Create Presentation
Download Presentation

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

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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 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 • … …**Bacteria Colony**Flocking of Birds Biological Systems Ant Colony Bee Colony**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**Assumption**Each agent • makes decision according to local information ; • has the tendency to behave as other agents do in its neighborhood.**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.**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:**Heading:**Position: Vicsek Model Neighbors:**Position:**Vicsek Model Neighbors: Heading:**Position:**Vicsek Model Neighbors: Heading: is the weighted average matrix.**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**(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**If i ~ j Adjacencymatrix: Otherwise Degree: Volume: Degree matrix: Average matrix: Laplacian:**Connectivity of The Graph**Connectivity: There is a path between any two vertices of the graph.**G1**G2 G1∪G2 JointConnectivity of Graphs Joint 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)**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.**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 Neighbors 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**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**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 )**Moreover, if**then we have Analysis of Matrices with Increasing Dimension Estimation 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