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Swarm Intelligence on Graphs Advanced Computer Networks: Part 2 1 Agenda Graph Theory (Brief) Swarm Intelligence Multi-agent Systems Consensus Protocol Example of Work 2 Graph Theory 3 Graph Theory Graph connection: nodes and links (undirected graph: balanced digraph) Identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere AIn = A Identity Matrix 4 Graph Theory Adjacency matrix a means of representing which or nodes of a graph are adjacent to which other nodes n1 n2 n3 n4 n5 n6 Node 1-6 Graph n1 n2 n3 n4 n5 n6 Adjacency Matrix 5 Graph Theory Degree matrix n1 n2 n3 n4 n5 n6 Node 1-6 Graph n1 n2 n3 n4 n5 n6 Degree Matrix 6 Graph Theory Laplacian matrix L= Graph 7 Swarm Behavior in Nature Collective Behavior Self-organized System 8 Swarm Intelligence Ant Colony Optimization Algorithms http://www.funpecrp.com.br/gmr/year2005/vol3-4/wob09_full_text.htm 9 Swarm Intelligence Ant Colony Optimization Algorithms The Traveling Salesman Problem • A set of cities is given and the distance between each of them is known. • The goal is to find the shortest tour that allows each city to be visited once and only once. 10 Swarm Intelligence Ant Colony Optimization Algorithms the Traveling Salesman Problem: An iterative algorithm At each iteration, a number of artificial ants are considered. Each of them builds a solution by walking from node to node on the graph with the constraint of not visiting any vertex that she has already visited in her walk. An ant selects the following node to be visited according to a stochastic mechanism that is biased by the pheromone: when in node i, the following node is selected stochastically among the previously unvisited ones if j has not been previously visited, it can be selected with a probability that is proportional to the pheromone associated with edge (i, j). the pheromone values are modified in order to bias ants in future iterations to construct solutions similar to the best ones previously constructed. 11 Swarm Intelligence Ant Colony Optimization Algorithms 12 Swarm Intelligence Ant Colony Optimization Algorithms ConstructAntSolutions: ApplyLocalSearch: A set of m artificial ants constructs solutions from elements of a finite set of available solution components. Once solutions have been constructed, and before updating the pheromone, it is common to improve the solutions obtained by the ants through a local search. UpdatePheromones: The aim of the pheromone update is to increase the pheromone values associated with good or promising solutions, and to decrease those that are associated with bad ones. Usually, this is achieved by decreasing all the pheromone values through pheromone evaporation by increasing the pheromone levels associated with a chosen set of good solutions. 13 Swarm Intelligence Particle Swarm Optimization Algorithms (PSO) PSO emulates the swarm behavior of insects, animals herding, birds flocking, and fish schooling where these swarms search for food in a collaborative manner. Each member in the swarm adapts its search patterns by learning from its own experience and other members’ experiences. A member in the swarm, called a particle, represents a potential solution which is a point in the search space. The global optimum is regarded as the location of food. Each particle has a fitness value and a velocity to adjust its flying direction according to the bestexperiences of the swarm to search for the global optimum in the solution space. http://science.howstuffworks.com/environmental/life/zoology/ insects-arachnids/termite3.htm 14 Swarm Intelligence Particle Swarm Optimization Algorithms (PSO) http://www.sciencedirect.com/science/article/pii/S09 60148109001232 15 Swarm Intelligence Application of Swarm Principles: Swarm of Robotics http://www.domesro.com/2008/08/swarm-robotics-for-domestic-use.html http://www.youtube.com/watch?feature=playe r_embedded&v=rYIkgG1nX4E#! 16 Multi-Agent Systems Multi-agent system Many agents: Interaction topology homogeneous heterogeneous complex network How to control the global behavior of the multi-agent system? How to apply the proposed model to solve the realistic problem? 