{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T16:04:12Z","timestamp":1781193852361,"version":"3.54.1"},"reference-count":51,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,18]],"date-time":"2021-05-18T00:00:00Z","timestamp":1621296000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006109","name":"Vedeck\u00e1 Grantov\u00e1 Agent\u00fara M\u0160VVa\u0160 SR a SAV","doi-asserted-by":"publisher","award":["No. 2\/0155\/19"],"award-info":[{"award-number":["No. 2\/0155\/19"]}],"id":[{"id":"10.13039\/501100006109","id-type":"DOI","asserted-by":"publisher"}]},{"name":"CHIST ERA III (SOON) \u201dSocial Network of Machines\u201d","award":["no number"],"award-info":[{"award-number":["no number"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Future Internet"],"abstract":"<jats:p>Determining the network size is a critical process in numerous areas (e.g., computer science, logistic, epidemiology, social networking services, mathematical modeling, demography, etc.). However, many modern real-world systems are so extensive that measuring their size poses a serious challenge. Therefore, the algorithms for determining\/estimating this parameter in an effective manner have been gaining popularity over the past decades. In the paper, we analyze five frequently applied distributed consensus gossip-based algorithms for network size estimation in multi-agent systems (namely, the Randomized gossip algorithm, the Geographic gossip algorithm, the Broadcast gossip algorithm, the Push-Sum protocol, and the Push-Pull protocol). We examine the performance of the mentioned algorithms with bounded execution over random geometric graphs by applying two metrics: the number of sent messages required for consensus achievement and the estimation precision quantified as the median deviation from the real value of the network size. The experimental part consists of two scenarios\u2014the consensus achievement is conditioned by either the values of the inner states or the network size estimates\u2014and, in both scenarios, either the best-connected or the worst-connected agent is chosen as the leader. The goal of this paper is to identify whether all the examined algorithms are applicable to estimating the network size, which algorithm provides the best performance, how the leader selection can affect the performance of the algorithms, and how to most effectively configure the applied stopping criterion.<\/jats:p>","DOI":"10.3390\/fi13050134","type":"journal-article","created":{"date-parts":[[2021,5,18]],"date-time":"2021-05-18T12:17:16Z","timestamp":1621340236000},"page":"134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Comparative Study of Distributed Consensus Gossip Algorithms for Network Size Estimation in Multi-Agent Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2430-1126","authenticated-orcid":false,"given":"Martin","family":"Kenyeres","sequence":"first","affiliation":[{"name":"Institute of Informatics, Slovak Academy of Sciences, D\u00fabravsk\u00e1 Cesta 9, 845 07 Bratislava, Slovakia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jozef","family":"Kenyeres","sequence":"additional","affiliation":[{"name":"EBCONT Proconsult GmbH, Millennium Tower, Handelskai 94-96, 1200 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.sysconle.2019.01.009","article-title":"Second-order consensus of hybrid multi-agent systems","volume":"125","author":"Zheng","year":"2019","journal-title":"Syst. Control Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1743","DOI":"10.1109\/TPWRS.2007.908471","article-title":"Multi-agent systems for power engineering applications\u2014Part I: Concepts, approaches, and technical challenges","volume":"22","author":"McArthur","year":"2007","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Shames, I., Charalambous, T., Hadjicostis, C.N., and Johansson, M. (2012, January 1\u20135). Distributed Network Size Estimation and Average Degree Estimation and Control in Networks Isomorphic to Directed Graphs. Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing, Allerton, Monticello, IL, USA.","DOI":"10.1109\/Allerton.2012.6483452"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Seda, P., Seda, M., and Hosek, J. (2020). On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments. Appl. Sci., 10.","DOI":"10.3390\/app10248853"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1017\/S0269888900008122","article-title":"Intelligent agents: Theory and practice","volume":"10","author":"Wooldridge","year":"1995","journal-title":"Knowl. Eng. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"28573","DOI":"10.1109\/ACCESS.2018.2831228","article-title":"Multi-Agent Systems: A Survey","volume":"6","author":"Dorri","year":"2018","journal-title":"IEEE Access"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Rocha, J., Boavida-Portugal, I., and Gomes, E. (2017). Introductory Chapter: Multi-Agent Systems. Multi-Agent Systems, IntechOpen.","DOI":"10.5772\/intechopen.70241"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, M., and Zhang, X. (2017). Information fusion in a multi-source incomplete information system based on information entropy. Entropy, 19.","DOI":"10.3390\/e19110570"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"704504","DOI":"10.1155\/2013\/704504","article-title":"A review of data fusion techniques","volume":"2013","author":"Castanedo","year":"2013","journal-title":"Sci. World J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Skorpil, V., and Stastny, J. (2006, January 16\u201320). Back-propagation and k-means algorithms comparison. Proceedings of the 2006 8th International Conference on Signal Processing, ICSP 2006, Guilin, China.","DOI":"10.1109\/ICOSP.2006.345838"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zacchigna, F.G., and Lutenberg, A. (2014, January 13\u201315). A novel consensus algorithm proposal: Measurement estimation by silent agreement (MESA). Proceedings of the 5th Argentine Symposium and Conference on Embedded Systems, SASE\/CASE 2014, Buenos Aires, Argentina.","DOI":"10.1109\/SASE-CASE.2014.6914462"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zacchigna, F.G., Lutenberg, A., and Vargas, F. (2015, January 12\u201314). MESA: A formal approach to compute consensus in WSNs. Proceedings of the 6th Argentine Conference on Embedded Systems, CASE 2015, Buenos Aires, Argentina.","DOI":"10.1109\/SASE-CASE.2015.7295841"},{"key":"ref_13","first-page":"167","article-title":"Consensus control of discrete-time multi-agent systems","volume":"79","author":"Merezeanu","year":"2017","journal-title":"U. Politeh. Buch. Ser. A"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Antal, C., Cioara, T., Anghel, I., Antal, M., and Salomie, I. (2021). Distributed Ledger Technology Review and Decentralized Applications Development Guidelines. Future Int., 13.","DOI":"10.3390\/fi13030062"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"489","DOI":"10.24846\/v25i4y201610","article-title":"Context-aware control platform for sensor network integration","volume":"25","author":"Merezeanu","year":"2016","journal-title":"Stud. Inform. Control"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Vladyko, A., Khakimov, A., Muthanna, A., Ateya, A.A., and Koucheryavy, A. (2019). Distributed Edge Computing to Assist Ultra-Low-Latency VANET Applications. Future Int., 11.","DOI":"10.3390\/fi11060128"},{"key":"ref_17","unstructured":"Xiao, L., Boyd, S., and Lall, S. (2005, January 25\u201327). A Scheme for robust distributed sensor fusion based on average consensus. Proceedings of the 4th International Symposium on Information Processing in Sensor Networks, IPSN 2005, Los Angeles, CA, USA."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4334","DOI":"10.1109\/TSP.2012.2196697","article-title":"Likelihood consensus and its application to distributed particle filtering","volume":"60","author":"Hlinka","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.sysconle.2004.02.022","article-title":"Fast linear iterations for distributed averaging","volume":"53","author":"Xiao","year":"2004","journal-title":"Syst. Control. