How Many Leaders Are Needed for Reaching a Consensus

Transcription

1 How Many Leaders Are Needed for Reaching a Consensus Xiaoming Hu KTH, Sweden Joint work with Zhixin Liu, Jing Han Academy of Mathematics and Systems Science, Chinese Academy of Sciences

2 Stockholm, September 10, 2009

3 Arizona, 1987

4 How Many Leaders Are Needed for Reaching a Consensus Xiaoming Hu KTH, Sweden Joint work with Zhixin Liu, Jing Han Academy of Mathematics and Systems Science, Chinese Academy of Sciences

5 Effective Leadership in Group Behavior 5 For a large group of animals, suppose a proportion of the individuals are given information about a desired direction x d. Then those individuals would modify their decision by u i = (1 w i ) X a i j (x j x i ) + w i k(x d x i ) j 2 N i While for the followers In practice, sensing and communication range is limited, so connectivity is largely based on distance.

6 6 Zoologists and biologists have found that For a given group size the accuracy of group motion (in a preferred direction) increased asymptotically as the proportion of informed individuals increased. As the group size became larger this relationship became increasingly nonlinear, meaning that the larger the group, the smaller the proportion of informed individuals needed to guide the group with a given accuracy.

7 7 Modeling issue: Mechanism for Information exchange. In a separate study it is found that in swarming honey bees Apis mellifera only about five percent bees (scouts) are involved in decision making. Now an interesting question is: as the population tends to infinity, what happens to the minimum percentage of the informed individuals for a given accuracy?

8 Multi-agent systems (MAS) Many locally interacting agents Collective behavior of the overall system A key issue in MAS: intervention of the MAS such that the system can exhibit the expected collective behavior?

9 MAS to be studied n agents Each agent has two state variables: position and heading. Position update equation: where is the heading of agent i at time t.

10 MAS to be studied (cont.) Heading update equation: with Control law I (Jadbabaie et al., 2003): M i (t): information set of the agent i at t Control law II (Vicsek et al., 1995):

11 Purpose of this paper Problem: How to guide the whole group to move with the same expected direction by adding leaders (information agents) knowing the expected direction Goal: Provide the number of leaders needed for the expected consensus

12 After adding leaders The position update equations of all agents: V 1 : set of followers V 2 : set of leaders The heading update equations of leaders:

13 After adding leaders (cont.) The neighbors of each follower are composed of two parts: Neighbor graphs are changed from undirected graph to directed graph.

14 After adding leaders (cont.) The heading update equations of followers under Control law I: Correspondingly, the heading update equations of followers under Control law II: with

15 Main Results

16 Assumption on the initial Assumption 1: states 1) The initial positions of all agents are independently and uniformly distributed in the unit square. 2) The initial headings of the agents are uniformly and independently distributed in [-π, π), and the initial headings of the leaders are 0. The headings and the positions are mutually independent.

17 Consensus under Control law I Theorem 1: For the MAS under Control law I, let the speed and the radius satisfy the following condition: If the proportion of leaders satisfies with 0 < a < 1. Then under Assumption 1, all agents can be guided to move with the same expected heading almost surely for large n. This paper mainly focuses on the consensus behavior of the MAS under Control law II.

18 Simulation example n=1000 Purpose: Guide all agents move with the heading Adding M leaders How many leaders are needed for consensus?

19 Three steps for analysis Step 1: Analysis of the system dynamics Step 2: Estimation of some characteristics concerning the initial states Step 3: Dealing with the nonlinearity

20 Step 1: Analysis of system dynamics Evolution of the distance between agents Lemma 1: For any two agents i and j, their distance satisfy the following inequality: where is the heading error of the agent i at time t, i.e.,

21 Step 1: Analysis of system dynamics (cont.) Dynamics of the heading errors For the heading error dynamics of followers, we have:

22 Step 1: Analysis of system dynamics (cont.) Evolution for the heading errors Lemma 2: For the heading error dynamics is determined of followers, by the we dynamical states of the agents have: where Arrange all eigenvalues of M(0) as are determined by the initial states of all agents with, T(0), N(0) and L(0) are ordinary degree matrix, leader degree matrix and the Laplacian of the initial neighbor graph.

23 If Then we have consensus! Note:

24 Step 2: Estimation of some characteristics concerning the initial states Estimation of the degrees of the initial neighbor graph Lemma 3: 1) For any agent and leader neighbors satisfy, its follower neighbors 2) The minimum number of neighbor of the initial neighbor Remark: The constant k in Lemma 2 can be taken graph as satisfies

25 Step 2: Estimation of some characteristics concerning the initial states (cont.) Lemma 4: For, we have for large n Lemma 5: Under the assumptions on the initial states, we have for large n where C 1 is a constant depending on r only.

26 Step 3: Dealing with the nonlinearity The nonlinearity includes two aspects: 1) The nonlinear coupled relationship of positions and headings of all agents 2) The inherent nonlinearity of the model Lemma 6: For the difference matrix between the average matrix and the weighted average matrix, we have for large n:

27 Main Result Theorem 2 For the MAS under control law II, let the velocity v=o(a n )> 0 and radius r > 0 be positive constants. If the proportion of the leaders satisfies M n n O 3/ 4 log 4 n then the headings of all agents will converge to 0 almost surely when the population size n is large enough. n

28 Concluding Remarks This talk: The number of leaders needed to guarantee the expected collective behavior for a class of MAS Many interesting problems: The critical number of leaders in the leader-follower model The intervention of other MAS, such as Boid model

29 Thank You!