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Complete circuit diagram with pheromone - Cristian Jimenez-Romero, David Sousa-Rodrigues, Jeffrey H. Johnson, Vitorino Ramos; Figure – Neural circuit controller of the virtual ant (page 3, fig. 2). [URL: http://arxiv.org/abs/1507.08467 ]

Intelligence and decision in foraging ants. Individual or Collective? Internal or External? What is the right balance between the two. Can one have internal intelligence without external intelligence? Can one take examples from nature to build in silico artificial lives that present us with interesting patterns? We explore a model of foraging ants in this paper that will be presented in early September in Exeter, UK, at UKCI 2015. (available on arXiv [PDF] and ResearchGate)

Cristian Jimenez-Romero, David Sousa-Rodrigues, Jeffrey H. Johnson, Vitorino Ramos; “A Model for Foraging Ants, Controlled by Spiking Neural Networks and Double Pheromones“, UKCI 2015 Computational Intelligence – University of Exeter, UK, September 2015.

Abstract: A model of an Ant System where ants are controlled by a spiking neural circuit and a second order pheromone mechanism in a foraging task is presented. A neural circuit is trained for individual ants and subsequently the ants are exposed to a virtual environment where a swarm of ants performed a resource foraging task. The model comprises an associative and unsupervised learning strategy for the neural circuit of the ant. The neural circuit adapts to the environment by means of classical conditioning. The initially unknown environment includes different types of stimuli representing food (rewarding) and obstacles (harmful) which, when they come in direct contact with the ant, elicit a reflex response in the motor neural system of the ant: moving towards or away from the source of the stimulus. The spiking neural circuits of the ant is trained to identify food and obstacles and move towards the former and avoid the latter. The ants are released on a landscape with multiple food sources where one ant alone would have difficulty harvesting the landscape to maximum efficiency. In this case the introduction of a double pheromone mechanism (positive and negative reinforcement feedback) yields better results than traditional ant colony optimization strategies. Traditional ant systems include mainly a positive reinforcement pheromone. This approach uses a second pheromone that acts as a marker for forbidden paths (negative feedback). This blockade is not permanent and is controlled by the evaporation rate of the pheromones. The combined action of both pheromones acts as a collective stigmergic memory of the swarm, which reduces the search space of the problem. This paper explores how the adaptation and learning abilities observed in biologically inspired cognitive architectures is synergistically enhanced by swarm optimization strategies. The model portraits two forms of artificial intelligent behaviour: at the individual level the spiking neural network is the main controller and at the collective level the pheromone distribution is a map towards the solution emerged by the colony. The presented model is an important pedagogical tool as it is also an easy to use library that allows access to the spiking neural network paradigm from inside a Netlogo—a language used mostly in agent based modelling and experimentation with complex systems.

References:

[1] C. G. Langton, “Studying artificial life with cellular automata,” Physica D: Nonlinear Phenomena, vol. 22, no. 1–3, pp. 120 – 149, 1986, proceedings of the Fifth Annual International Conference. [Online]. Available: http://www.sciencedirect.com/ science/article/pii/016727898690237X
[2] A. Abraham and V. Ramos, “Web usage mining using artificial ant colony clustering and linear genetic programming,” in Proceedings of the Congress on Evolutionary Computation. Australia: IEEE Press, 2003, pp. 1384–1391.
[3] V. Ramos, F. Muge, and P. Pina, “Self-organized data and image retrieval as a consequence of inter-dynamic synergistic relationships in artificial ant colonies,” Hybrid Intelligent Systems, vol. 87, 2002.
[4] V. Ramos and J. J. Merelo, “Self-organized stigmergic document maps: Environment as a mechanism for context learning,” in Proceddings of the AEB, Merida, Spain, February 2002. ´
[5] D. Sousa-Rodrigues and V. Ramos, “Traversing news with ant colony optimisation and negative pheromones,” in European Conference in Complex Systems, Lucca, Italy, Sep 2014.
[6] E. Bonabeau, G. Theraulaz, and M. Dorigo, Swarm Intelligence: From Natural to Artificial Systems, 1st ed., ser. Santa Fe Insitute Studies In The Sciences of Complexity. 198 Madison Avenue, New York: Oxford University Press, USA, Sep. 1999.
[7] M. Dorigo and L. M. Gambardella, “Ant colony system: A cooperative learning approach to the traveling salesman problem,” Universite Libre de Bruxelles, Tech. Rep. TR/IRIDIA/1996-5, ´ 1996.
[8] M. Dorigo, G. Di Caro, and L. M. Gambardella, “Ant algorithms for discrete optimization,” Artif. Life, vol. 5, no. 2, pp. 137– 172, Apr. 1999. [Online]. Available: http://dx.doi.org/10.1162/ 106454699568728
[9] L. M. Gambardella and M. Dorigo, “Ant-q: A reinforcement learning approach to the travelling salesman problem,” in Proceedings of the ML-95, Twelfth Intern. Conf. on Machine Learning, M. Kaufman, Ed., 1995, pp. 252–260.
[10] A. Gupta, V. Nagarajan, and R. Ravi, “Approximation algorithms for optimal decision trees and adaptive tsp problems,” in Proceedings of the 37th international colloquium conference on Automata, languages and programming, ser. ICALP’10. Berlin, Heidelberg: Springer-Verlag, 2010, pp. 690–701. [Online]. Available: http://dl.acm.org/citation.cfm?id=1880918.1880993
[11] V. Ramos, D. Sousa-Rodrigues, and J. Louçã, “Second order ˜ swarm intelligence,” in HAIS’13. 8th International Conference on Hybrid Artificial Intelligence Systems, ser. Lecture Notes in Computer Science, J.-S. Pan, M. Polycarpou, M. Wozniak, A. Carvalho, ´ H. Quintian, and E. Corchado, Eds. Salamanca, Spain: Springer ´ Berlin Heidelberg, Sep 2013, vol. 8073, pp. 411–420.
[12] W. Maass and C. M. Bishop, Pulsed Neural Networks. Cambridge, Massachusetts: MIT Press, 1998.
[13] E. M. Izhikevich and E. M. Izhikevich, “Simple model of spiking neurons.” IEEE transactions on neural networks / a publication of the IEEE Neural Networks Council, vol. 14, no. 6, pp. 1569–72, 2003. [Online]. Available: http://www.ncbi.nlm.nih. gov/pubmed/18244602
