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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

Swarm Intelligence on Graphs

Advanced Computer Networks: Part 2


  • Graph Theory (Brief)

  • Swarm Intelligence

  • Multi-agent Systems

  • Consensus Protocol

  • Example of Work

Graph theory1
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

Graph theory2
Graph Theory

  • Adjacency matrix

    • a means of representing which or nodes of a graph are adjacent to which other nodes










Node 1-6





Adjacency Matrix

Graph theory3
Graph Theory

  • Degree matrix










Node 1-6





Degree Matrix

Graph theory4
Graph Theory

  • Laplacian matrix

L =


Swarm behavior in nature
Swarm Behavior in Nature

Collective Behavior

Self-organized System

Swarm intelligence
Swarm Intelligence

  • Ant Colony OptimizationAlgorithms


Swarm intelligence1
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.

Swarm intelligence2
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.

Swarm intelligence3
Swarm Intelligence

  • Ant Colony OptimizationAlgorithms

Swarm intelligence4
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.

Swarm intelligence5
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.


Swarm intelligence6
Swarm Intelligence

  • Particle Swarm OptimizationAlgorithms (PSO)


Swarm intelligence7
Swarm Intelligence

  • Application of Swarm Principles: Swarm of Robotics

  • http://www.youtube.com/watch?feature=player_embedded&v=rYIkgG1nX4E#!


Multi agent systems
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?


Consensus protocols
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


Leader following discrete time consensus protocol
Leader-Following Discrete-time Consensus Protocol

  • Effective leadership and decision making in animal groups on the move


Leader following discrete time consensus protocol1
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


W ren 2007
W. Ren, 2007

  • Multi-vehicle consensus with a time-varying reference state



W ren 20071
W. Ren, 2007





Y cao 2009
Y. Cao, 2009

  • Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication



Y cao 20091
Y. Cao, 2009


ζ1(0)=3, ζ2(0)=1, ζ3(0)=-1, ζ4(0)=-2

ζ1(-1)=0, ζ2(-1)=0, ζ3(-1)=0, ζ4(-1)=0


Example of work leader following behavior
Example of Work:Leader-Following Behavior


Leader following behavior leader connects to node 1 2 3 4 respectively
Leader-Following Behavior leader connects to node 1, 2, 3, 4 respectively

Compared with



Further work
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



  • 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.