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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!
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