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mahmoud, M.S., Oyedeji, M.O., and Xia, Y. (2020). Advanced Distributed Consensus for Multiagent Systems, Academic Press.","DOI":"10.1016\/B978-0-12-821186-1.00012-X"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kenyeres, M., and Kenyeres, J. (2020). Average consensus over mobile wireless sensor networks: Weight matrix guaranteeing convergence without reconfiguration of edge weights. Sensors, 20.","DOI":"10.3390\/s20133677"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gutierrez-Gutierrez, J., Zarraga-Rodriguez, M., and Insausti, X. (2018). Analysis of Known Linear Distributed Average Consensus Algorithms on Cycles and Paths. Sensors, 18.","DOI":"10.3390\/s18040968"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yu, J.Y., and Rabbat, M. (2013, January 15\u201318). Performance comparison of randomized gossip, broadcast gossip and collection tree protocol for distributed averaging. Proceedings of the 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2013, Montreal, QC, Canada.","DOI":"10.1109\/CAMSAP.2013.6714015"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1634","DOI":"10.1002\/asjc.525","article-title":"Quantized consensus of multi-agent systems via broadcast gossip algorithms","volume":"14","author":"Liu","year":"2012","journal-title":"Asian J. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/978-94-007-0638-5_1","article-title":"Performance of gossip algorithms in wireless sensor networks","volume":"81","author":"Baldi","year":"2011","journal-title":"Lect. Notes Electr. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2748","DOI":"10.1109\/TSP.2009.2016247","article-title":"Broadcast gossip algorithms for consensus","volume":"57","author":"Aysal","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Aysal, T.C., Yildiz, M.E., Sarwate, A.D., and Scaglione, A. (2008, January 5\u20139). Broadcast gossip algorithms. Proceedings of the IEEE Information Theory Workshop, ITW, Porto, Portugal.","DOI":"10.1109\/ITW.2008.4578682"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1109\/TSP.2007.908946","article-title":"Geographic gossip: Efficient averaging for sensor networks","volume":"56","author":"Dimakis","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Aysal, T.C., Yildiz, M.E., Sarwate, A.D., and Scaglione, A. (2008, January 9\u201311). Broadcast gossip algorithms: Design and analysis for consensus. Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008, Cancun, Mexico.","DOI":"10.1109\/CDC.2008.4739315"},{"key":"ref_30","unstructured":"Jesus, P., Baquero, C., and Almeida, P.S. (2010). Dependability in Aggregation by Averaging. arXiv."},{"key":"ref_31","unstructured":"Jesus, P., Baquero, C., and Almeida, P.S. (2007, January 21\u201323). A study on aggregation by averaging algorithms (poster). Proceedings of the EuroSys 2007\u20132nd EuroSys Conference, Lisbon, Portugal."},{"key":"ref_32","unstructured":"Blasa, F., Cafiero, S., Fortino, G., and Di Fatta, G. (2011, January 20\u201325). Symmetric push-sum protocol for decentralised aggregation. Proceedings of the 3rd International Conference on Advances in P2P Systems, AP2PS 2011, Lisbon, Portugal."},{"key":"ref_33","unstructured":"Huang, W., Wang, Y., and Provan, G. (2021, May 16). Comparing Asynchronous Distributed Averaging Gossip Algorithms Over Scale-free Graphs. Available online: http:\/\/www.cs.ucc.ie\/~gprovan\/Provan\/comparegossip.pdf."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Cardoso, J.C.S., Baquero, C., and Almeida, P.S. (2009, January 1\u20134). Probabilistic Estimation of Network Size and Diameter. Proceedings of the 4th Latin-American Symposium on Dependable Computing, LADC 2009, Joao Pessoa, Brazil.","DOI":"10.1109\/LADC.2009.19"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.comnet.2018.03.014","article-title":"ABSCEV: An agent-based simulation framework about smart transportation for reducing waiting times in charging electric vehicles","volume":"138","author":"Lacuesta","year":"2018","journal-title":"Comput. Netw."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5927","DOI":"10.1109\/JIOT.2018.2871255","article-title":"Security in Vehicles with IoT by Prioritization Rules, Vehicle Certificates, and Trust Management","volume":"6","author":"Sendra","year":"2019","journal-title":"IEEE Internet Things J."