1. Lecture III: Collective Behavior of Multi -Agent Systems: Analysis
Zhixin Liu
Complex Systems Research Center,
Academy of Mathematics and Systems Sciences, CAS

2. 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

3. Lecture III: Collective Behavior of Multi -Agent Systems: Analysis
Zhixin Liu
Complex Systems Research Center,
Academy of Mathematics and Systems Sciences, CAS

4. Outline
Introduction
Model
Theoretical analysis
Concluding remarks

5. What Is The Agent?
From Jing Han’s PPT

6. 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

7. 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
… …

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

9. Engineering Systems

10. 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

11. Outline
Introduction
Model
Theoretical analysis
Concluding remarks

12. Modeling of MAS Distributed/Autonomous Local interactions/rules
Neighbors may be dynamic
May have no physical connections

13. This lecture will mainly discuss
A Basic Model

14. Assumption Each agent makes decision according to local information ;
has the tendency to behave as other agents do in its neighborhood.

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

16. 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:

17. Vicsek Model
Neighbors:
Position:
Heading:

18. Vicsek Model
Neighbors:
Position:
Heading:

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

20. Vicsek Model

21. 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= small density, small noise
d) ρ= 12, ε= higher density, small noise

22. 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?

23. Outline
Introduction
Model
Theoretical analysis
Concluding remarks

24. 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

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

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

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

28. 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

29. The Linearized Vicsek Model
A. Jadbabaie , J. Lin, and S. Morse, IEEE Trans. Auto. Control, 2003.

30. 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

31. 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.

32. Example 1:

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

34. Example2:

35. The neighbor graph does not converge
May not likely to happen for LVM

36. How to guarantee connectivity?
What kind of conditions on model parameters are needed ?

37. 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, ).

38. 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)

39. 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.

40. 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 )

41. 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?

42. 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

43. Estimating the Rate of Synchronization
The rate of synchronization depends on the spectral gap.
Normalized Laplacian:
Spectrum :
Spectral gap:
Rayleigh quotient

44. 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 )

45. 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 )

46. The Lower Bound of
( G.G.Tang, L.Guo, 2007 )

47. Estimating The Spectral Gap of G(0)
Proposition 1: For G(n,r(n)) with large n
( G.G.Tang, L.Guo, 2007 )

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

49. Analysis of Matrices with Increasing Dimension
Using the above corollary, we have for large n

50. Dealing With Inherent Nonlinearity
A key Lemma: There exists a positive constantη, such that for large n, we have : 2
with

51. 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.

52. Theorem 8
High density with short distance interaction
Let and the velocity
satisfy
Then for large population, the MAS will synchronize almost surely.

53. 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.

54. 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? ……

55. 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

56. 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

57. Boid Model: Craig Reynolds(1987):
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

58. 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, , 2006.
Z.X. Liu, How many leaders are required for consensus? Proc. the 27th Chinese Control Conference, pp , 2008.

59. Thank you!