17 Consensus Protocols Consensus problem A group of agents To make a decision To reach an agreement Depend on their shared state information Information exchange among the agents To design a suitable protocol for the group to reach a consensus Shared information among agents is converged to the group decision value but do not allow to reach a particular value 18 Consensus Protocols 19 Consensus Protocols 20 Calculation Examination 1 1 0 0 0 0 1 3 1 1 0 0 0 1 4 1 1 1 L 0 1 1 3 1 0 0 0 1 1 2 0 0 0 1 0 0 1 21 Leader-Following Discrete-time Consensus Protocol Effective leadership and decision making in animal groups on the move 22 Leader-Following Discrete-time Consensus Protocol Leader-following consensus models Leader agreement of a group based on specific quantities of interest an external input to control the global behavior of the system determine the final state of the system unaffected by the followers send the information to the followers only Followers reach consensus following the leader's state influenced by the leader directly no feedback information from the followers to the leader 23 W. Ren, 2007 Multi-vehicle consensus with a time-varying reference state 1 24 W. Ren, 2007 2 3 c 25 Y. Cao, 2009 Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication 4 26 Y. Cao, 2009 5 ζ1(0)=3, ζ2(0)=1, ζ3(0)=-1, ζ4(0)=-2 ζ1(-1)=0, ζ2(-1)=0, ζ3(-1)=0, ζ4(-1)=0 27 Example of Work: Leader-Following Behavior 28 28 Proposed work: Leader-Following Behavior 1.5 node 1 node 2 node 3 node 4 LEADER Information State 1 0.5 0 -0.5 0 5 10 15 20 25 Times 30 35 40 45 6 29 50 Leader-Following Behavior 30 Leader-Following Behavior leader connects to node 1, 2, 3, 4 respectively Compared with 1 31 Leader-Following Behavior 5 6 32 Leader-Following Behavior 45 node 1 node 2 node 3 node 4 LEADER 40 35 Information State 30 25 20 15 10 5 0 -5 0 5 10 15 20 Times 25 30 35 40 33 Further Work Large scale multi-agent networks with dynamical topologies Partial information exchange between followers and leader How to identify the leader? How the leader control the group behavior? Consensus on large scale multiagent networks 34 References www.wikipedia.com Marco Dorigo, Mauro Birattari, and Thomas St¨utzle, “Ant Colony Optimization”, IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE, NOVEMBER, 2006. J. J. Liang, A. K. Qin, “Comprehensive Learning Particle Swarm Optimizer for Global Optimization of Multimodal Functions”, IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 10, NO. 3, JUNE 2006. J. A. Fax and R. M. Murray, "Information flow and cooperative control of vehicle formations," IEEE Trans. Autom. Control, vol. 49, pp.1465-1476, 2004. D. B. Kingston, R. W. Beard, "Discrete-time average-consensus under switching network topologies," in Proc. American Control Conf.,14-16 June 2006. W. Ren, "Multi-vehicle consensus with a time -varying reference state, “Systems & Control Letters, vol. 56, pp. 474-483, 2007. Y. Cao, W. Ren, Y. Li, "Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication," Automatica, vol. 45, pp. 1299-1305, 2009. J. Hu, Y. Hong, "Leader-follower coordination of multi-agent systems with coupling time delays," Physica A: Statistical Mechanics and its Applications., vol. 374, iss. 2, pp.853-863, 2007. D. Bauso, L. Giarr'e, R. Pesenti, "Distributed consensus protocols for coordinating buyers," Proc. IEEE Decision and Control Conf., December, 2003. R. E. Kranton, D. F. Minehart, "A theory of buyer-seller networks," The American Economic Review, vol. 91, no. 3, pp. 485508, 2001. I.D. Couzin, J. Krause, N.R. Franks, S. A. Levin, “Effective leadership and decision making in animal groups on the move,” Nature, iss. 433, pp. 513-516, 2005. R.O. Saber, R.M. Murray, “Flocking with obstacle avoidance: cooperation with limited communication in mobile networks,” in Proc. IEEE Decision and Control Conf., vol.2, pp. 2022-2028, 2003. E. Semsar-Kazerooni, K. Khorasani, “Optimal consensus algorithms for cooperative team of agents subject to partial information,” Automatica, 2008. J. Zhou, W. Yu, X. Wu, M. Small, J. Lu, “Flocking of multi-agent dynamical systems based on pseudo-leader mechanism,” Chaotic Dynamics, 2009. 35