By:-Omkar Thakoor - 100050009Prakhar Jain - 100050024Utkarsh Diwaker - 100050087

Swarm Intelligence

Swarm Intelligence : DefinitionSI as a discipline of AIAnt Colony Optimization(ACO) :

IntroductionThe ACO Algorithm

ACO for subset-problemMaximum Independent Set Problem(MISP)Solving MISP using ACOSummaryReferences

Overview

What is Swarm Intelligence

• According to Bonabeau et al, it is

“The emergent collective intelligence of groups of simple agents”

• Refers to the collective behaviors that result from the local interactions of the individuals with each other and with their environment.

Swarm Intelligence (continued)

Swarm intelligence as a discipline of Artificial Intelligence, deals with natural and artificial systems composed of many individuals that coordinate using decentralized control and self-organization.

Basic Philosophy of Swarm Intelligence : Observe the behaviour of social animals and try to mimic those animals on computer systems

Examples of Swarms :

Ants taking prey Heards of animals

Examples of Swarms

flocks of birdsschools of fish

Properties of a Swarm Intelligence System

• Composed of many individuals

• the individuals are relatively homogeneous (i.e., their computing behaviour is governed by same set of rules.)

• the interactions among the individuals are based on simple behavioral rules that exploit only local information that the individuals exchange directly or via the environment

• the overall behaviour of the system results from the interactions of individuals with each other and with their environment, that is, the group behavior self-organizes

More About SI Systems• The characterizing property of a swarm

intelligence system is its ability to act in a

coordinated way without the presence of a

coordinator or of an external controller.

• Many examples can be observed in nature of

swarms that perform some collective behavior

without any individual controlling the group, or

being aware of the overall group behavior.

• Notwithstanding the lack of individuals in

charge of the group, the swarm as a whole can

show an intelligent behavior.

More about SI systems (continued)

• This is the result of the interaction of spatially

neighboring individuals that act on the basis of

simple rules.

• Most often, the behavior of each individual of

the swarm is described in probabilistic terms:

Each individual has a stochastic behavior that

depends on his local perception of the

neighborhood.

Ant Colony Optimization - Biological Inspiration

Inspired by foraging behavior of ants.Ants find shortest path to food source from

nest.Ants deposit pheromone along traveled path

which is used by other ants to follow the trail.This kind of indirect communication via the

local environment is called stigmergy.Has adaptability, robustness and redundancy.

ANTSWhy are ants interesting?ants solve complex tasks by simple local

meansant productivity is better than the sum of

their single activitiesants are ‘grand masters’ in search and

exploitation

Which mechanisms are important?cooperation and division of labourpheromones

Foraging behavior of Ants

2 ants start with equal probability of going on either path.

Foraging behavior of Ants

The ant on shorter path has a shorter to-and-fro time from it’s nest to the food.

Foraging behavior of Ants

The density of pheromone on the shorter path is higher because of 2 passes by the ant (as compared to 1 by the other).

Foraging behavior of Ants

The next ant takes the shorter route.

Foraging behavior of Ants

Over many iterations, more ants begin using the path with higher pheromone, thereby further reinforcing it.

Foraging behavior of Ants

After some time, the shorter path is almost exclusively used.

Ant Colony OptimizationProbabilistic technique for solving

computational problems which can be reduced to finding good paths through graphs.

An ant corresponds to a simple computational agent in the ACO algorithm.

It iteratively constructs a solution for the problem at hand.

The intermediate solutions are referred to as solution states.

At each iteration of the algorithm, each ant moves from a state x to state y, corresponding to a more complete intermediate solution

 Thus, each ant  computes a set  of feasible expansions to its current state in each iteration, and moves to one of these in probability

The ACO Algorithm

For ant k ,   pk

xy  = probability of moving from state x to state y

which depends on the combination of two values, viz.

The attractiveness  Ƞxy of the move, computed by some heuristic indicating the a priori desirability of that move and

The trail level  τ xy of the move, indicating how proficient it has been in the past to make that particular move.

Trails are updated usually when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively.

ACO Algorithm (continued)

 The kth ant moves from state  to state  with probability pkxy

τ xy is the  amount of pheromone deposited for transition from state x to state y

Ƞxy  is the desirability of state transition xy (a priori knowledge)0 ≤ α is a parameter to control the influence of τ xy

β ≥ 1  is a parameter to control the influence of  Ƞxy

ACO Algorithm (continued)

In each iteration, the pheromone values are updated by all the ants that have built a solution in the iteration. The pheromone τ ij on the edge joining node i and node j is updated as follows

τ ij = (1 – σ) τ ij + ∑k ∆τijk

σ is the pheromone evaporation coefficient,The summation is over the no of ants ∆τij

k is the pheromone quantity laid by ant k on the edge joining node i and node j and is given by

∆τijk = Q(Lk)

(Lk is the cost of the kth ant's tour (typically length) and Q is a constant.)

ACO Algorithm (continued)

procedure ACO_MetaHeuristicwhile(not_termination)

generateSolutions()daemonActions()pheromoneUpdate()

end while

end procedure

The ACO Algorithm

In the Ant System seen so far, the pheromone is laid on paths while for subset problems no path exists connecting the items.

A subset-based Ant System adapts the central idea in the following way: “the more pheromone on a particular item, the more profitable that item is. ”

In other words, we move the pheromone from paths to items.