[14] C. Liu and J. Shapiro, “Implementing classical conditioning with spiking neurons,” in Artificial Neural Networks ICANN 2007, ser. Lecture Notes in Computer Science, J. de S, L. Alexandre, W. Duch, and D. Mandic, Eds. Springer Berlin Heidelberg, 2007, vol. 4668, pp. 400–410. [Online]. Available: http://dx.doi.org/10.1007/978-3-540-74690-4 41
[15] J. Haenicke, E. Pamir, and M. P. Nawrot, “A spiking neuronal network model of fast associative learning in the honeybee,” Frontiers in Computational Neuroscience, no. 149, 2012. [Online]. Available: http://www.frontiersin.org/computational neuroscience/10.3389/conf.fncom.2012.55.00149/full
[16] L. I. Helgadottir, J. Haenicke, T. Landgraf, R. Rojas, and M. P. Nawrot, “Conditioned behavior in a robot controlled by a spiking neural network,” in International IEEE/EMBS Conference on Neural Engineering, NER, 2013, pp. 891–894.
[17] A. Cyr and M. Boukadoum, “Classical conditioning in different temporal constraints: an STDP learning rule for robots controlled by spiking neural networks,” pp. 257–272, 2012.
[18] X. Wang, Z. G. Hou, F. Lv, M. Tan, and Y. Wang, “Mobile robots’ modular navigation controller using spiking neural networks,” Neurocomputing, vol. 134, pp. 230–238, 2014.
[19] C. Hausler, M. P. Nawrot, and M. Schmuker, “A spiking neuron classifier network with a deep architecture inspired by the olfactory system of the honeybee,” in 2011 5th International IEEE/EMBS Conference on Neural Engineering, NER 2011, 2011, pp. 198–202.
[20] U. Wilensky, “Netlogo,” Evanston IL, USA, 1999. [Online]. Available: http://ccl.northwestern.edu/netlogo/
[21] C. Jimenez-Romero and J. Johnson, “Accepted abstract: Simulation of agents and robots controlled by spiking neural networks using netlogo,” in International Conference on Brain Engineering and Neuro-computing, Mykonos, Greece, Oct 2015.
[22] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge: Cambridge University Press, 2002.
[23] J. v. H. W Gerstner, R Kempter and H. Wagner, “A neuronal learning rule for sub-millisecond temporal coding,” Nature, vol. 386, pp. 76–78, 1996.
[24] I. P. Pavlov, “Conditioned reflexes: An investigation of the activity of the cerebral cortex,” New York, 1927.
[25] E. J. H. Robinson, D. E. Jackson, M. Holcombe, and F. L. W. Ratnieks, “Insect communication: ‘no entry’ signal in ant foraging,” Nature, vol. 438, no. 7067, pp. 442–442, 11 2005. [Online]. Available: http://dx.doi.org/10.1038/438442a
[26] E. J. Robinson, D. Jackson, M. Holcombe, and F. L. Ratnieks, “No entry signal in ant foraging (hymenoptera: Formicidae): new insights from an agent-based model,” Myrmecological News, vol. 10, no. 120, 2007.
[27] D. Sousa-Rodrigues, J. Louçã, and V. Ramos, “From standard ˜ to second-order swarm intelligence phase-space maps,” in 8th European Conference on Complex Systems, S. Thurner, Ed., Vienna, Austria, Sep 2011.
[28] V. Ramos, D. Sousa-Rodrigues, and J. Louçã, “Spatio-temporal ˜ dynamics on co-evolved stigmergy,” in 8th European Conference on Complex Systems, S. Thurner, Ed., Vienna, Austria, 9 2011.
[29] S. Tisue and U. Wilensky, “Netlogo: A simple environment for modeling complexity,” in International conference on complex systems. Boston, MA, 2004, pp. 16–21.