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1109\/TPDS.2011.209","article-title":"Extrema Propagation: Fast Distributed Estimation of Sums and Network Sizes","volume":"23","author":"Baquero","year":"2012","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Kennedy, O., Koch, C., and Demers, A. (April, January 29). Dynamic Approaches to In-Network Aggregation. Proceedings of the 25th IEEE International Conference on Data Engineering, ICDE 2009, Shanghai, China.","DOI":"10.1109\/ICDE.2009.233"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/COMST.2014.2354398","article-title":"A Survey of Distributed Data Aggregation Algorithms","volume":"17","author":"Jesus","year":"2015","journal-title":"IEEE Commun. Surveys Tuts."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4091","DOI":"10.1002\/cpe.3549","article-title":"A comparative study of spanning tree and gossip protocols for aggregation","volume":"27","author":"Nyers","year":"2015","journal-title":"Concurr. Comp."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Nyers, L., and Jelasity, M. (2014, January 25\u201329). Spanning tree or gossip for aggregation: A comparative study. Proceedings of the European Conference on Parallel Processing, Euro-Par 2014, Porto, Portugal.","DOI":"10.1007\/978-3-319-09873-9_32"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/978-3-540-92157-8_7","article-title":"A practical approach to network size estimation for structured overlays","volume":"5343","author":"Nyers","year":"2008","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"48929","DOI":"10.1109\/ACCESS.2020.2977949","article-title":"An Analytic Latency Model for a Next-Hop Data-Ferrying Swarm on Random Geometric Graphs","volume":"8","author":"Fraser","year":"2020","journal-title":"IEEE Access"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Gulzar, M.M., Rizvi, S.T.H., Javed, M.Y., Munir, U., and Asif, H. (2018). Multi-agent cooperative control consensus: A comparative review. Electronics, 7.","DOI":"10.3390\/electronics7020022"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Qurashi, M.A., Angelopoulos, C.M., Katos, V., Munir, U., and Asif, H. (2020, January 7\u201311). An Architecture for Resilient Intrusion Detection in IoT Networks. Proceedings of the 2020 IEEE International Conference on Communications, ICC 2020, Dublin, Ireland.","DOI":"10.1109\/ICC40277.2020.9148868"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"23153","DOI":"10.1109\/ACCESS.2021.3053254","article-title":"Dynamic Spectrum Sensing under Crash and Byzantine Failure Environments for Distributed Convergence in Cognitive Radio Networks","volume":"9","author":"Mustafa","year":"2021","journal-title":"IEEE Access"},{"key":"ref_47","unstructured":"Kempe, D., Dobra, A., and Gehrke, J. (2003, January 11\u201314). Gossip-based computation of aggregate information. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, Cambridge, MA, USA."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2508","DOI":"10.1109\/TIT.2006.874516","article-title":"Randomized gossip algorithms","volume":"52","author":"Boyd","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Avrachenkov, K., Chamie, M.E., and Neglia, G. (2011, January 27\u201329). A local average consensus algorithm for wireless sensor networks. Proceedings of the 7th IEEE International Conference on Distributed Computing in Sensor Systems, DCOSS\u201911, Barcelona, Spain.","DOI":"10.1109\/DCOSS.2011.5982199"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/978-3-030-61659-5_2","article-title":"On Performance Evaluation of Distributed System Size Estimation Executed by Average Consensus Weights","volume":"403","author":"Kenyeres","year":"2021","journal-title":"Fuzziness Soft Comput."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjb\/e2014-50094-4","article-title":"Consensus reaching in swarms ruled by a hybrid metric-topological distance","volume":"87","author":"Shang","year":"2014","journal-title":"Eur. Phys. J. B"}],"container-title":["Future Internet"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-5903\/13\/5\/134\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:03:33Z","timestamp":1760162613000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-5903\/13\/5\/134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,18]]},"references-count":51,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["fi13050134"],"URL":"https:\/\/doi.org\/10.3390\/fi13050134","relation":{},"ISSN":["1999-5903"],"issn-type":[{"value":"1999-5903","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,18]]}}}