{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T21:07:12Z","timestamp":1761599232970,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,9]],"date-time":"2022-04-09T00:00:00Z","timestamp":1649462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the field of data science and data mining, the problem associated with clustering features and determining its optimum number is still under research consideration. This paper presents a new 2D clustering algorithm based on a mathematical topological theory that uses a pseudometric space and takes into account the local and global topological properties of the data to be clustered. Taking into account cluster symmetry property, from a metric and mathematical-topological point of view, the analysis was carried out only in the positive region, reducing the number of calculations in the clustering process. The new clustering theory is inspired by the thermodynamics principle of energy. Thus, both topologies are recursively taken into account. The proposed model is based on the interaction of particles defined through measuring homogeneous-energy criterion. Based on the energy concept, both general and local topologies are taken into account for clustering. The effect of the integration of a new element into the cluster on homogeneous-energy criterion is analyzed. If the new element does not alter the homogeneous-energy of a group, then it is added; otherwise, a new cluster is created. The mathematical-topological theory and the results of its application on public benchmark datasets are presented.<\/jats:p>","DOI":"10.3390\/sym14040781","type":"journal-article","created":{"date-parts":[[2022,4,10]],"date-time":"2022-04-10T06:02:54Z","timestamp":1649570574000},"page":"781","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Novel 2D Clustering Algorithm Based on Recursive Topological Data Structure"],"prefix":"10.3390","volume":"14","author":[{"given":"Ismael","family":"Osuna-Gal\u00e1n","sequence":"first","affiliation":[{"name":"Departamento de Electr\u00f3nica, Polytechnic University of Chiapas, Carretera Tuxtla Gutierrez-Portillo Zaragoza Km 21+500, Suchiapa 29150, Mexico"}]},{"given":"Yolanda","family":"P\u00e9rez-Pimentel","sequence":"additional","affiliation":[{"name":"Departamento de Electr\u00f3nica, Polytechnic University of Chiapas, Carretera Tuxtla Gutierrez-Portillo Zaragoza Km 21+500, Suchiapa 29150, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2323-9335","authenticated-orcid":false,"given":"Carlos","family":"Aviles-Cruz","sequence":"additional","affiliation":[{"name":"Departamento de Electr\u00f3nica, Universidad Aut\u00f3noma Metropolitana, Av. San Pablo 180 Col. Reynosa Tamaulipas, Ciudad de M\u00e9xico 02200, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1016\/j.neucom.2017.06.053","article-title":"A review of clustering techniques and developments","volume":"267","author":"Saxena","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s11704-019-8208-z","article-title":"A survey on ensemble learning","volume":"14","author":"Dong","year":"2020","journal-title":"Front. Comput. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.ins.2019.10.019","article-title":"Privacy-preserving clustering for big data in cyber-physical-social systems: Survey and perspectives","volume":"515","author":"Zhao","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4743","DOI":"10.1007\/s10489-018-1238-7","article-title":"K-means properties on six clustering benchmark datasets","volume":"48","author":"Sieranoja","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"116328","DOI":"10.1016\/j.eswa.2021.116328","article-title":"Personalized individual semantics based consensus reaching process for large-scale group decision making with probabilistic linguistic preference relations and application to COVID-19 surveillance","volume":"191","author":"Wan","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s40745-015-0040-1","article-title":"A Comprehensive Survey of Clustering Algorithms","volume":"2","author":"Xu","year":"2015","journal-title":"Ann. Data Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Vishwakarma, H., and Akashe, S. (2017). Clustering Algorithms: Experiment and Improvements. Computing and Network Sustainability, Springer.","DOI":"10.1007\/978-981-10-3935-5"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bhateja, V., Coello Coello, C.A., Satapathy, S.C., and Pattnaik, P.K. (2018). Survey on Clustering Algorithms for Unstructured Data. Intelligent Engineering Informatics, Springer.","DOI":"10.1007\/978-981-10-7566-7"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1007\/s11276-016-1417-6","article-title":"A survey of clustering algorithms for cognitive radio ad hoc networks","volume":"24","author":"Osman","year":"2018","journal-title":"Wirel. Netw."