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×n square matrix w

ith 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

Graph Adjacency Matrix

Node 1-6

n1 n2 n3 n4 n5 n6n1n2n3n4n5n6

6

Graph Theory

Degree matrix

Graph

n1 n2 n3 n4 n5 n6

Node 1-6

n1n2n3n4n5n6

Degree Matrix

7

Graph Theory

Laplacian matrix

Graph

L =

8

Swarm Behavior in Nature

Collective Behavior

Self-organized System

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Swarm Intelligence

Ant Colony Optimization Algorithms

http://www.funpecrp.com.br/gmr/year2005/vol3-4/wob09_full_text.htm

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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.

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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.

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Swarm Intelligence

Ant Colony Optimization Algorithms

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Swarm Intelligence Ant Colony Optimization Algorithms

ConstructAntSolutions: A set of m artificial ants constructs solutions from elements of a finite set of availab

le solution components.

ApplyLocalSearch: Once solutions have been constructed, and before updating the pheromone, it is c

ommon 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.

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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

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Swarm Intelligence

Particle Swarm Optimization Algorithms (PSO)

http://www.sciencedirect.com/science/article/pii/S0960148109001232

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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

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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?

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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

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Consensus Protocols

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Consensus Protocols

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Calculation Examination

100100021100013110111410001131000011

L

2222

Leader-Following Discrete-time Consensus Protocol Effective leadership

and decision making in animal groups on the move

2323

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

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W. Ren, 2007

Multi-vehicle consensus with a time-varying reference state

1

2525

W. Ren, 2007

2 3

c

2626

Y. Cao, 2009

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

4

2727

Y. Cao, 2009

5ζ1(0)=3, ζ2(0)=1, ζ3(0)=-1, ζ4(0)=-2

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

2828

Example of Work: Leader-Following

Behavior

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Proposed work: Leader-Following Behavior

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

1

1.5

Times

Info

rmat

ion

Sta

te

node 1node 2node 3node 4LEADER

6

3030

Leader-Following Behavior

3131

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

1Compared with

3232

Leader-Following Behavior 5

6

3333

Leader-Following Behavior

0 5 10 15 20 25 30 35 40-5

0

5

10

15

20

25

30

35

40

45

Times

Info

rmat

ion

Sta

te

node 1node 2node 3node 4LEADER

3434

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

3535

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.