For the subset problem, the Ant system considers a special type of local heuristic which takes into account both, problem knowledge and the partial solution being built by a particular ant k

Ant System for subset problems

The intensity of pheromone trail on item i at time t+1 is given by :-

τ i(t+1) = (1 – σ) τ i(t) + ∑k ∆τik(t)

∆τik(t) is the quantity of pheromone trail laid on

item i by the k-th ant at time tThis quantity is given by :-∆τi

k(t) = G(Lk) , if k-th ant incorporates item i

0 , otherwise

Application of ACO to subset problems

{

The function Q depends upon the problem and gives the amount of pheromone added to item i

Usually Q(Lk) = M/Lk or M*Lk for minimization and maximization problems respectivelyM is a constant.Lk depends on the objective.

The heuristic value for the item i ∊ S - Ŝk(t) , is defined as a function of the partial solution Ŝk(t) built by ant k at time t.

(continued)

Now, for a partial solution Ŝk (t) = {i1, …, ij} , the

probability of selecting ip as the next item (p ∊ {j+1, j+2, …, n}) is given by :-

allowedk(t) ⊆ S – Ŝk (t) is the set of remaining feasible

itemsτ ip

(t) is the amount of pheromone on item i

Ƞip(Ŝk

(t)) represents the heuristic value for item i based

on the solution being built by the k-th ant

Thus, the higher the value of τ ip(t) and Ƞip

(Ŝk (t)), the

more profitable it is to include ip in the partial solution

(continued)

The maximum independent set problem (MISP) consists of finding the largest subset of vertices of a graph such that none of them are connected by an edge (i.e., all vertices are independent of each other).

If G = <V, E> denotes a graph where V is the set of nodes and E the set of edges, the problem is to determine a set V* ⊆ V such that ∀ i,j ∊ V* the edge <i,j> ∉ E and | V* | is maximum.

The maximum independent set problem (MISP)

Let Fk(t) be the set of remaining feasible items with respect to Ŝk(t) : the solution being built by ant k at time t.

The local heuristic for the MISP can be defined as Ƞi(Ŝk

(t)) = |Fi|

where Fi represents Fk(t+1) in case item i is added to Ŝk(t)

Then the local heuristic aims at assigning higher scores to that item (say i) which yields a larger Fi. Thus, larger the value of Fk(t+1), the larger the set of remaining items for completing Ŝk after the inclusion of item i.

MISP (continued)

The probability for item selection was given previously where

allowedk(t) = V - Ŝk (t) – Uk(t)

Uk(t) = { j | ( (j,i) E ∊ ∨ (i,j) E) ∊ ∧ i ∊ Ŝk (t) } ,

i.e., the set of infeasible items with respect to Ŝk

(t).

Function Q is defined as Q(Lk) = M*Lk,

where M = 1/|v| and Lk, the objective value, is the cardinality of the set of vertices conforming the solution obtained by the ant k

MISP(continued)

Let us consider the following example concerning the heuristic defined above

Figure shows a small MISP instance where |V| = 8.

Example

Let us suppose that in time the partial solution being built by the k-th ant is Ŝk

(t) = {2}, then

Fk(t) = V – {2} – {1,8} = {3, 7, 4, 5, 6}

the set {1,8} represents the subset of infeasible items due to the inclusion of item 2 in the partial solution

Now the subset {3, 7, 4, 5, 6} is the set of current feasible items and the corresponding heuristic values are as follows :

Example (continued)

Ƞ3(Ŝk(t)) = |F3| = |{4, 5}| = 2

Ƞ4(Sk(t)) = |F4| = |{3, 6, 7}| = 3

Ƞ5(Sk(t)) = |F5| = |{3}| = 1

Ƞ6(Sk(t)) = |F6| = |{4, 7}| = 2

Ƞ7(Sk(t)) = |F7| = |{4, 6}| = 2

Therefore, the highest score is obtained by item i = 4possessing the biggest set of feasible items for the next selection step

Example (continued)

Let Ŝk(t) be the solution being built by ant k at time t.

Define Ƞv(Ŝk(t)) =

dG(v) is the degree of vertex v, and NG(v) is the neighbour set of vertex v

It can be seen that this heuristic is different from the previous in that it doesn’t depend on the solution being built.

It can be shown that higher the value of Ƞv(Ŝk(t)), better

the chances of ‘v’ being present in the optimal solution.Many other such heuristics can be and in fact are used inpractice.

A different Heuristic

Nature is very Intelligent and we can still learn a lot of intelligent things from nature

Individual agents in the group seem to have no intelligence but group as a whole shows some intelligence

The intelligence of the group come from various simple rules followed by individual agents.

Has wide variety of applications.

Summary

 G. Leguizamon, Z. Michalewicz and Martin Schutz, "An ant system for the maximum independent set problem," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001

Hwayong Choi, Namsu Ahn, Sungsoo Park, “An Ant Colony Optimization Approach for the Maximum Independent Set Problem”, Computational Intelligence and Multimedia Applications, 2003.ICCIMA 2003. Proceedings. Fifth International Conference

http://en.wikipedia.org/wiki/Swarm_intelligencehttp://www.scholarpedia.org/article/

Swarm_intelligence

References