David MS Rodrigues Reading the News Through its Structure New Hybrid Connectivity Based ApproachesFigure – Two simplicies a and b connected by the 2-dimensional face, the triangle {1;2;3}. In the analysis of the time-line of The Guardian newspaper (link) the system used feature vectors based on frequency of words and them computed similarity between documents based on those feature vectors. This is a purely statistical approach that requires great computational power and that is difficult for problems that have large feature vectors and many documents. Feature vectors with 100,000 or more items are common and computing similarities between these documents becomes cumbersome. Instead of computing distance (or similarity) matrices between documents from feature vectors, the present approach explores the possibility of inferring the distance between documents from the Q-analysis description. Q-analysis is a very natural notion of connectivity between the simplicies of the structure and in the relation studied, documents are connected to each other through shared sets of tags entered by the journalists. Also in this framework, eccentricity is defined as a measure of the relatedness of one simplex in relation to another [7].

David M.S. Rodrigues and Vitorino Ramos, “Traversing News with Ant Colony Optimisation and Negative Pheromones” [PDF], accepted as preprint for oral presentation at the European Conference on Complex SystemsECCS14 in Lucca, Sept. 22-26, 2014, Italy.

Abstract: The past decade has seen the rapid development of the online newsroom. News published online are the main outlet of news surpassing traditional printed newspapers. This poses challenges to the production and to the consumption of those news. With those many sources of information available it is important to find ways to cluster and organise the documents if one wants to understand this new system. Traditional approaches to the problem of clustering documents usually embed the documents in a suitable similarity space. Previous studies have reported on the impact of the similarity measures used for clustering of textual corpora [1]. These similarity measures usually are calculated for bag of words representations of the documents. This makes the final document-word matrix high dimensional. Feature vectors with more than 10,000 dimensions are common and algorithms have severe problems with the high dimensionality of the data. A novel bio inspired approach to the problem of traversing the news is presented. It finds Hamiltonian cycles over documents published by the newspaper The Guardian. A Second Order Swarm Intelligence algorithm based on Ant Colony Optimisation was developed [2, 3] that uses a negative pheromone to mark unrewarding paths with a “no-entry” signal. This approach follows recent findings of negative pheromone usage in real ants [4].

In this case study the corpus of data is represented as a bipartite relation between documents and keywords entered by the journalists to characterise the news. A new similarity measure between documents is presented based on the Q-analysis description [5, 6, 7] of the simplicial complex formed between documents and keywords. The eccentricity between documents (two simplicies) is then used as a novel measure of similarity between documents. The results prove that the Second Order Swarm Intelligence algorithm performs better in benchmark problems of the travelling salesman problem, with faster convergence and optimal results. The addition of the negative pheromone as a non-entry signal improves the quality of the results. The application of the algorithm to the corpus of news of The Guardian creates a coherent navigation system among the news. This allows the users to navigate the news published during a certain period of time in a semantic sequence instead of a time sequence. This work as broader application as it can be applied to many cases where the data is mapped to bipartite relations (e.g. protein expressions in cells, sentiment analysis, brand awareness in social media, routing problems), as it highlights the connectivity of the underlying complex system.

Keywords: Self-Organization, Stigmergy, Co-Evolution, Swarm Intelligence, Dynamic Optimization, Foraging, Cooperative Learning, Hamiltonian cycles, Text Mining, Textual Corpora, Information Retrieval, Knowledge Discovery, Sentiment Analysis, Q-Analysis, Data Mining, Journalism, The Guardian.

References:

[1] Alexander Strehl, Joydeep Ghosh, and Raymond Mooney. Impact of similarity measures on web-page clustering.  In Workshop on Artifcial Intelligence for Web Search (AAAI 2000), pages 58-64, 2000.
[2] David M. S. Rodrigues, Jorge Louçã, and Vitorino Ramos. From standard to second-order Swarm Intelligence  phase-space maps. In Stefan Thurner, editor, 8th European Conference on Complex Systems, Vienna, Austria,  9 2011.
[3] Vitorino Ramos, David M. S. Rodrigues, and Jorge Louçã. Second order Swarm Intelligence. In Jeng-Shyang  Pan, Marios M. Polycarpou, Micha l Wozniak, André C.P.L.F. Carvalho, Hector Quintian, and Emilio Corchado,  editors, HAIS’13. 8th International Conference on Hybrid Artificial Intelligence Systems, volume 8073 of Lecture  Notes in Computer Science, pages 411-420. Springer Berlin Heidelberg, Salamanca, Spain, 9 2013.
[4] Elva J.H. Robinson, Duncan Jackson, Mike Holcombe, and Francis L.W. Ratnieks. No entry signal in ant  foraging (hymenoptera: Formicidae): new insights from an agent-based model. Myrmecological News, 10(120), 2007.
[5] Ronald Harry Atkin. Mathematical Structure in Human A ffairs. Heinemann Educational Publishers, 48 Charles  Street, London, 1 edition, 1974.
[6] J. H. Johnson. A survey of Q-analysis, part 1: The past and present. In Proceedings of the Seminar on Q-analysis  and the Social Sciences, Universty of Leeds, 9 1983.
[7] David M. S. Rodrigues. Identifying news clusters using Q-analysis and modularity. In Albert Diaz-Guilera,  Alex Arenas, and Alvaro Corral, editors, Proceedings of the European Conference on Complex Systems 2013, Barcelona, 9 2013.

