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Swarm Intelligence on Graphs

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Swarm Intelligence on Graphs

Advanced Computer Networks: Part 2

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

- 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

- or unit matrix of size n is the n×nsquare matrix with ones on the main diagonal and zeros elsewhere

Identity Matrix

- 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

- Degree matrix

n1

n2

n3

n4

n5

n6

n1

n2

n3

Node 1-6

n4

n5

n6

Graph

Degree Matrix

- Laplacian matrix

L =

Graph

Collective Behavior

Self-organized System

- Ant Colony OptimizationAlgorithms

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

- 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.

- 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.

- the Traveling Salesman Problem: An iterative algorithm

- Ant Colony OptimizationAlgorithms

- 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.

- ConstructAntSolutions:

- 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

- Particle Swarm OptimizationAlgorithms (PSO)

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

- 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

- Multi-agent system
- Many agents:
- homogeneous
- heterogeneous

- Interaction topology
- complex network

- Many agents:
- 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 problem
- A group of agents
- To make a decision
- To reach an agreement
- Depend on their shared state information
- Information exchange among the agents

- A group of 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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- Effective leadership and decision making in animal groups on the move

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

1

24

2

3

c

25

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

4

26

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

6

29

30

Compared with

1

31

5

6

32

33

- 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

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- 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.

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