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Lecture III: Collective Behavior of Multi -Agent Systems: Analysis. Zhixin Liu Complex Systems Research Center, Academy of Mathematics and Systems Sciences, CAS. In the last lecture, we talked about. Complex Networks Introduction Network topology Average path length

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lecture iii collective behavior of multi agent systems analysis

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

Zhixin Liu

Complex Systems Research Center,

Academy of Mathematics and Systems Sciences, CAS

in the last lecture we talked about
In the last lecture, we talked about

Complex Networks

  • Introduction
  • Network topology

Average path length

Clustering 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 analysis3

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

Zhixin Liu

Complex Systems Research Center,

Academy of Mathematics and Systems Sciences, CAS

outline
Outline
  • Introduction
  • Model
  • Theoretical analysis
  • Concluding remarks
what is the agent
What Is The Agent?

From Jing Han’s PPT

what is the agent6
What Is The Agent?
  • Agent:system with two important capabilities:
    • Autonomy: capable ofautonomous 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
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, clustering, aggregation, ……
  • Examples:
    • Physical systems
    • Biological systems
    • Social and economic systems
    • Engineering systems
    • … …
flocking of birds

Bacteria Colony

Flocking of Birds

Biological Systems

Ant Colony

Bee Colony

from local rules to collective behavior
From Local Rules to Collective Behavior

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

scale-free, small-world

pattern

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

swarm intelligence

Crowd Panic

outline11
Outline
  • Introduction
  • Model
  • Theoretical analysis
  • Concluding remarks
modeling of mas
Modeling of MAS
  • Distributed/Autonomous
  • Local interactions/rules
  • Neighbors may be dynamic
  • May have no physical connections
a basic model
A Basic Model

This lecture will mainly discuss

slide14

Assumption

Each agent

  • makes decision according to local information ;
  • has the tendency to behave as other agents do in its neighborhood.
vicsek model t vicsek et al prl 1995

r

Alignment:steer towards the average heading of neighbors

A bird’s Neighborhood

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

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

Motivation:

 to investigate properties in nonequilibrium systems

 A simplified Boid model for flocking behavior.

notations

r

Notations

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

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

v: moving speed of each agent

r: neighborhood radius of each agent

Neighbors:

vicsek model

Heading:

Position:

Vicsek Model

Neighbors:

vicsek model18

Position:

Vicsek Model

Neighbors:

Heading:

vicsek model19

Position:

Vicsek Model

Neighbors:

Heading:

is the weighted average matrix.

vicsek model20
Vicsek Model

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

some phenomena observed vicsek et al physical review letters 1995
Some Phenomena Observed(Vicsek, et al. Physical Review Letters, 1995)

a)ρ= 6, ε= 1 high density, large noisec )

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 = 1

Random initial conditions

synchronization
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?
outline23
Outline
  • Introduction
  • Model
  • Theoretical analysis
  • Concluding remarks
slide24

(0)

(1)

(2)

(t-1)

(t)

G(0)

G(1)

G(2)

G(t-1)

……

……

x(0)

x(1)

x (2)

x (t-1)

x (t)

Interaction and Evolution

……

……

  • Positions and headings are strongly coupled
  • Neighbor graphs may change with time
some basic concepts
Some Basic Concepts

If i ~ j

Adjacencymatrix:

Otherwise

Degree:

Volume:

Degree matrix:

Average matrix:

Laplacian:

connectivity of the graph
Connectivity of The Graph

Connectivity:

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

joint connectivity of graphs

G1

G2

G1∪G2

JointConnectivity of Graphs

Joint Connectivity:

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

product of stochastic matrices
Product of Stochastic Matrices

Stochasticmatrix A=[aij]: If ∑jaij=1; and aij≥0

SIA (Stochastic, Indecomposable, Aperiodic) matrix A

If where

Theorem 1:(J. Wolfowitz, 1963)

Let A={A1,A2,…,Am}, if for each sequence Ai1, Ai2, …Aik of positive length, the matrix product AikAi(k-1)…Ai1 is SIA. Then there exists a vector c, such that

the l ineariz ed vicsek model
The Linearized Vicsek Model

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

slide30

Theorem 2(Jadbabaie et al. , 2003)

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

Synchronization of the linearized Vicsek model

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

the vicsek model
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.
slide33

Connected all the time, but synchronization does not happen.

  • Differences between with VM and LVM.
slide35

The neighbor graph does not converge

May not likely to happen for LVM

slide36
How to guarantee connectivity?
  • What kind of conditions on model parameters are needed ?
random framework
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
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.

, then

Theorem5Let

Corollary:

Not applicable to neighbor graph !

P.Erdős,and A. Rényi (1959)

random geometric graph

Random geometric graph:

If are i.i.d. in unit cube

uniformly, then geometric graph is called a random geometric graph

Random Geometric Graph

Geometric graphG(V,E):

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

slide40

Connectivity of Random Geometric Graph

Theorem6

Graph with is connected with

probability one as if and only if

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

analysis of vicsek model
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

Projection onto the subspace spanned by

Dealing With Graphs With Changing Neighbors

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

3) Estimation of the number of agents in a ring

estimating the rate of synchronization
Estimating the Rate of Synchronization

The rate of synchronization depends on the spectral gap.

Normalized Laplacian:

Spectrum :

Spectral gap:

Rayleigh quotient

the upper bound of

Example:

=

+

Lemma 2:For large n, we have

The Upper Bound of

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.

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

the lower bound of
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

Lemma 4: For random geometric graphs with large n ,

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

the lower bound of46
The Lower Bound of

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

estimating the spectral gap of g 0
Estimating The Spectral Gap of G(0)

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

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

analysis of matrices with increasing dimension

Moreover, if

then we have

Analysis of Matrices with Increasing Dimension

Estimation of multi-array martingales

where

analysis of matrices with increasing dimension49
Analysis of Matrices with Increasing Dimension

Using the above corollary, we have for large n

dealing with inherent nonlinearity
Dealing With Inherent Nonlinearity

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

with

slide51

Theorem 7

High Density Implies Synchronization

For any given system parameters

and when the number of agnets n

is large, the Vicsek model will synchronize almost surely.

This theorem is consistent with the simulation result.

slide52

Theorem 8

High density with short distance interaction

Let and the velocity

satisfy

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

concluding remarks
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.
slide54
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?
  • ……
slide55

Edges formed by the neighborhood

Random connections are allowed

2. The Rule of Global Information

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

slide56

Each node is connected with the nearest neighbors

Remark:

Forto be asymptotically connected, neighbors are necessary and sufficient.

F.Xue, P.R.Kumar, 2004

3. Other MAS beyond the Vicsek Model

Nearest Neighbor Model

boid model craig reynolds 1987

Separation: steer to avoid crowding neighbors

A bird’s Neighborhood

Alignment: steer towards the average heading of neighbors

Cohesion: steer to move toward theaverage position of neighbors

Boid Model:Craig Reynolds(1987):

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

in the next lecture we will talk about
In the next lecture, we will talk about
  • 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.