Lecture III:Collective Behavior of Multi -Agent Systems: Analysis

Zhixin Liu

Complex Systems Research Center, Complex Systems Research Center, Academy of Mathematics and Systems Academy of Mathematics and Systems

Sciences, CASSciences, CAS

In the last lecture, we talked about

Complex NetworksIntroduction Network topology

Average path lengthClustering coefficient

Degree distribution

Some basic models Regular graphs: complete graph, ring graph Random graphs: ER model Small-world networks: WS model, NW model Scale free networks: BA model

Concluding remarks

Lecture III:Collective Behavior of Multi -Agent Systems: Analysis

Zhixin Liu

Complex Systems Research Center, Complex Systems Research Center, Academy of Mathematics and Systems Academy of Mathematics and Systems

Sciences, CASSciences, CAS

Outline

Introduction Model Theoretical analysis Concluding remarks

What Is The Agent?

From Jing Han’s PPT

What Is The Agent?

Agent: system with two important capabilities: Autonomy: capable of autonomous action – of deciding for themselves what they need to do in order to satisfy their objectives ；

Interactions: capable of interacting with other agents - the kind of social activity that we all engage in every day of our lives: cooperation, competition, negotiation, and the like.

Examples: Individual, insect, bird, fish, people, robot, …

From Jing Han’s PPT

Multi-Agent System (MAS)

MAS Many agents Local interactions between agents Collective behavior in the population level

More is different.---Philp Anderson, 1972 e.g., Phase transition, coordination, synchronization, consensus, c

lustering, aggregation, ……

Examples: Physical systems Biological systems Social and economic systems Engineering systems … …

Flocking of Birds

Bee Colony

Ant Colony

Biological Systems

Bacteria Colony

Engineering Systems

From Local Rules to Collective Behavior

Phase transition, coordination, synchronization, consensus, clustering, aggregation, ……

scale-free, small-world

Crowd Panic

pattern

swarm intelligence

A basic problem: How locally interacting agents lead to the collective behavior of the overall systems?

Outline

Introduction Model Theoretical analysis Concluding remarks

Modeling of MAS

Distributed/Autonomous Local interactions/rules Neighbors may be dynamic May have no physical connections

A Basic Model

This lecture will mainly discuss

Each agent

• has the tendency to behave as other agents do in its neighborhood.

Assumption

• makes decision according to local information ;

Vicsek Model (T. Vicsek et al. , PRL, 1995)

http://angel.elte.hu/~vicsek/http://angel.elte.hu/~vicsek/

r

A bird’s Neighborhood Alignment: steer towards the average heading of neighbors

Motivation: to investigate properties in nonequilibrium systems

A simplified Boid model for flocking behavior.

Notations

})()(:{)( rtxtxjtN jii

Neighbors:

xi(t) : position of agent i in the plane at time t

v: moving speed of each agent

r: neighborhood radius of each agent

)(ti : heading of agent i, i= 1,…,n. t=1,2, ……

r

Vicsek Model

})()(:{)( rtxtxjtN jii

Neighbors:

Position: ))1(sin),1((cos)()1( ttvtxtx iiii

Heading:

)()(cos

)()(sin

arctan)1(

ti

Njt

j

ti

Njt

j

ti

Vicsek Model

})()(:{)( rtxtxjtN jii

Neighbors:

Position: ))1(sin),1((cos)()1( ttvtxtx iiii

Heading:

)(tan)(cos

)(cos)1(tan

)()(

tt

tt i

tNjtNj

j

ji

i

i

Vicsek Model

})()(:{)( rtxtxjtN jii

Neighbors:

Position: ))1(sin),1((cos)()1( ttvtxtx iiii

Heading:

),(tan)(~

)1(tan ttPt

)(~

tP is the weighted average matrix.

otherwise

jiift

t

tp

tptP

tNj j

j

ij

ij

i

0

~)(cos

)(cos

)(~

)},(~{)(~

)(

Vicsek Model

http://angel.elte.hu/~vicsek/

Some Phenomena Observed (Vicsek, et al. Physical Review Letters, 1995)

a) ρ= 6, ε= 1 high density, large noise c )

b) ρ= 0.48, ε= 0.05 small density, small noise

d) ρ= 12, ε= 0.05 higher density, small noise

n = 300

v = 0.03

r = 1Random

initial conditions

Synchronization

Def. 1: We say that a MAS reach synchronization if there exists θ, such that the following equations hold for all i,

Question: Under what conditions, the whole system can reach synchronization?

Outline

Introduction Model Theoretical analysis Concluding remarks

(0)

x (0) x (1) x (2)

G(0)

(1) (2)

G(1)

…… ……

G(2)

(t-1) (t)

x (t-1) x (t)

G(t-1)

…… ……

• Positions and headings are strongly coupled • Neighbor graphs may change with time

Interaction and Evolution

),,,( 21 nddddiagT

ii

ii

i ddddnid min,max,,,1, minmax

1P T A

Degree:

Volume:1

( )n

jj

Vol G d

Average matrix:

Degree matrix:

Laplacian: ATL

Adjacency matrix:

0

1ija

If i ~ j

Otherwise

Some Basic Concepts

},{ ijaA

Connectivity:

There is a path between any two vertices of the graph.

