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

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

Biological Systems Ant Colony Bee Colony Bacteria Colony Flocking of Birds Bacteria Colony Bee Colony

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

This lecture will mainly discuss A Basic Model

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

Vicsek Model Neighbors: Position: Heading:

Vicsek Model Neighbors: Position: Heading:

Vicsek Model Neighbors: Position: Heading: is the weighted average matrix.

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

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

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

Interaction and Evolution …… ……  (0) x (0)  (1)  (2) G(2)  (t-1)  (t) x (t-1) x (t) G(t-1) …… x (1) x (2) G(0) G(1) Positions and headings are strongly coupled Neighbor graphs may change with time

Some Basic Concepts Adjacency matrix: Degree: Volume: Degree matrix: If i ~ j Otherwise Degree: Volume: Degree matrix: Average matrix: Laplacian:

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

Joint Connectivity of Graphs G1∪G2 Joint Connectivity: The union of {G1,G2,……,Gm} is a connected graph.

Product of Stochastic Matrices Stochastic matrix A=[aij]: If ∑j aij=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 Aik Ai(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.

Synchronization of the linearized Vicsek model 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.

Example 1:

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

Example2:

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, ).

Not applicable to neighbor 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 Theorem 5 Let Corollary: Not applicable to neighbor graph ! P.Erdős,and A. Rényi (1959)

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

Connectivity of Random Geometric Graph Theorem 6 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?

Dealing With Graphs With Changing Neighbors Projection onto the subspace spanned by 2) 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

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. = + Example: 2 Lemma 2: For large n, we have ( 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 )

Analysis of Matrices with Increasing Dimension Estimation of multi-array martingales where Moreover, if then we have

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 : 2 with

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.

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 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.

How to interpret the phase transition of the model? …… 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 If some sort of global interactions are exist for the agents, will that be helpful? Edges formed by the neighborhood Random connections are allowed

Nearest Neighbor Model 3. Other MAS beyond the Vicsek Model Nearest Neighbor Model Each node is connected with the nearest neighbors How many neighbors are needed for consensus of the MAS ? Remark: For to be asymptotically connected, neighbors are necessary and sufficient. F.Xue, P.R.Kumar, 2004

Boid Model: Craig Reynolds(1987): http://www.red3d.com/cwr/boids/applet A bird’s Neighborhood Alignment: steer towards the average heading of neighbors Separation: steer to avoid crowding neighbors Cohesion: steer to move toward the average position of neighbors

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.

Thank you!