Graph Consensus: Autonomus and Controlled

Prepared by Abhijit Das

Many of the beautiful pictures are from a lecture by Ron Chen, City U. Hong KongPinning Control of Graphs

Natural and biological structures

Airline Route Systems

Distribution of galaxies in the universe

Motions of biological groups

Fishschool

Birdsflock

Locustsswarm

Firefliessynchronize

J.J. Finnigan, Complex science for a complex world

The internet

ecosystem ProfessionalCollaboration network

Barcelona rail network

Graph

Directed Graph or Diagraph

Un-directed Graph

04/12/23 7ARRI, UTA

Two properties of diagraph nodes

• Out-degree: Number of connections going out from a node

• In-degree: Number of connections going in to a node

• Edge: Connection between any two nodes

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Important types of Diagraphs

Balanced

Strongly Connected

Tree

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What is Consensus among nodesConsensus in the English language is defined firstly as unanimous or general agreement

1h

2h

3h

4h

h h h h

Before Consensus After Consensus04/12/23 10ARRI, UTA

Graph Dynamics (Diagraph)

1

2

3

4

5

Adjacency Matrix

1 0 0 0 1 0

2 1 0 1 0 0

3 1 0 0 0 0

4 0 0 1 0 1

5 1 1 0 0 0

A

14

21 23

31

43 45

51 52

0 0 0 01

0 0 02

0 0 0 03

0 0 04

0 0 05

w

w w

A w

w w

w w

or

1 2 3 4 5 1 2 3 4 5

Diagonal Matrix

1 0 0 0 0

0 2 0 0 0

0 0 1 0 0

0 0 0 2 0

0 0 0 0 2

D

Laplacian matrix

L D A

1 0 0 1 0

1 2 1 0 0

1 0 1 0 0

0 0 1 2 1

1 1 0 0 2

L

21w

31w

51w

14w

43w

45w

23w

Note that is row stochastic I L04/12/23 11ARRI, UTA

Continuous Time System

• Each node if assumed to have simple integrator dynamics, for -th node,

• Input

• Resultant Dynamics of the graph with all node

i ix ui

i

i ij j ij

u a x x

x A D x Lx

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CommentAs is row stochastic

The first eigenvalue of will be 0

The right eigenvector corresponding to 0 eigenvalue will be

At steady state all state values will be equal

I L

L

1 1 1 1T

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State solution

Eigen decomposition and Left and right eigenvector

Right eigenvector Left eigenvector

R RLX X

0( ) Lt

x Lx

x t e x

L LX L XRX LX

1 1 1

L R L R

L L R R L L

X LX X X

L X X X X X X

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State solution (Contd..)

11 1

0 0 0! ! !

n n nL L L

L L L Ln n n

L X Xe X X X e X

n n n

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State solution (Contd..)

1

0

0

10

L L

Lt

X X t

tL L

x e x

x e x

x X e X x

At Steady state 0

1

1

1

tL c LX x e X x

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State solution (Contd..)1020

1 2 3 1 2 3 30

1 1

0

1

1

1

n ntc

L L

n

x

x

x e xX X

x

1

0 0 0

0 0

0 0 n

with

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Finding consensus value for SC graph

Considering only the first line of the equation

0

0

ii

i ic i i c

i i ii

xx x x

For balanced graph0i

ic

xx

n

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Simulation results (SC graph)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

With

nor

mal

pro

toco

l

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What if there is one leader in the graph

Assuming rest of the graph is connected

The Laplacian matrix of a graph with a leader

1

0 0 0L

L

with 1L may be anything

Left eigenvector 1

1 0 0 0L

L

XX

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Consensus value for one leader graph

102030

1 1

0

1

1 0 0 0 1 0 0 01

1

tc

L L

n

x

x

x e xX X

x

10cx x

Note that if there is more than one leaders then no single solution is possible

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Simulation result (one leader case)

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

With

nor

mal

pro

toco

l

For tree network the result will be equivalent04/12/23 22ARRI, UTA

Graph contains a spanning tree

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

With

nor

mal

pro

toco

l

How the value of cx can be determined ?04/12/23 23ARRI, UTA

Eigenvalue properties

• For stability all the eigenvalues should be in the left half of the plane

• The second largest eigenvalue is of a standard laplacian matrix is known as Fiedler eigenvalue

• Fiedler eigenvalue determines the speed of the whole network, thus it is important to maximize its value

• Note that Fiedler eigenvalue in general can not be determined from the dominant eigenvalue of the inverse of laplacian matrix

s

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Gershgorin disk of a network