Connectivity of The Graph

Joint Connectivity:

The union of {G1,G2,……,Gm} is a connected graph.

Joint Connectivity of Graphs

G1 G2 G1∪G2

Product of Stochastic Matrices Stochastic matrix A=[aij]: If ∑j aij=1; and aij≥0

SIA (Stochastic, Indecomposable, Aperiodic) matrix A

If where ,1lim cA nt

t

Theorem 1: (J. Wolfowitz, 1963)Let A={A1,A2,…,Am}, if for each sequence Ai1, Ai2, …Aik of posit

ive length, the matrix product Aik Ai(k-1) … Ai1 is SIA. Then the

re exists a vector c, such that .1lim 12

cAAA niiikk

.]1,1[1 n

.))1(sin),1((cos)()1(

,)()(

1)1(

)(

ttvtxtx

ttN

t

iiii

tNjj

ii

i

.))1(sin),1((cos)()1(

),()()1(

ttvtxtx

ttPt

otherwise

jiiftNtp

tptP

iij

ij

0

~|)(|

1)(

)},({)(

The Linearized Vicsek Model

A. Jadbabaie , J. Lin, and S. Morse, IEEE Trans. Auto. Control, 2003.

Related result: J.N.Tsitsiklis, et al., IEEE TAC, 1984

Joint connectivity of the neighbor graphs on each time interval [th, (t+1)h] with h >0

Synchronization of the linearized Vicsek model

Theorem 2 (Jadbabaie et al. , 2003)

The Vicsek Model

Theorem 3: If the initial headings belong to (-/2, /2),

and the neighbor graphs are connected, then the system will synchronize.

Liu and Guo (2006CCC), Hendrickx and Blondel (2006).

The constraint on the initial heading can not be removed.

Example 1: ,1.00,8.0,12 vrn

67)0(),21,23()0(

;3)0(),23,21()0(

23)0(),1,0()0(

;32)0(),23,21()0(

;611)0(),21,23()0(

)0(),0,1()0(

;6)0(),21,23()0(

;34)0(),23,21()0(

2)0(),1,0()0(

;35)0(),23,21()0(

;65)0(),21,23()0(

;0)0(),0,1()0(

112

1111

1010

99

88

77

66

55

44

33

22

11

x

x

x

x

x

x

x

x

x

x

x

x

• Connected all the time, but synchronization does not happen.• Differences between with VM and LVM.

Example2: ,1.0,3.0,24 vrn

;1211)0();259.0,966.0()0(

;611)0();21,23()0(

;43)0();22,22()0(

;35)0();23,21()0(

;127)0();966.0,259.0()0(

;23)0();0,1()0(

;125)0();966.0,259.0()0(

;35)0();23,21()0(

;4)0();22,22()0(

;67)0();21,23()0(

;12)0();259.0,966.0()0(

;)0();0,1()0(

;1223)0();259.0,966.0()0(

;65)0();21,23()0(

;47)0();22,22()0(

;32)0();23,21()0(

;1219)0();966.0,259.0()0(

;2)0();1,0()0(

;1217)0();966.0,259.0()0(

;3)0();23,21()0(

;45)0();22,22()0(

;6)0();21,23()0(

;1213)0();259.0,966.0()0(

;0)0();0,1()0(

2424

2323

2222

2121

2020

1919

1818

1717

1616

1515

1414

1313

112

111

1010

99

18

77

66

55

44

33

12

11

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

The neighbor graph does not convergeMay not likely to happen for LVM

How to guarantee connectivity?

What kind of conditions on model parameters are needed ?

Random Framework

Random initial states:

1) The initial positions of all agents are uniformly and independently distributed in the unit square;

2) The initial headings of all agents are uniformly and independently distributed in [-+ε, -ε] with ε∈ (0, ).

Random Graph

G(n,p): all graphs with vertex set V={1,…,n} in which the edges are chosen uniformly and independently with probability p.

P.Erdős,and A. Rényi (1959)

Not applicable to neighbor graph !

Corollary: c

cep econnectedisGP

)(

Theorem 5 Let , then

Random Geometric Graph

Geometric graph G(V,E) :

Random geometric graph:

If are i.i.d. in unit cube

uniformly, then geometric graph is

called a random geometric graph ( , )nG V r

}1,{ nixi

*M.Penrose, Random Geometric Graphs, Oxford University Press,2003.

},,:),{(

},,,2,1{

VjirxxjiE

nV

ji

Connectivity of Random Geometric Graph

Theorem 6Graph with is connected with

probability one as if and only ifn

ncnnr

)()log(

)(

( , ( ))G n r n

.)( ncn

( , ( ))G n r n

( P.Gupta, P.R.Kumar,1998 )

Analysis of Vicsek Model

How to deal with changing neighbor graphs ? How to estimate the rate of the synchronization? How to deal with matrices with increasing

dimension? How to deal with the nonlinearity of the model?

