Swarm Intelligence

1. Swarm Intelligence By:- Omkar Thakoor - 100050009 Prakhar Jain - 100050024 Utkarsh Diwaker - 100050087

2. Overview • Swarm Intelligence : Definition • SI as a discipline of AI • Ant Colony Optimization(ACO) : • Introduction • The ACO Algorithm • ACO for subset-problem • Maximum Independent Set Problem(MISP) • Solving MISP using ACO • Summary • References

3. 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.

4. 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

5. Examples of Swarms : Ants taking prey Heards of animals

6. Examples of Swarms flocks of birds schools of fish

7. 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

8. 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.

9. 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.

10. 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.

11. ANTS Why are ants interesting? • ants solve complex tasks by simple local means • ant productivity is better than the sum of their single activities • ants are ‘grand masters’ in search and exploitation Which mechanisms are important? • cooperation and division of labour • pheromones

12. Foraging behavior of Ants • 2 ants start with equal probability of going on either path.

13. Foraging behavior of Ants • The ant on shorter path has a shorter to-and-fro time from it’s nest to the food.

14. 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).

15. Foraging behavior of Ants • The next ant takes the shorter route.

16. Foraging behavior of Ants • Over many iterations, more ants begin using the path with higher pheromone, thereby further reinforcing it.

17. Foraging behavior of Ants • After some time, the shorter path is almost exclusively used.

18. Ant Colony Optimization • Probabilistic 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.

19. The ACO Algorithm • 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

20. ACO Algorithm (continued) • For ant k , pkxy = probability of moving from state xto statey which depends on the combination of two values, viz. • TheattractivenessȠxyof the move, computed by some heuristic indicating the a priori desirability of that move and • The trail level τ xyof 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.

21. ACO Algorithm (continued) • Thekthant moves from state  to state  with probability pkxy • τ xyis 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

22. 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 • ∆τijkis the pheromone quantity laid by ant k on the edge joining node i and node j and is given by ∆τijk = Q(Lk) (Lkis the cost of the kthant's tour (typically length) and Q is a constant.)

23. The ACO Algorithm procedure ACO_MetaHeuristic while(not_termination) generateSolutions() daemonActions() pheromoneUpdate() end while end procedure

24. Ant System for subset problems • 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

25. Application of ACO to 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 itemi by the k-thant at time t This quantity is given by :- ∆τik(t) = G(Lk) , if k-th ant incorporates item i 0, otherwise {

26. (continued) • 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 respectively • M is a constant. • Lkdepends 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.

27. (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 items • τ 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

28. The maximum independent set problem (MISP) • 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.

29. MISP (continued) • 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.

30. 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

31. Example • Let us consider the following example concerning the heuristic defined above • Figure shows a small MISP instance where |V| = 8.

32. Example (continued) • 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 :

33. 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 = 4 possessing the biggest set of feasible items for the next selection step

34. A different Heuristic 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 in practice.

35. Summary • 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.

36. References •  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, NamsuAhn, 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_intelligence • http://www.scholarpedia.org/article/Swarm_intelligence