In order to solve hard combinatorial optimization problems (e.g. optimally scheduling students and teachers along a week plan on several different classes and classrooms), one way is to computationally mimic how ants forage the vicinity of their habitats searching for food. On a myriad of endless possibilities to find the optimal route (minimizing the travel distance), ants, collectively emerge the solution by using stigmergic signal traces, or pheromones, which also dynamically change under evaporation.

Current algorithms, however, make only use of a positive feedback type of pheromone along their search, that is, if they collectively visit a good low-distance route (a minimal pseudo-solution to the problem) they tend to reinforce that signal, for their colleagues. Nothing wrong with that, on the contrary, but no one knows however if a lower-distance alternative route is there also, just at the corner. On his global search endeavour, like a snowballing effect, positive feedbacks tend up to give credit to the exploitation of solutions but not on the – also useful – exploration side. The upcoming potential solutions can thus get crystallized, and freeze, while a small change on some parts of the whole route, could on the other-hand successfully increase the global result.

Influence of Negative Pheromone in Swarm IntelligenceFigure – Influence of negative pheromone on kroA100.tsp problem (fig.1 – page 6) (values on lines represent 1-ALPHA). A typical standard ACS (Ant Colony System) is represented here by the line with value 0.0, while better results could be found by our approach, when using positive feedbacks (0.95) along with negative feedbacks (0.05). Not only we obtain better results, as we found them earlier.

There is, however, an advantage when a second type of pheromone (a negative feedback one) co-evolves with the first type. And we decided to research for his impact. What we found out, is that by using a second type of global feedback, we can indeed increase a faster search while achieving better results. In a way, it’s like using two different types of evaporative traffic lights, in green and red, co-evolving together. And as a conclusion, we should indeed use a negative no-entry signal pheromone. In small amounts (0.05), but use it. Not only this prevents the whole system to freeze on some solutions, to soon, as it enhances a better compromise on the search space of potential routes. The pre-print article is available here at arXiv. Follows the abstract and keywords:

Vitorino Ramos, David M. S. Rodrigues, Jorge Louçã, “Second Order Swarm Intelligence” [PDF], in Hybrid Artificial Intelligent Systems, Lecture Notes in Computer Science, Springer-Verlag, Volume 8073, pp. 411-420, 2013.

Abstract: An artificial Ant Colony System (ACS) algorithm to solve general purpose combinatorial Optimization Problems (COP) that extends previous AC models [21] by the inclusion of a negative pheromone, is here described. Several Travelling Salesman Problem‘s (TSP) were used as benchmark. We show that by using two different sets of pheromones, a second-order co-evolved compromise between positive and negative feedbacks achieves better results than single positive feedback systems. The algorithm was tested against known NP complete combinatorial Optimization Problems, running on symmetrical TSPs. We show that the new algorithm compares favourably against these benchmarks, accordingly to recent biological findings by Robinson [26,27], and Grüter [28] where “No entry” signals and negative feedback allows a colony to quickly reallocate the majority of its foragers to superior food patches. This is the first time an extended ACS algorithm is implemented with these successful characteristics.

Keywords: Self-Organization, Stigmergy, Co-Evolution, Swarm Intelligence, Dynamic Optimization, Foraging, Cooperative Learning, Combinatorial Optimization problems, Symmetrical Travelling Salesman Problems (TSP).

Hybrid Artificial Intelligent Systems HAIS 2013 (pp. 411-420 Second Order Swarm Intelligence)Figure – Hybrid Artificial Intelligent Systems new LNAI (Lecture Notes on Artificial Intelligence) series volume 8073, Springer-Verlag Book [original photo by my colleague David M.S. Rodrigues].

New work, new book. Last week one of our latest works come out published on Springer. Edited by Jeng-Shyang Pan, Marios M. Polycarpou, Emilio Corchado et al. “Hybrid Artificial Intelligent Systems” comprises a full set of new papers on this hybrid area on Intelligent Computing (check the full articles list at Springer). Our new paper “Second Order Swarm Intelligence” (pp. 411-420, Springer books link) was published on the Bio-inspired Models and Evolutionary Computation section.