Dealing With Graphs With Changing Neighbors

3) Estimation of the number of agents in a ring

r)1(

r)1(

r})1()1(:{ rxxrjC ii

1) Projection onto the subspace spanned by .]1,1[1 n

2) Stability analysis of TV systems (Guo, 1994)

Estimating the Rate of Synchronization

The rate of synchronization depends on the spectral gap.

Normalized Laplacian: 2/12/1 LTT

1100 n Spectrum :

)1,1max( 11 nSpectral gap:

Rayleigh quotient

Vj jj

ji ji

Tz dz

zz

n2

~

2

11

)(inf

Vj jj

ji ji

zn dz

zz2

~

2

1

)(sup

Lemma1: Let edges of all triangles be “extracted” from a complete graph. Then there exists an algorithm such that the number of residual edges at each vertex is no more than three.

The Upper Bound of

))1(1(

)321(4

112)0(

21 on

Lemma 2: For large n, we have

= +

Example:

)0(1n

( G.G.Tang, L.Guo, JSSC, 2007 )

The Lower Bound of

Lemma 3: For an undirected graph G, suppose there exists a path set P joining all pairs of vertexes such that each path in P has a length at most l, and each edge of G is contained in at most m paths in P. Then we have

mldnd 2maxmin1 /

)0(1

Lemma 4: For random geometric graphs with large n ,

)).1(1(4

)),1(1(2

min

2max

orn

d

ornd

n

n

( G.G.Tang, L.Guo, 2007 )

The Lower Bound of )0(1

))1(1()6(512

)0(4

2

1 or

r

( G.G.Tang, L.Guo, 2007 )

Proposition 1: For G(n,r(n)) with large n

Estimating The Spectral Gap of G(0)

))1(1(

)321(4

112)0(

21 on

))1(1()6(512

)0(4

2

1 or

r

))1(1()6(512

1)0(4

2

or

r

( G.G.Tang, L.Guo, 2007 )

Analysis of Matrices with Increasing Dimension

Estimation of multi-array martingales

..,log34

3),(maxmax

11

11sanS

Cwnkf n

wm

jjj

nknm

.),(sup,),(max 21

,11

2

1nkFwECnkfS jj

njkw

n

jj

nkn

where

..log3),(maxmax1

111

sanSCwnkf nw

m

jjj

nknm

,log4 1 nCS wnMoreover, if then we have

..,log1)1(cosmax)4

.;.,log)1(tanmax)3

.;.,logsin

)0(cosmax)2

.;.,log)0(sinmax)1

2

1

2

1

2

)0(1

2

)0(1

sanrnO

sanrnO

sannrO

sannrO

nini

nini

nNj

jni

nNj

jni

i

i

Using the above corollary, we have for large n

Analysis of Matrices with Increasing Dimension

Dealing With Inherent Nonlinearity

A key Lemma: There exists a positive constantη, such that for large n, we have :

)).1(1(

)6(5123

4/,64/min1

)()(~

sup)0()2

)),1(1()0()()1

4

22

1

or

rr

sPsP

ordtd

ts

ijij

with 4

22

2 )6(5123

4/,/64min,

4,

64max

r

rr

r

For any given system parameters

and when the number of agnets n

is large, the Vicsek model will synchronize almost surely.

0v,0r

Theorem 7

High Density Implies Synchronization

This theorem is consistent with the simulation result.

Let and the velocity

satisfy

Then for large population, the MAS will synchronize almost surely.

),(log

),1(61

nn ron

nor

.

log 2/3

6

n

nrOv n

n

Theorem 8High density with short distance interaction

Concluding Remarks

In this lecture, we presented the synchronization analysis of the Vicsek model and the related models under deterministic framework and stochastic framework.

The synchronization of three dimensional Vicsek model can be derived.

There are a lot of problems deserved to be further investigated.

1. Deeper understanding of self-organization,

What is the critical population size for synchronization with given radius and velocity ?

Under random framework, dealing with the noise effect is a challenging work.

How to interpret the phase transition of the model?

……

2. The Rule of Global Information

Edges formed by the neighborhoodRandom connections

are allowed

If some sort of global interactions are exist for the agents, will that be helpful?

3. Other MAS beyond the Vicsek Model

Nearest Neighbor Model( , ( ))G n n

)(nEach node is connected with the nearest neighbors

Remark:

For to be asymptotically connected, neighbors

are necessary and sufficient. F.Xue, P.R.Kumar, 2004

))(,( nnG )(log n

http://www.red3d.com/cwr/boids/applethttp://www.red3d.com/cwr/boids/applet

A bird’s Neighborhood

Cohesion: steer to move toward theaverage position of neighbors

Separation: steer to avoid crowding neighbors

Alignment: steer towards the average heading of neighbors

Boid Model: Craig Reynolds(1987):

Collective Behavior of Multi-Agent Systems: Intervention

References:

J. Han, M. Li M, L. Guo, Soft control on collective behavior of a group of autonomous agents by a shill agent, J. Systems Science and Complexity, vol.19, no.1, 54-62, 2006.

Z.X. Liu, How many leaders are required for consensus? Proc. the 27th Chinese Control Conference, pp. 2-566-2-570, 2008.

In the next lecture, we will talk about

Thank you!