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==Neighbourhoods and topologies==
The topology of the swarm defines the subset of particles with which each particle can exchange information.<ref name=kennedy2002population/> The basic version of the algorithm uses the global topology as the swarm communication structure.<ref name=bratton2007/> This topology allows all particles to communicate with all the other particles, thus the whole swarm share the same best position '''g''' from a single particle. However, this approach might lead the swarm to be trapped into a local minimum,<ref>Mendes, R. (2004). [https://pdfs.semanticscholar.org/d224/80b09d1f0759fb20e0fb0bd2de205457c8bc.pdf Population Topologies and Their Influence in Particle Swarm Performance]{{Dead link|date=August 2025 |bot=InternetArchiveBot |fix-attempted=yes }} (PhD thesis). Universidade do Minho.</ref> thus different topologies have been used to control the flow of information among particles. For instance, in local topologies, particles only share information with a subset of particles.<ref name=bratton2007/> This subset can be a geometrical one<ref>Suganthan, Ponnuthurai N. "[https://ieeexplore.ieee.org/abstract/document/785514/ Particle swarm optimiser with neighbourhood operator]." Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on. Vol. 3. IEEE, 1999.</ref> – for example "the ''m'' nearest particles" – or, more often, a social one, i.e. a set of particles that is not depending on any distance. In such cases, the PSO variant is said to be local best (vs global best for the basic PSO).
A commonly used swarm topology is the ring, in which each particle has just two neighbours, but there are many others.<ref name=bratton2007/> The topology is not necessarily static. In fact, since the topology is related to the diversity of communication of the particles,<ref name=oliveira2016communication/> some efforts have been done to create adaptive topologies (SPSO,<ref>SPSO [http://www.particleswarm.info Particle Swarm Central]</ref> APSO,<ref> Almasi, O. N. and Khooban, M. H. (2017). A parsimonious SVM model selection criterion for classification of real-world data sets via an adaptive population-based algorithm. Neural Computing and Applications, 1-9. [https://link.springer.com/article/10.1007/s00521-017-2930-y https://doi.org/10.1007/s00521-017-2930-y]</ref> stochastic star,<ref>Miranda, V., Keko, H. and Duque, Á. J. (2008). [https://repositorio.inesctec.pt/bitstream/123456789/1561/1/PS-05818.pdf Stochastic Star Communication Topology in Evolutionary Particle Swarms (EPSO)]. International Journal of Computational Intelligence Research (IJCIR), Volume 4, Number 2, pp. 105-116</ref> TRIBES,<ref>Clerc, M. (2006). Particle Swarm Optimization. ISTE (International Scientific and Technical Encyclopedia), 2006</ref> Cyber Swarm,<ref>Yin, P., Glover, F., Laguna, M., & Zhu, J. (2011). [http://leeds-faculty.colorado.edu/glover/fred%20pubs/428%20-%20A_complementary_cyber_swarm_algorithm_pub%20version%20w%20pen%20et%20al.pdf A Complementary Cyber Swarm Algorithm]. International Journal of Swarm Intelligence Research (IJSIR), 2(2), 22-41</ref> and C-PSO<ref name=elshamy07sis/>)
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===Binary, discrete, and combinatorial===
As the PSO equations given above work on real numbers, a commonly used method to solve discrete problems is to map the discrete search space to a continuous ___domain, to apply a classical PSO, and then to demap the result. Such a mapping can be very simple (for example by just using rounded values) or more sophisticated.<ref>Roy, R., Dehuri, S., & Cho, S. B. (2012). [http://sclab.yonsei.ac.kr/publications/Papers/IJ/A%20Novel%20Particle%20Swarm%20Optimization%20Algorithm%20for%20Multi-Objective%20Combinatorial%20Optimization%20Problem.pdf A Novel Particle Swarm Optimization Algorithm for Multi-Objective Combinatorial Optimization Problem] {{Webarchive|url=https://web.archive.org/web/20220120210030/http://sclab.yonsei.ac.kr/publications/Papers/IJ/A%20Novel%20Particle%20Swarm%20Optimization%20Algorithm%20for%20Multi-Objective%20Combinatorial%20Optimization%20Problem.pdf |date=2022-01-20 }}. 'International Journal of Applied Metaheuristic Computing (IJAMC)', 2(4), 41-57</ref>
However, it can be noted that the equations of movement make use of operators that perform four actions:
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<ref name="taherkhani2016inertia">{{cite journal | first1=M. | last1=Taherkhani | title=A novel stability-based adaptive inertia weight for particle swarm optimization | last2=Safabakhsh | first2=R. | journal=Applied Soft Computing | year=2016 | volume=38 | pages=281–295 | doi=10.1016/j.asoc.2015.10.004}}</ref>
<ref name="bratton2007">{{cite book | first1=Daniel | last1=Bratton | url=http://www.cil.pku.edu.cn/resources/pso_paper/src/2007SPSO.pdf | last2=Kennedy | first2=James | title=2007 IEEE Swarm Intelligence Symposium | chapter=Defining a Standard for Particle Swarm Optimization | pages=120–127 | year=2007 | doi=10.1109/SIS.2007.368035 | isbn=978-1-4244-0708-8 | s2cid=6217309 | archive-date=2016-01-27 | access-date=2016-01-22 | archive-url=https://web.archive.org/web/20160127030145/http://www.cil.pku.edu.cn/resources/pso_paper/src/2007SPSO.pdf | url-status=dead }}</ref>
<ref name="Zambrano-Bigiarini2013">{{cite book | first1=M. | last1=Zambrano-Bigiarini | last2=Clerc | first2=M. | last3=Rojas | first3=R. | title=2013 IEEE Congress on Evolutionary Computation | chapter=Standard Particle Swarm Optimisation 2011 at CEC-2013: A baseline for future PSO improvements | publisher=Evolutionary Computation (CEC), 2013 IEEE Congress on | pages=2337–2344 | year=2013| doi=10.1109/CEC.2013.6557848 | isbn=978-1-4799-0454-9 | s2cid=206553432 }}</ref>
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*[http://www.particleswarm.info Particle Swarm Central] is a repository for information on PSO. Several source codes are freely available.
*[http://vimeo.com/17407010 A brief video] of particle swarms optimizing three benchmark functions.
*[http://www.mathworks.com/matlabcentral/fileexchange/11559-particle-swarm-optimization-simulation Simulation of PSO convergence in a two-dimensional space (Matlab).] {{Webarchive|url=https://web.archive.org/web/20240414152449/http://www.mathworks.com/matlabcentral/fileexchange/11559-particle-swarm-optimization-simulation |date=2024-04-14 }}
*[http://www.vocal.com/particle-swarm-optimization/ Applications] of PSO.
*{{cite journal|doi=10.1016/j.eswa.2008.10.086|title=Automatic calibration of a rainfall–runoff model using a fast and elitist multi-objective particle swarm algorithm|journal=Expert Systems with Applications|volume=36|issue=5|pages=9533–9538|year=2009|last1=Liu|first1=Yang}}
*[http://www.adaptivebox.net/research/bookmark/psocodes_link.html Links to PSO source code] {{Webarchive|url=https://web.archive.org/web/20210415022817/http://www.adaptivebox.net/research/bookmark/psocodes_link.html |date=2021-04-15 }}
{{Major subfields of optimization}}
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