1

j

1 0 0 1

1 1 0 0

0 1 1 0

0 0 1 1

balL

0 0 0 0 0

1 1 0 0 0

1 0 1 0 0

0 1 0 1 0

0 1 0 0 1

treeL

1

j

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More comments

• Fiedler eigenvalue is also known as algebraic connectivity or spectral gap of a graph

• Algebraic connectivity is different from connectivity or vertex-connectivity

• Network synchronization speed does NOT depend on vertex-connectivity

• Number of zero eigenvalues in a laplacian matrix reveals, number of connected components in a graph

k

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Reducibility

Consider a matrix with . If is reducible, there exist an integer anda Permutation matrix such that

r rA 2r A

1n T

11

21 22

31 32 33

1 2 3

0 0 0

0 0

0T

n n n nn

B

B B

B B BT AT

B B B B

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Irreducibility

Consider a matrix . Then, is irreducible if and only if For any scalar .

r rA A

10

r

r rcI A

0c

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Comment on reducibility

• A connected graph (strongly/balanced) is generally have irreducible adjacency and laplacian matrix

• A tree network generally posses a reducible adjacency and laplacian matrix

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Discrete time system Murray-Saber, 2004

x Lx Continuous time system

max

( 1) ( ) ( ) ( )

( 1) ( )

0,1/

i

i i ij j ij N

x k x k a x k x k

x k P x k

P I L

d

maxd Max out-degree

Discretized

P Perron matrix

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Definition

1

Stochastic matrix: row sum =

Primitive matrix: If the matrix has one eigenvalue with maximum modulus

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Perron-Frobenius Theorem

Let be a primitive non-negetive matrix

with left and right eigenvectors and

Assumptions:

1. and

2. 1

Then, lim

T

k Tk

P

w v

Pv v wP w

vw

P vw

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Comment

max

When a perron matrix become

non-negetive, stochastic and primitive?

Hint:

1. Graph is a diagraph non-negetive

and row-stochastic

2. Graph is a SC diagraph with 0 1/

Primitive

P

G

G d

P

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State Solution- DT system

( ) (0) lim

lim ( ) (0) 1

(0)

(0)

k kk

k d

d i ii

ii

d

x k P x P

x k x v wx v

x w x

xx

n

with exist

with

For balanced graph

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Comparison

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Courtesy: Fax-Murray-Saber, 2006

Performance – Murray-Saber 2007

1

( ) ( )

( 1) ( )

0 ( ) ( 1)

c dx x x

t L t

k P k

L P k k

Error vector: where, = or

CT:

DT:

Note that, and

2

2 21

Algebric connectivity:

CT Graph:

DT Graph:

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Theorems

2 2

2 2

T

T

L

P

For balanced graph:

CT:

for all

DT:

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Alternative Laplacian-Structure: Fax-Murray 2004

1

1

1

1

(1 )

1, ( 1) ( )

i

i

i j ij Ni

i ij ij N

x x xN

N a d

x Qx

Q I D A L

P I L I D A

x k D Ax k

with

For does not converge

for every diagraphs (For example bipartite graph)04/12/23 38ARRI, UTA

Based on Vicsek model: Jadbabaie-Lin-Morse

1

1( 1) ( ) ( )

1

( 1) ( )

i

i i jj Ni

P

x k x k x kN

x k I D I A x k

Perron matrix

This Perron matrix is stable! How?

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Example: Bipartite graph

11

1

1

2

2

0 0 1 1

0 0 1 1

1 1 0 0

1 1 0 0

1 1.

A

P D A

P

P I D I A

P

Fax-Murray Formulation: contains

two eigenvalues at and So, is not Primitive

Jadbabaie-Lin-Morse:

is Primitive04/12/23 40ARRI, UTA

Trust Consensus: Ballal-Lewis-2008

1 2..

n

i

i

Tni i ii ii ii

i i

i ij j ij

ij ij

i ij j ij

t t t

u

u w

w c i

j

u

and

Baras-Jiang-2006

the confidence that node has

it its trust openion of node

Ballal-Lewis Bilinear Trust04/12/23 41ARRI, UTA

Bilinear trust Dynamics

1

( )

( )

1( 1) ( )

1

( 1) ( ) ( )

( ) ( ) ( )

i i

i

i ij j ij i ij j

n

i i ij j iji

n

u L t

L t I

k kn

k F k I k

F k I I D k L k

CT system:

For DT system (based on Vicsek model):

where, 04/12/23 42ARRI, UTA

Simulations

04/12/23 ARRI, UTA 43

1

2

4

3

Comment

1

( 1) ( ) ( )n

n n n

k F k I k

I D L I L I

CT and DT system described by Ballal-Lewis,

are not equivalent.