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ray, K., Sharma, T.K., Rawat, S., Saini, R.K., and Bandyopadhyay, A. (2019). Effective Data Clustering Algorithms. Soft Computing: Theories and Applications, Springer.","DOI":"10.1007\/978-981-13-0589-4"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Djouzi, K., and Beghdad-Bey, K. (2019, January 26\u201327). A Review of Clustering Algorithms for Big Data. Proceedings of the 2019 International Conference on Networking and Advanced Systems (ICNAS), Annaba, Algeria.","DOI":"10.1109\/ICNAS.2019.8807822"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"31883","DOI":"10.1109\/ACCESS.2019.2903568","article-title":"Survey of State-of-the-Art Mixed Data Clustering Algorithms","volume":"7","author":"Ahmad","year":"2019","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"107776","DOI":"10.1016\/j.topol.2021.107776","article-title":"Uniformities on strongly topological gyrogroups","volume":"302","author":"Zhang","year":"2021","journal-title":"Topol. Its Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1145\/3467477","article-title":"Evolutionary Machine Learning: A Survey","volume":"54","author":"Telikani","year":"2022","journal-title":"ACM Comput. Surv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.asoc.2017.04.031","article-title":"A novel cluster center fast determination clustering algorithm","volume":"57","author":"Jinyin","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Schubert, E., and Rousseeuw, P. (2019). Faster k-Medoids Clustering: Improving the PAM, CLARA, and CLARANS Algorithms. Lecture Notes in Computer Science, Springer.","DOI":"10.1007\/978-3-030-32047-8_16"},{"key":"ref_17","first-page":"1191","article-title":"Multiple Kernel k-means with Incomplete Kernels","volume":"42","author":"Liu","year":"2019","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_18","unstructured":"Rani, K. (2019, January 4\u20136). Visual Analytics for Comparing the Impact of Outliers in k-Means and k-Medoids Algorithm. Proceedings of the 2019 Amity International Conference on Artificial Intelligence (AICAI), Dubai, United Arab Emirates."},{"key":"ref_19","first-page":"4766","article-title":"A Comparison of K-Means Clustering Algorithm and CLARA Clustering Algorithm on Iris Dataset","volume":"7","author":"Gupta","year":"2019","journal-title":"Int. J. Eng. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"74683","DOI":"10.1109\/ACCESS.2019.2921320","article-title":"A Novel Algorithm for Initial Cluster Center Selection","volume":"7","author":"Li","year":"2019","journal-title":"IEEE Access"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1109\/TFUZZ.2018.2883033","article-title":"Deviation-Sparse Fuzzy C-Means With Neighbor Information Constraint","volume":"27","author":"Zhang","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"105928","DOI":"10.1016\/j.asoc.2019.105928","article-title":"Fuzzy C-Means clustering through SSIM and patch for image segmentation","volume":"87","author":"Tang","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"107411","DOI":"10.1016\/j.topol.2020.107411","article-title":"Continuously triangulating the continuous cluster category","volume":"285","author":"Garcia","year":"2020","journal-title":"Topol. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4540731","DOI":"10.1155\/2019\/4540731","article-title":"Topology: A Theory of a Pseudometric-Based Clustering Model and Its Application in Content-Based Image Retrieval","volume":"2019","year":"2019","journal-title":"Math. Probl. Eng."},{"key":"ref_25","unstructured":"Lim, J., Jun, J., Kim, S.H., and McLeod, D. (2012, January 23\u201325). A Framework for Clustering Mixed Attribute Type Datasets. Proceedings of the 4th International Conference on Emerging Databases-Technologies, Applications, and Theory (EDB 2012), Seoul, Korea."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Nazari, Z., Kang, D., Asharif, M., Sung, Y., and Ogawa, S. (2015, January 28\u201330). A new hierarchical clustering algorithm. Proceedings of the 2015 International Conference on Intelligent Informatics and Biomedical Sciences (ICIIBMS), Okinawa, Japan.","DOI":"10.1109\/ICIIBMS.2015.7439517"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"281","DOI":"10.3233\/IDA-160805","article-title":"Optimized aggregation function in hierarchical clustering combination","volume":"20","author":"Rashedi","year":"2016","journal-title":"Intell. Data Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1109\/JSTARS.2016.2537548","article-title":"Semi-supervised Hierarchical Clustering for Semantic SAR Image Annotation","volume":"9","author":"Yao","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Pitolli, G., Aniello, L., Laurenza, G., Querzoni, L., and Baldoni, R. (2017, January 23\u201326). Malware family identification with BIRCH clustering. Proceedings of the 2017 International Carnahan Conference on Security Technology (ICCST), Madrid, Spain.","DOI":"10.1109\/CCST.2017.8167802"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cao, X., Su, T., Wang, P., Wang, G., Lv, Z., and Li, X. (2018, January 26\u201328). An Optimized Chameleon Algorithm Based on Local Features. Proceedings of the 2018 10th International Conference on Machine Learning and Computing (ICMLC 2018), Macau, China.","DOI":"10.1145\/3195106.3195118"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Yokoyama, S., Bogardi-Meszoly, A., and Ishikawa, H. (2015, January 3\u20136). EBSCAN: An entanglement-based algorithm for discovering dense regions in large geo-social data streams with noise. Proceedings of the 8th ACM SIGSPATIAL International Workshop on Location-Based Social Networks, Bellevue, WA, USA.","DOI":"10.1145\/2830657.2830661"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1016\/j.procs.2016.04.265","article-title":"DENCLUE-IM: A New Approach for Big Data Clustering","volume":"83","author":"Rehioui","year":"2016","journal-title":"Procedia Comput. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.patcog.2016.03.008","article-title":"A fast DBSCAN clustering algorithm by accelerating neighbor searching using Groups method","volume":"58","author":"Kumar","year":"2016","journal-title":"Pattern Recognit."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Hartmann, S., Ma, H., Hameurlain, A., Pernul, G., and Wagner, R.R. (2018). Parameter Free Mixed-Type Density-Based Clustering. Database and Expert Systems Applications, Springer International Publishing.","DOI":"10.1007\/978-3-319-98812-2"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1080\/02664763.2016.1277191","article-title":"A new algorithm for clustering based on kernel density estimation","volume":"45","author":"Matioli","year":"2018","journal-title":"J. Appl. Stat."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Shu, Z., Yang, S., Wu, H., Xin, S., Pang, C., Kavan, L., and Liu, L. (2022). 3D Shape Segmentation Using Soft Density Peak Clustering and Semi-Supervised Learning. CAD Comput.-Aided Des., 145.","DOI":"10.1016\/j.cad.2021.103181"},{"key":"ref_37","unstructured":"Elleithy, K., and Sobh, T. (2015). Document Classification Using Enhanced Grid Based Clustering Algorithm. New Trends in Networking, Computing, E-Learning, Systems Sciences, and Engineering, Springer International Publishing."},{"key":"ref_38","unstructured":"Wagner, T., Feger, R., and Stelzer, A. (2016, January 5\u20137). A fast grid-based clustering algorithm for range\/Doppler\/DoA measurements. Proceedings of the 2016 European Radar Conference (EuRAD), London, UK."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1007\/s11277-017-4554-z","article-title":"GCCR: An Efficient Grid Based Clustering and Combinational Routing in Wireless Sensor Networks","volume":"97","author":"Lalitha","year":"2017","journal-title":"Wirel. Pers. Commun."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"29623","DOI":"10.1007\/s11042-017-5441-z","article-title":"Gridwave: A grid-based clustering algorithm for market transaction data based on spatial-temporal density-waves and synchronization","volume":"77","author":"Deng","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1228","DOI":"10.1007\/s10489-018-1324-x","article-title":"FGCH: A fast and grid based clustering algorithm for hybrid data stream","volume":"49","author":"Chen","year":"2019","journal-title":"Appl. Intell."},{"key":"ref_42","unstructured":"Kr\u00f6mer, P., Zhang, H., Liang, Y., and Pan, J.S. (2019). A Fast and Efficient Grid-Based K-means++ Clustering Algorithm for Large-Scale Datasets. The Fifth Euro-China Conference on Intelligent Data Analysis and Applications, Springer International Publishing."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Menendez, H., and Camacho, D. (2015, January 25\u201328). GANY: A genetic spectral-based Clustering algorithm for Large Data Analysis. Proceedings of the 2015 IEEE Congress on Evolutionary Computation (CEC), Sendai, Japan.","DOI":"10.1109\/CEC.2015.7256951"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.patcog.2016.01.035","article-title":"Global discriminative-based nonnegative spectral clustering","volume":"55","author":"Shang","year":"2016","journal-title":"Pattern Recognit."