{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T10:04:27Z","timestamp":1766138667416,"version":"build-2065373602"},"reference-count":64,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,22]],"date-time":"2024-04-22T00:00:00Z","timestamp":1713744000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Community detection is a widely studied topic in network structure analysis. We propose a community detection method based on the search for the local maxima of an objective function. This objective function reflects the quality of candidate communities in the network structure. The objective function can be constructed from a probability matrix that describes interactions in a network. Different models, such as network structure models and network flow models, can be used to build the probability matrix, and it acts as a link between network models and community detection models. In our influence-spreading model, the probability matrix is called an influence-spreading matrix, which describes the directed influence between all pairs of nodes in the network. By using the local maxima of an objective function, our method can standardise and help in comparing different definitions and approaches of community detection. Our proposed approach can detect overlapping and hierarchical communities and their building blocks within a network. To compare different structures in the network, we define a cohesion measure. The objective function can be expressed as a sum of these cohesion measures. We also discuss the probability of community formation to analyse a different aspect of group behaviour in a network. It is essential to recognise that this concept is separate from the notion of community cohesion, which emphasises the need for varying objective functions in different applications. Furthermore, we demonstrate that normalising objective functions by the size of detected communities can alter their rankings.<\/jats:p>","DOI":"10.3390\/computation12040085","type":"journal-article","created":{"date-parts":[[2024,4,22]],"date-time":"2024-04-22T12:38:41Z","timestamp":1713789521000},"page":"85","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Detecting Overlapping Communities Based on Influence-Spreading Matrix and Local Maxima of a Quality Function"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3677-816X","authenticated-orcid":false,"given":"Vesa","family":"Kuikka","sequence":"first","affiliation":[{"name":"Finnish Defence Research Agency, Tykkikent\u00e4ntie 1, P.O. Box 10, 11311 Riihim\u00e4ki, Finland"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1038\/s41567-022-01716-7","article-title":"20 years of network community detection","volume":"18","author":"Fortunato","year":"2022","journal-title":"Nat. Phys."},{"key":"ref_2","first-page":"20120375","article-title":"Network science","volume":"371","year":"2013","journal-title":"Philos. Trans. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Newman, M.E.J. (2018). Networks: An introduction, Oxford University Press.","DOI":"10.1093\/oso\/9780198805090.003.0001"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.09.002","article-title":"Community detection in networks: A user guide","volume":"659","author":"Fortunato","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_5","unstructured":"Rosvall, M., Delvenne, J.C., Schaub, M.T., and Lambiotte, R. (2019). Advances in Network Clustering and Blockmodeling, John Wiley & Sons Ltd."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7821","DOI":"10.1073\/pnas.122653799","article-title":"Community structure in social and biological networks","volume":"99","author":"Girvan","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"8577","DOI":"10.1073\/pnas.0601602103","article-title":"Modularity and community structure in networks","volume":"103","author":"Newman","year":"2006","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"118703","DOI":"10.1103\/PhysRevLett.100.118703","article-title":"Community structure in directed networks","volume":"100","author":"Leicht","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1073\/pnas.0605965104","article-title":"Resolution limit in community detection","volume":"104","author":"Fortunato","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s41109-020-00289-9","article-title":"A comparative study of overlapping community detection methods from the perspective of the structural properties","volume":"5","author":"Vieira","year":"2020","journal-title":"Appl. Netw. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1152243","DOI":"10.3389\/fphy.2023.1152243","article-title":"Considering weights in real social networks: A review","volume":"11","author":"Bellingeri","year":"2023","journal-title":"Front. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Long, J.C., Cunningham, F.C., and Braithwaite, J. (2013). Bridges, brokers and boundary spanners in collaborative networks: A systematic review. BMC Health Serv. Res., 13.","DOI":"10.1186\/1472-6963-13-158"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.socnet.2004.11.008","article-title":"Centrality and network flow","volume":"27","author":"Borgatti","year":"2005","journal-title":"Soc. Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0378-8733(91)90017-N","article-title":"Centrality in valued graphs: A measure of betweenness based on network flow","volume":"13","author":"Freeman","year":"1991","journal-title":"Soc. Netw."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"20220774","DOI":"10.1098\/rspa.2022.0774","article-title":"Message passing methods on complex networks","volume":"479","author":"Newman","year":"2023","journal-title":"Proc. R. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1186\/s40649-018-0060-z","article-title":"Influence spreading model used to analyse social networks and detect sub-communities","volume":"5","author":"Kuikka","year":"2018","journal-title":"Comput. Soc. Netw."