ECCS11 Spatio-Temporal Dynamics on Co-Evolved Stigmergy Vitorino Ramos David M.S. Rodrigues Jorge Louçã

Ever tried to solve a problem where its own problem statement is changing constantly? Have a look on our approach:

Vitorino Ramos, David M.S. Rodrigues, Jorge LouçãSpatio-Temporal Dynamics on Co-Evolved Stigmergy“, in European Conference on Complex Systems, ECCS’11, Vienna, Austria, Sept. 12-16 2011.

Abstract: Research over hard NP-complete Combinatorial Optimization Problems (COP’s) has been focused in recent years, on several robust bio-inspired meta-heuristics, like those involving Evolutionary Computation (EC) algorithmic paradigms. One particularly successful well-know meta-heuristic approach is based on Swarm Intelligence (SI), i.e., the self-organized stigmergic-based property of a complex system whereby the collective behaviors of (unsophisticated) entities interacting locally with their environment cause coherent functional global patterns to emerge. This line of research recognized as Ant Colony Optimization (ACO), uses a set of stochastic cooperating ant-like agents to find good solutions, using self-organized stigmergy as an indirect form of communication mediated by artificial pheromone, whereas agents deposit pheromone-signs on the edges of the problem-related graph complex network, encompassing a family of successful algorithmic variations such as: Ant Systems (AS), Ant Colony Systems (ACS), Max-Min Ant Systems (MaxMin AS) and Ant-Q.

Albeit being extremely successful these algorithms mostly rely on positive feedback’s, causing excessive algorithmic exploitation over the entire combinatorial search space. This is particularly evident over well known benchmarks as the symmetrical Traveling Salesman Problem (TSP). Being these systems comprised of a large number of frequently similar components or events, the principal challenge is to understand how the components interact to produce a complex pattern feasible solution (in our case study, an optimal robust solution for hard NP-complete dynamic TSP-like combinatorial problems). A suitable approach is to first understand the role of two basic modes of interaction among the components of Self-Organizing (SO) Swarm-Intelligent-like systems: positive and negative feedback. While positive feedback promotes a snowballing auto-catalytic effect (e.g. trail pheromone upgrading over the network; exploitation of the search space), taking an initial change in a system and reinforcing that change in the same direction as the initial deviation (self-enhancement and amplification) allowing the entire colony to exploit some past and present solutions (environmental dynamic memory), negative feedback such as pheromone evaporation ensure that the overall learning system does not stables or freezes itself on a particular configuration (innovation; search space exploration). Although this kind of (global) delayed negative feedback is important (evaporation), for the many reasons given above, there is however strong assumptions that other negative feedbacks are present in nature, which could also play a role over increased convergence, namely implicit-like negative feedbacks. As in the case for positive feedbacks, there is no reason not to explore increasingly distributed and adaptive algorithmic variations where negative feedback is also imposed implicitly (not only explicitly) over each network edge, while the entire colony seeks for better answers in due time.

In order to overcome this hard search space exploitation-exploration compromise, our present algorithmic approach follows the route of very recent biological findings showing that forager ants lay attractive trail pheromones to guide nest mates to food, but where, the effectiveness of foraging networks were improved if pheromones could also be used to repel foragers from unrewarding routes. Increasing empirical evidences for such a negative trail pheromone exists, deployed by Pharaoh’s ants (Monomorium pharaonis) as a ‘no entry‘ signal to mark unrewarding foraging paths. The new algorithm comprises a second order approach to Swarm Intelligence, as pheromone-based no entry-signals cues, were introduced, co-evolving with the standard pheromone distributions (collective cognitive maps) in the aforementioned known algorithms.

To exhaustively test his adaptive response and robustness, we have recurred to different dynamic optimization problems. Medium-size and large-sized dynamic TSP problems were created. Settings and parameters such as, environmental upgrade frequencies, landscape changing or network topological speed severity, and type of dynamic were tested. Results prove that the present co-evolved two-type pheromone swarm intelligence algorithm is able to quickly track increasing swift changes on the dynamic TSP complex network, compared to standard algorithms.