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.ymssp.2016.10.033","article-title":"A spectral-based clustering for structural health monitoring of the Sydney Harbour Bridge","volume":"87","author":"Alamdari","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Tian, L., Du, Q., Kopriva, I., and Younan, N. (2018, January 22\u201327). Spatial-spectral Based Multi-view Low-rank Sparse Sbuspace Clustering for Hyperspectral Imagery. Proceedings of the IGARSS 2018\u20142018 IEEE International Geoscience and Remote Sensing Symposium, Valencia, Spain.","DOI":"10.1109\/IGARSS.2018.8519284"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Nemade, V., Shastri, A., Ahuja, K., and Tiwari, A. (2018, January 18\u201321). Scaled and Projected Spectral Clustering with Vector Quantization for Handling Big Data. Proceedings of the 2018 IEEE Symposium Series on Computational Intelligence (SSCI), Bangalore, India.","DOI":"10.1109\/SSCI.2018.8628915"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Ma, L., Zhang, Y., Leiva, V., Liu, S., and Ma, T. (2022). A new clustering algorithm based on a radar scanning strategy with applications to machine learning data. Expert Syst. Appl., 191.","DOI":"10.1016\/j.eswa.2021.116143"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.engappai.2014.07.016","article-title":"GGSA: A Grouping Gravitational Search Algorithm for data clustering","volume":"36","author":"Dowlatshahi","year":"2014","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.engappai.2013.11.008","article-title":"Automatic cluster evolution using gravitational search algorithm and its application on image segmentation","volume":"29","author":"Kumar","year":"2014","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Nikbakht, H., and Mirvaziri, H. (2015, January 3\u20135). A new algorithm for data clustering based on gravitational search algorithm and genetic operators. Proceedings of the 2015 The International Symposium on Artificial Intelligence and Signal Processing (AISP), Mashhad, Iran.","DOI":"10.1109\/AISP.2015.7123532"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Sheshasaayee, A., and Sridevi, D. (2016, January 26\u201327). Fuzzy C-means algorithm with gravitational search algorithm in spatial data mining. Proceedings of the 2016 International Conference on Inventive Computation Technologies (ICICT), Coimbatore, India.","DOI":"10.1109\/INVENTIVE.2016.7823259"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Deng, Z., Qian, G., Chen, Z., and Su, H. (2017, January 26\u201327). Identifying Tor Anonymous Traffic Based on Gravitational Clustering Analysis. Proceedings of the 2017 9th International Conference on Intelligent Human-Machine Systems and Cybernetics (IHMSC), Hangzhou, China.","DOI":"10.1109\/IHMSC.2017.133"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.engappai.2018.05.004","article-title":"Optimized gravitational-based data clustering algorithm","volume":"73","author":"Alswaitti","year":"2018","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Yuqing, S., Junfei, Q., and Honggui, H. (2016, January 28\u201330). Structure design for RBF neural network based on improved K-means algorithm. Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China.","DOI":"10.1109\/CCDC.2016.7532265"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Amin, H., Deabes, W., and Bouazza, K. (2017, January 4\u20137). Clustering of user activities based on adaptive threshold spiking neural networks. Proceedings of the 2017 Ninth International Conference on Ubiquitous and Future Networks (ICUFN), Milan, Italy.","DOI":"10.1109\/ICUFN.2017.7993735"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1601","DOI":"10.1109\/JSTSP.2018.2875385","article-title":"Deep Multimodal Subspace Clustering Networks","volume":"12","author":"Abavisani","year":"2018","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Ren, Z., Chen, J., Ye, L., Wang, C., Liu, Y., and Zhou, W. (2018, January 7\u201310). Application of RBF Neural Network Optimized Based on K-Means Cluster Algorithm in Fault Diagnosis. Proceedings of the 2018 21st International Conference on Electrical Machines and Systems (ICEMS), Jeju, Korea.","DOI":"10.23919\/ICEMS.2018.8549274"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Kimura, M. (2019, January 17\u201320). AutoClustering: A feed-forward neural network based clustering algorithm. Proceedings of the 2018 IEEE International Conference on Data Mining Workshops (ICDMW), Singapore.","DOI":"10.1109\/ICDMW.2018.00102"},{"key":"ref_60","first-page":"3742536","article-title":"Representation Learning Based on Autoencoder and Deep Adaptive Clustering for Image Clustering","volume":"2021","author":"Cheng","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_61","unstructured":"Engelking, R. (1989). General Topology, Springer International Publishing."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.topol.2018.11.022","article-title":"On the relationship between ideal cluster points and ideal limit points","volume":"252","author":"Balcerzak","year":"2019","journal-title":"Topol. Its Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/781\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:50:52Z","timestamp":1760136652000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/781"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,9]]},"references-count":62,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040781"],"URL":"https:\/\/doi.org\/10.3390\/sym14040781","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,4,9]]}}}