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

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

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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

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

Zhixin Liu

Complex Systems Research Center,

Academy of Mathematics and Systems Sciences, 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 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)

- 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
- … …

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

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

- makes decision according to local information ;

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

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.

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 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

- 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

(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

If i ~ j

Adjacencymatrix:

Otherwise

Degree:

Volume:

Degree matrix:

Average matrix:

Laplacian:

G2

G1∪G2

JointConnectivity of GraphsJoint Connectivity:

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

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 Linearized Vicsek Model

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

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

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.

Connected all the time, but synchronization does not happen.

- Differences between with VM and LVM.

The neighbor graph does not converge

May 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.

, then

Theorem5Let

Corollary:

Not applicable to neighbor graph !

P.Erdős,and A. Rényi (1959)

If are i.i.d. in unit cube

uniformly, then geometric graph is called a random geometric graph

Random Geometric GraphGeometric graphG(V,E):

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

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

- 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?

Projection onto the subspace spanned by

Dealing With Graphs With Changing Neighbors2) Stability analysis of TV systems (Guo, 1994)

3) Estimation of the number of agents in a ring

Estimating the Rate of Synchronization

The rate of synchronization depends on the spectral gap.

Normalized Laplacian:

Spectrum :

Spectral gap:

Rayleigh quotient

=

+

Lemma 2:For large n, we have

The Upper Bound ofLemma1: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

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 of

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

Estimating The Spectral Gap of G(0)

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

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

then we have

Analysis of Matrices with Increasing DimensionEstimation of multi-array martingales

where

Analysis of Matrices with Increasing Dimension

Using the above corollary, we have for large n

Dealing With Inherent Nonlinearity

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

with

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.

High density with short distance interaction

Let and the velocity

satisfy

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

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?
- ……

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?

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

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

- 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.

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