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"052306","DOI":"10.1103\/PhysRevE.101.052306","article-title":"Consistency of community structure in complex networks","volume":"101","author":"Riolo","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1038\/nature03607","article-title":"Uncovering the overlapping community structure of complex networks in nature and society","volume":"435","author":"Palla","year":"2005","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2023","DOI":"10.1007\/s10115-022-01704-6","article-title":"A review of clique-based overlapping community detection algorithms","volume":"64","author":"Gupta","year":"2022","journal-title":"Knowl. Inf. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.datak.2018.07.009","article-title":"An overlapping community detection algorithm in complex networks based on information theory","volume":"117","author":"Zhou","year":"2018","journal-title":"Data Knowl. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"033015","DOI":"10.1088\/1367-2630\/11\/3\/033015","article-title":"Detecting the overlapping and hierarchical community structure in complex networks","volume":"11","author":"Lancichinetti","year":"2009","journal-title":"New. J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Prokop, P., Dr\u00e1\u017edilov\u00e1, P., and Plato\u0161, J. (2023, January 28\u201330). Hierarchical Overlapping Community Detection for Weighted Networks. Proceedings of the International Conference on Complex Networks and Their Applications, Menton, France.","DOI":"10.1007\/978-3-031-53499-7_13"},{"key":"ref_23","unstructured":"Henderson, D., Jacobson, S.H., and Johnson, A.W. (2003). Handbook of Metaheuristics, Springer."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Land, A.H., and Doig, A.G. (1960). An Automatic Method for Solving Discrete Programming Problems, Springer.","DOI":"10.2307\/1910129"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4694","DOI":"10.1038\/s41598-024-55190-7","article-title":"Community detection with greedy modularity disassembly strategy","volume":"14","author":"Rustamaji","year":"2024","journal-title":"Sci. Rep."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Yang, J., Sun, Y., Cheng, S., Bian, K., Liu, Z., Sun, X., and Cao, Y. (2022, January 28\u201331). A Memetic Algorithm Based on Adaptive Simulated Annealing for Community Detection. Proceedings of the International Conference on Intelligence Science, Xi\u2019an, China.","DOI":"10.1007\/978-3-031-14903-0_3"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21449","DOI":"10.1007\/s11042-022-12745-y","article-title":"Discovering community structure in social networks based on the synergy of label propagation and simulated annealing","volume":"81","author":"Jokar","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"042822","DOI":"10.1103\/PhysRevE.88.042822","article-title":"Spectral methods for community detection and graph partitioning","volume":"88","author":"Newman","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Kuikka, V., Aalto, H., Ij\u00e4s, M., and Kaski, K.K. (2022). Efficiency of Algorithms for Computing Influence and Information Spreading on Social Networks. Algorithms, 15.","DOI":"10.3390\/a15080262"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"056117","DOI":"10.1103\/PhysRevE.80.056117","article-title":"Community detection algorithms: A comparative analysis","volume":"80","author":"Lancichinetti","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1007\/s11831-020-09421-5","article-title":"Classification and comparative evaluation of community detection algorithms","volume":"28","author":"Mittal","year":"2021","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_32","unstructured":"Khan, B.S., and Niazi, M.A. (2017). Network community detection: A review and visual survey. arXiv."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"30750","DOI":"10.1038\/srep30750","article-title":"A comparative analysis of community detection algorithms on artificial networks","volume":"6","author":"Yang","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Lancichinetti, A., Radicchi, F., Ramasco, J.J., and Fortunato, S. (2011). Finding statistically significant communities in networks. PLoS ONE, 6.","DOI":"10.1371\/journal.pone.0018961"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"062805","DOI":"10.1103\/PhysRevE.90.062805","article-title":"Community detection in networks: Structural communities versus ground truth","volume":"90","author":"Hric","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s41109-019-0238-9","article-title":"On community structure in complex networks: Challenges and opportunities","volume":"4","author":"Cherifi","year":"2019","journal-title":"Appl. Netw. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1145\/2501654.2501657","article-title":"Overlapping community detection in networks: The state-of-the-art and comparative study","volume":"45","author":"Xie","year":"2013","journal-title":"ACM Comput. Surv. (Csur)"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"016118","DOI":"10.1103\/PhysRevE.80.016118","article-title":"Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities","volume":"80","author":"Lancichinetti","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"24115","DOI":"10.1038\/srep24115","article-title":"Overlapping community detection based on network decomposition","volume":"6","author":"Ding","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1073\/pnas.0706851105","article-title":"Maps of random walks on complex networks reveal community structure","volume":"105","author":"Rosvall","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"11073","DOI":"10.1007\/s00521-023-08284-8","article-title":"KO: Modularity optimization in community detection","volume":"35","year":"2023","journal-title":"Neural Comput. Appl."