So, they have different consensus value.

For example, the equivalent CT system for

is

not

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Zhihua Qu’s formulation

1 1

1 1

( ) ( )

( ) ( )

( ) ( )

i i

n nij ij ij ij

i j i i jn nj j

il il il ill l

n

ij

x u

s t w s t wu x x x x

s t w s t w

x I D t x L t x

S s I A

D

where,

Note that, is a stochastic matrix.

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Comment

ij

D

w

If is irreducible (strongly connected/balanced)

then the algebric connectivity of the graph depends

on

Although, graph consensus can be achieved

successfully with the proposed control law

for irreduc Dible as well as reducible

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Passive system: Definition

1

0 0

( ) ( )

( )

( ), ( ) (0) 0

. . 0,

( ( )) ( (0)) ( ) ( ) ( ( ))

( ) ( ) ( ) ( )

t tT

Tf g

x f x g x u

y h x

V x S x C V

s t t

V x t V x u s y s ds S x s ds

L V x S x L V x h x

Consider a nonlinear system

is passive iff

and positive with

also, and

04/12/23 47ARRI, UTA

Mark Spong’s Lyapunov formulation

1

1 1

2

2 2 ( )

0

i i

i

N

ii

N NT

f i g i i i i ii i

i ij j ij N

N

V V

VV x L V L Vu S x y u

x

u K y y V

Number of agent:

If then can be proved

for only balanced graph.

04/12/23 48ARRI, UTA

Can we change for which iu

1

1 1

2 2

1 1

2 2

1 1

2 2

i ij j ij i ji ij j j

c r

T Ti ij j i i i i j j

j

u K y K y K y

u D D A

u K y y y y y y y

Some example:

Another one:

0?V

04/12/23 49ARRI, UTA

Zhihua Qu’s Lyapunov formulation

2

1 1

1

( 1)

2

( ) ( ) ,

1

i i i

i

i

n n

c i j j ii j

nT

c i c c ci

T Tc i i

T n nc i i

n

x I D x Lx

V x x

V e Q e

Q G P I D I D P G

e x x G i

I

If then it can be shown that

where,

and eleminating th

column from 04/12/23 50ARRI, UTA

Comments: Zhihua Qu

D

D D

This Lyapunov formulation can successfully

be done for irreducible and reducible matrix

For reducible matrix, should be

lower-triangular complete

04/12/23 51ARRI, UTA

Lihua Xie’s Lyapunov formulation

04/12/23 ARRI, UTA 52

01

n

i ij i j i ij

u

e a x x b x x

1. Considering a one leader network

2. Define a input based on terminal sliding mode

control surface (see addendum)

3. Define error as

Lihua Xie’s formulation contd…

04/12/23 ARRI, UTA 53

1 2 0

0

, ,......, 1

2. ( )

T

nx x x x

T x

If the conditions of the previous slide exist

Then,

1. The network will achieve consensus and

The consensus will achieve in finite time

(see addendum)

Scale free network

04/12/23 54ARRI, UTA

Courtesy Wikipedia

Ron Chen’s pinning control

04/12/23 55ARRI, UTA

Ron Chen’s Lyapunov formulation

04/12/23 ARRI, UTA 56

1

1

( ) ( )

1,2,3,.......,

( ) ( )

1,.....,

k k k k k k

k

k k k k k

k

N

i i i i j j i ijj i

N

i i i i j j ijj i

x f x c a x x u

k l

x f x c a x x

k l N

Consider a scale-free undi-rected network

Pinned

with

NOT pinned

with

Ron Chen’s formulation contd…

04/12/23 ARRI, UTA 57

( )

( )

k k k k k

k k k

i i i i i

i i i

T

u c d x x

c d

E x X

g x E U V E

U

Define a input

with some condition imposed on and

Error is defined as =

Then, if a lyapunov candidate is defined as

with, some symmetric and atleast semi

V definite

some positive definite matrix

Ron Chen’s formulation contd…

04/12/23 ARRI, UTA 58

If is symmetric then the whole network

can be stablilized ( ) 0 following some

conditions such as

0

where is a matrix such that ( ) is

uniformly decrasing

g x

U V G D I T

T f x Tx

V

Ron Chen’s formulation contd…

04/12/23 ARRI, UTA 59

min

min

( )

( ) 0

( , ( ))c

f x

G D

f L

Moreover, if is Lipschitz continuous

then, it can be shown that for the combination

network

with

Controlled consensus

If, and 1 then

algebric connectivity is increased by

i.e. one leader is connected to every node

with a weight

For all other case,

if the Graph is SC, then adding a leader to few node

Tc

x Lx Bu

u c B

c

c

s

decrease the algebric connectiMAY vity

04/12/23 60ARRI, UTA

Some case studies

1

42

3

Consensus time approx 7.5 sec

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Time

Sta

tes

with

diff

. In

i. co

nd.