Keywords: Self-Organization, Stigmergy, Co-Evolution, Swarm Intelligence, Dynamic Optimization, Foraging, Cooperative Learning, Combinatorial Optimization problems, Dynamical Symmetrical Traveling Salesman Problems (TSP).


Fig. – Recovery times over several dynamical stress tests at the fl1577 TSP problem (1577 node graph) – 460 iter max – Swift changes at every 150 iterations (20% = 314 nodes, 40% = 630 nodes, 60% = 946 nodes, 80% = 1260 nodes, 100% = 1576 nodes). [click to enlarge]

ECCS11 From Standard to Second Order Swarm Intelligence Phase-Space Maps David Rodrigues Jorge Louçã Vitorino Ramos

David M.S. Rodrigues, Jorge Louçã, Vitorino Ramos, “From Standard to Second Order Swarm Intelligence Phase-space maps“, in European Conference on Complex Systems, ECCS’11, Vienna, Austria, Sept. 12-16 2011.

Abstract: Standard Stigmergic approaches to Swarm Intelligence encompasses the use of a set of stochastic cooperating ant-like agents to find optimal solutions, using self-organized Stigmergy as an indirect form of communication mediated by a singular artificial pheromone. Agents deposit pheromone-signs on the edges of the problem-related graph to give rise to a family of successful algorithmic approaches entitled Ant Systems (AS), Ant Colony Systems (ACS), among others. These mainly rely on positive feedback’s, to search for an optimal solution in a large combinatorial space. The present work shows how, using two different sets of pheromones, a second-order co-evolved compromise between positive and negative feedback’s achieves better results than single positive feedback systems. This follows the route of very recent biological findings showing that forager ants, while laying attractive trail pheromones to guide nest mates to food, also gained foraging effectiveness by the use of pheromones that repelled foragers from unrewarding routes. The algorithm presented here takes inspiration precisely from this biological observation.

The new algorithm was exhaustively tested on a series of well-known benchmarks over hard NP-complete Combinatorial Optimization Problems (COP’s), running on symmetrical Traveling Salesman Problems (TSP). Different network topologies and stress tests were conducted over low-size TSP’s (eil51.tsp; eil78.tsp; kroA100.tsp), medium-size (d198.tsp; lin318.tsp; pcb442.tsp; att532.tsp; rat783.tsp) as well as large sized ones (fl1577.tsp; d2103.tsp) [numbers here referring to the number of nodes in the network]. We show that the new co-evolved stigmergic algorithm compared favorably against the benchmark. The algorithm was able to equal or majorly improve every instance of those standard algorithms, not only in the realm of the Swarm Intelligent AS, ACS approach, as in other computational paradigms like Genetic Algorithms (GA), Evolutionary Programming (EP), as well as SOM (Self-Organizing Maps) and SA (Simulated Annealing). In order to deeply understand how a second co-evolved pheromone was useful to track the collective system into such results, a refined phase-space map was produced mapping the pheromones ratio between a pure Ant Colony System (where no negative feedback besides pheromone evaporation is present) and the present second-order approach. The evaporation rate between different pheromones was also studied and its influence in the outcomes of the algorithm is shown. A final discussion on the phase-map is included. This work has implications in the way large combinatorial problems are addressed as the double feedback mechanism shows improvements over the single-positive feedback mechanisms in terms of convergence speed and on major results.

Keywords: Stigmergy, Co-Evolution, Self-Organization, Swarm Intelligence, Foraging, Cooperative Learning, Combinatorial Optimization problems, Symmetrical Traveling Salesman Problems (TSP), phase-space.

Fig. – Comparing convergence results between Standard algorithms vs. Second Order Swarm Intelligence, over TSP fl1577 (click to enlarge).

[...] People should learn how to play Lego with their minds. Concepts are building bricks [...] V. Ramos, 2002.

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