},{"key":"ref_42","unstructured":"Jerdee, M., Kirkley, A., and Newman, M. (2023). Normalized mutual information is a biased measure for classification and community detection. arXiv."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"P10008","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","article-title":"Fast unfolding of communities in large networks","volume":"2008","author":"Blondel","year":"2008","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"5233","DOI":"10.1038\/s41598-019-41695-z","article-title":"From Louvain to Leiden: Guaranteeing well-connected communities","volume":"9","author":"Traag","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"13689","DOI":"10.1007\/s00500-023-08067-2","article-title":"Overlapping community detection algorithm based on similarity of node relationship","volume":"27","author":"Liu","year":"2023","journal-title":"Soft Comput."},{"key":"ref_46","unstructured":"Hieu, D.D., and Duong, P.T.H. (2024). Overlapping community detection algorithms using Modularity and the cosine. arXiv."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"P07042","DOI":"10.1088\/1742-5468\/2009\/07\/P07042","article-title":"Quantifying and identifying the overlapping community structure in networks","volume":"2009","author":"Shen","year":"2009","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"016107","DOI":"10.1103\/PhysRevE.77.016107","article-title":"Fuzzy communities and the concept of bridgeness in complex networks","volume":"77","author":"Nepusz","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"P02017","DOI":"10.1088\/1742-5468\/2011\/02\/P02017","article-title":"Fuzzy overlapping communities in networks","volume":"2011","author":"Gregory","year":"2011","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1038\/s41567-019-0459-y","article-title":"From networks to optimal higher-order models of complex systems","volume":"15","author":"Lambiotte","year":"2019","journal-title":"Nat. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"129288","DOI":"10.1016\/j.physleta.2023.129288","article-title":"Higher-order interactions disturb community detection in complex networks","volume":"494","author":"Liu","year":"2023","journal-title":"Phys. Lett. A"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Bakshy, E., Rosenn, I., Marlow, C., and Adamic, L. (2012, January 16\u201320). The role of social networks in information diffusion. Proceedings of the 21st International Conference on World Wide Web, Lyon, France.","DOI":"10.1145\/2187836.2187907"},{"key":"ref_53","unstructured":"Lamberson, P.J. (2016). The Oxford Handbook of the Economics of Networks, Oxford University Press."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Centola, D. (2018). How Behavior Spreads: The Science of Complex Contagions, Princeton University Press.","DOI":"10.23943\/9781400890095"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1186\/s40649-021-00094-z","article-title":"Modelling community structure and temporal spreading on complex networks","volume":"8","author":"Kuikka","year":"2020","journal-title":"Comput. Soc. Netw."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"173","DOI":"10.3934\/biophy.2023011","article-title":"Similarity of epidemic spreading and information network connectivity mechanisms demonstrated by analysis of two probabilistic models","volume":"10","author":"Almiala","year":"2023","journal-title":"AIMS Biophys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1086\/521848","article-title":"Complex Contagions and the Weakness of Long Ties","volume":"113","author":"Centola","year":"2007","journal-title":"Am. J. Sociol."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"129244","DOI":"10.1016\/j.physa.2023.129244","article-title":"Influence spreading model for partial breakthrough effects on complex networks","volume":"630","author":"Almiala","year":"2023","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1086\/jar.33.4.3629752","article-title":"An Information Flow Model for Conflict and Fission in Small Groups","volume":"33","author":"Zachary","year":"1977","journal-title":"J. Anthropol. Res."},{"key":"ref_60","first-page":"S477","article-title":"Identifying the Role That Individual Animal Play in Their Social Network","volume":"271","author":"Newman","year":"2004","journal-title":"Proc. R. Soc. Lond. B"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1007\/s00265-003-0651-y","article-title":"The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations: Can geographic isolation explain this unique trait?","volume":"54","author":"Lusseau","year":"2003","journal-title":"Behav. Ecol. Sociobiol."},{"key":"ref_62","unstructured":"Leskovec, J., and Krevl, A. (2020, January 01). SNAP Datasets: Stanford Large Network Dataset Collection. Available online: http:\/\/snap.stanford.edu\/data."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Rossi, R.A., and Ahmed, N.K. (2015, January 25\u201330). The Network Data Repository with Interactive Graph Analytics and Visualization. Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, Austin, TX, USA.","DOI":"10.1609\/aaai.v29i1.9277"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Bastian, M., Heymann, S., and Jacomy, M. (2009, January 17\u201320). Gephi: An Open Source Software for Exploring and Manipulating Networks. Proceedings of the International AAAI Conference on Web and Social Media, San Jose, CA, USA.","DOI":"10.1609\/icwsm.v3i1.13937"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/4\/85\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:32:05Z","timestamp":1760106725000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/4\/85"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,22]]},"references-count":64,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["computation12040085"],"URL":"https:\/\/doi.org\/10.3390\/computation12040085","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,4,22]]}}}