0  0 
t 


   
0  0 0


and
lim e tL  p1q1T
t 
where p1 and q1T are, respectively, the first column of P and
T
the first row of P-1, that is, where p1q1  1.
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Theorem: For a directed network D containing a rooted
out-branching, the state trajectory generated by the
consensus dynamic model, initialized from u 0 , satisfies


lim ut   p1q1T u 0
t 
where p1 and q1T , are, respectively, the right and left
eigenvectors associated with the zero eigenvalue of L(D),
normalized such that p1q1T  1 . As a result, one has
ut   A for all initial conditions if and only if D contains
a rooted out-branching.
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