04/12/23 61ARRI, UTA

Some case studies contd…

1

42

3

L Consensus time approx 8 sec

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Time

Sta

tes

with

diff

. In

i. co

nd.

04/12/23 62ARRI, UTA

Some case studies contd…

1

42

3

LConsensus time approx: 3 sec

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

Time

Sta

tes

with

diff

. In

i. co

nd.

04/12/23 63ARRI, UTA

A special case

2 1.3472

L

L

2 2

04/12/23 64ARRI, UTA

A special case contd…

L

2 1.2451

04/12/23 65ARRI, UTA

Mathematical formulation: Lewis, 09

1

( ) 0,

n

L D A

D D

L

L L

x Lx

Define new laplacian matrix

Note that the new laplacian has diagonal dominance

property over irreducibility.

So, is nonsingular and

i.e. is a AS system.04/12/23 66ARRI, UTA

Controlled consensus: Lewis-’09

1

0 0

( )

When there are more than one leader or a

leader network is present

ss

G l n

n l L

x L B x

u u

x L Bu

L CL

C L

04/12/23 67ARRI, UTA

Leader-Graph network

Leader network

Graph network

Connection may be from both way

04/12/23 68ARRI, UTA

One case study: based on Z. Qu’s Laplacian

11

21 22

31 32 33

1 2 3

0 0 0

0 0

0

n n n nn

d

d d

d d dD

d d d d

Consider a reducible graph (Ex: Tree)

N1

N3

N2

Lower triangularly complete

04/12/23 69ARRI, UTA

One case study

1 10 0

0 0 0 0

0 0

n nd d

D

Now we add a leader/ leader

It is now possible to show that the new graph has

better algebric connectivity from Lyapun

virtual cl

ov anal

one

ysis

04/12/23 70ARRI, UTA

Case study: contd…

1

1

0

Tn n

T T T Tn n n n n n n n

V

T T Tn n n n n n

V

V e Pe

V e PW I D G G I D W P e

e PW DG G DW P e

04/12/23 71ARRI, UTA

Jadbabaie-Lin-Morse’s leader network

04/12/23 ARRI, UTA 72

0( )

0

1( 1) ( ) ( ) ( )

1 ( )

it can be shown that

lim ( ) 1

i

i i j ij N ki i

t

x k x k x k b k xN b k

x k x

Noisy information exchange: Ren-Beard-Kingston-2005

04/12/23 ARRI, UTA 73

*

*

* *

Noise on the edge: ,

with ~ 0,

Unknown consensus value:

Process noise: , with ~ 0,

Error Covariance:

( )( )

j i

ij j ij ij ij

Ti i i

v v

z x v v R

x

x w w Q

P E x x x x

Estimator dynamics

04/12/23 ARRI, UTA 74

1

( ) ( ) ( ) ( )

with Kalman gain:

and

( )

i

i

i ij ij ij ij

T

ij i j ij

i i ij j ij ij N

x t w K t z t x t

K P P R

P P w t P R P Q

Das-Lewis contribution

04/12/23 ARRI, UTA 75

( ) ( )i i i i ix f x w t u ˆ ( ) ( , )i i i iu f x v x t

ˆ ˆ ( )Ti i i i if x W x

ˆ ˆ( )Ti i i i i i i i iW F e p d b FW

0r

Select from Lyapunov

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

2

Synch. Motion

Control Node

04/12/23 ARRI, UTA 76

Thank you

Addendum: Zhihong Man

04/12/23 ARRI, UTA 77

1 2

2

11

1 2

2 1

( ) ( ) ( )

( ) ( ) ( )sgn( )

,0 ( ) , ( ) 0, 0, 0

q

p

q

p

x x

x f x g x b x u

qu b x f x x x l s

p

s x x l g x b x p q

Define a system as

Then TSM control law generally have the form

Addendum: Lihua Xie

04/12/23 ARRI, UTA 78

1 2

0 0 0

0

, ,......,

( ) ( ) ( ) ( )2 2

10

2( ) ( , , , )

T

n

t t

T

E e e e

S E t E t E t dt E t dt

V S S V V V

T x f V

Define, as error vector and

as sliding surface

If then

Addendum: Courtesy Fang-Antsaklis

04/12/23 ARRI, UTA 79