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Swarm Intelligence on Graphs

Swarm Intelligence on Graphs. Advanced Computer Networks: Part 2. Agenda. Graph Theory (Brief) Swarm Intelligence Multi-agent Systems Consensus Protocol Example of Work. Graph Theory. Graph Theory. Graph connection: nodes and links (undirected graph: balanced digraph) Identity matrix

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Swarm Intelligence on Graphs

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  1. Swarm Intelligence on Graphs Advanced Computer Networks: Part 2

  2. Agenda • Graph Theory (Brief) • Swarm Intelligence • Multi-agent Systems • Consensus Protocol • Example of Work

  3. Graph Theory

  4. Graph Theory • Graph connection: nodes and links (undirected graph: balanced digraph) • Identity matrix • or unit matrix of size n is the n×nsquare matrix with ones on the main diagonal and zeros elsewhere • AIn= A Identity Matrix

  5. 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 n1 n2 n3 Node 1-6 n4 n5 n6 Graph Adjacency Matrix

  6. Graph Theory • Degree matrix n1 n2 n3 n4 n5 n6 n1 n2 n3 Node 1-6 n4 n5 n6 Graph Degree Matrix

  7. Graph Theory • Laplacian matrix L = Graph

  8. Swarm Behavior in Nature Collective Behavior Self-organized System

  9. Swarm Intelligence • Ant Colony OptimizationAlgorithms http://www.funpecrp.com.br/gmr/year2005/vol3-4/wob09_full_text.htm

  10. Swarm Intelligence • Ant Colony OptimizationAlgorithms • 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.

  11. Swarm Intelligence • Ant Colony OptimizationAlgorithms • 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.

  12. Swarm Intelligence • Ant Colony OptimizationAlgorithms

  13. Swarm Intelligence • Ant Colony OptimizationAlgorithms • ConstructAntSolutions: • A set of m artificial ants constructs solutions from elements of a finite set of available solution components. • ApplyLocalSearch: • 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.

  14. Swarm Intelligence • Particle Swarm OptimizationAlgorithms (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

  15. Swarm Intelligence • Particle Swarm OptimizationAlgorithms (PSO) http://www.sciencedirect.com/science/article/pii/S0960148109001232

  16. Swarm Intelligence • Application of Swarm Principles: Swarm of Robotics • http://www.youtube.com/watch?feature=player_embedded&v=rYIkgG1nX4E#! http://www.domesro.com/2008/08/swarm-robotics-for-domestic-use.html

  17. Multi-Agent Systems • Multi-agent system • Many agents: • homogeneous • heterogeneous • Interaction topology • 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

  18. 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

  19. Consensus Protocols

  20. Consensus Protocols

  21. Calculation Examination

  22. Leader-Following Discrete-time Consensus Protocol • Effective leadership and decision making in animal groups on the move 22

  23. Leader-Following Discrete-time Consensus Protocol • Leader-following consensus models • agreement of a group based on specific quantities of interest • Leader • 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

  24. W. Ren, 2007 • Multi-vehicle consensus with a time-varying reference state 1 24

  25. W. Ren, 2007 2 3 c 25

  26. Y. Cao, 2009 • Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication 4 26

  27. 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

  28. Example of Work:Leader-Following Behavior 28

  29. Proposed work: Leader-Following Behavior 6 29

  30. Leader-Following Behavior 30

  31. Leader-Following Behavior leader connects to node 1, 2, 3, 4 respectively Compared with 1 31

  32. Leader-Following Behavior 5 6 32

  33. Leader-Following Behavior 33

  34. 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 multi-agent networks 34

  35. 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. 485-508, 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

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