{"supplementarymaterial":"no","author":[{"full_name":"D'Archivio, Niccolรฒ","last_name":"D'Archivio","first_name":"Niccolรฒ"},{"full_name":"Almahmoud, Hind","first_name":"Hind","last_name":"Almahmoud"},{"first_name":"Emanuele","last_name":"Natale","full_name":"Natale, Emanuele"},{"id":"68748c44-84d5-11f1-b4f6-ca083374e553","first_name":"Frederik","last_name":"Mallmann-Trenn","full_name":"Mallmann-Trenn, Frederik"}],"date_published":"2026-07-01T00:00:00Z","title":"Order statistics in population protocols via simple dynamics","date_updated":"2026-07-21T07:34:49Z","article_processing_charge":"No","researchdata_availability":"no","department":[{"_id":"MoHe"}],"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"das_tickbox":"0","oa":1,"language":[{"iso":"eng"}],"year":"2026","page":"425-436","month":"07","publisher":"Association for Computing Machinery","abstract":[{"text":"We study simple dynamics in the population protocol model, in\r\nwhich ๐ agents start with totally ordered initial opinions ๐ฅ1, ๐ฅ2, . . . ,\r\n๐ฅ๐ and, in each round, a randomly chosen agent changes its opinion\r\nas a function of the opinion of other randomly chosen agents. Such\r\ndynamics often converge to consensus on a single fixation value ๐ห.\r\nThis paper asks how to control the distribution of ๐ห as a randomised\r\nchoice among the initial opinions by designing suitable simple\r\ndynamics. Writing the sorted initial values as ๐ฅ(1) โค ยท ยท ยท โค ๐ฅ(๐)\r\n,\r\nwe design two protocols that realise natural target laws over order\r\nstatistics.\r\nFirst, for a parameter ๐ โ (0, 1), our geometric protocol biases\r\ntoward larger opinions and satisfies P\r\n\r\n๐ห = ๐ฅ(๐)\r\n\r\nโ ๐\r\n๐โ๐\r\n, for\r\n๐ = 1, . . . , ๐. Equivalently, P\r\n\r\n๐ห = ๐ฅ(๐)\r\n\r\n= (1 โ ๐)๐\r\n๐โ๐\r\n/(1 โ ๐\r\n๐\r\n).\r\nSecond, our binomial protocol assigns a shifted binomial law to the\r\nranks in ascending order: if ๐พ โ1 โผ Bin(๐โ1, 1โ๐), then ๐ห = ๐ฅ(๐พ)\r\n,\r\ni.e., P\r\n\r\n๐ห = ๐ฅ(๐)\r\n\r\n=\r\n๐โ1\r\n๐โ1\r\n\u0001\r\n๐\r\n๐โ๐\r\n(1 โ ๐)\r\n๐โ1\r\n, for ๐ = 1, . . . , ๐.\r\nApplications of this include computing the Top-๐ values for\r\nsmall ๐ on general interaction graphs. A central contribution of\r\nthis work is that, in contrast to most population protocols, we can\r\ncharacterise the fixation distribution in closed form. This is enabled\r\nby a novel analysis technique, which also yields applications: we\r\nderive new results for the Median protocol that extend the state of\r\nthe art.","lang":"eng"}],"ddc":["000"],"acknowledgement":"This work has been supported by the AID INRIA-DGA project\r\nnยฐ2023000872 โBioSwarmโ, the French government National Research Agency (ANR) through the UCA JEDI (ANR-15-IDEX-01),\r\nthe EUR DS4H (ANR-17-EURE-004) and the 3IA Cote dโAzur Investments ANR-23-IACL-0001, and EPSRC grant EP/W005573/1","OA_type":"gold","has_accepted_license":"1","scopus_import":"1","publication_identifier":{"isbn":["9798400725128"]},"_id":"22368","day":"01","OA_place":"publisher","publication_status":"published","conference":{"end_date":"2026-07-10","location":"Egham, United Kingdom","start_date":"2026-07-06","name":"PODC: Symposium on Principles of Distributed Computing"},"type":"conference","status":"public","oa_version":"Published Version","publication":"Proceedings of the Annual ACM Symposium on Principles of Distributed Computing","corr_author":"1","date_created":"2026-07-19T22:01:47Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"file_name":"2026_ACMPODC_dArchivio.pdf","creator":"dernst","date_created":"2026-07-21T07:31:29Z","success":1,"access_level":"open_access","file_id":"22379","relation":"main_file","content_type":"application/pdf","checksum":"e8208393a016d8e71d7b26c402045488","date_updated":"2026-07-21T07:31:29Z","file_size":824239}],"quality_controlled":"1","file_date_updated":"2026-07-21T07:31:29Z","citation":{"ieee":"N. DโArchivio, H. Almahmoud, E. Natale, and F. Mallmann-Trenn, โOrder statistics in population protocols via simple dynamics,โ in Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Egham, United Kingdom, 2026, pp. 425โ436.","chicago":"DโArchivio, Niccolรฒ, Hind Almahmoud, Emanuele Natale, and Frederik Mallmann-Trenn. โOrder Statistics in Population Protocols via Simple Dynamics.โ In Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, 425โ36. Association for Computing Machinery, 2026. https://doi.org/10.1145/3796701.3815922.","short":"N. DโArchivio, H. Almahmoud, E. Natale, F. Mallmann-Trenn, in:, Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Association for Computing Machinery, 2026, pp. 425โ436.","ista":"DโArchivio N, Almahmoud H, Natale E, Mallmann-Trenn F. 2026. Order statistics in population protocols via simple dynamics. Proceedings of the Annual ACM Symposium on Principles of Distributed Computing. PODC: Symposium on Principles of Distributed Computing, 425โ436.","apa":"DโArchivio, N., Almahmoud, H., Natale, E., & Mallmann-Trenn, F. (2026). Order statistics in population protocols via simple dynamics. In Proceedings of the Annual ACM Symposium on Principles of Distributed Computing (pp. 425โ436). Egham, United Kingdom: Association for Computing Machinery. https://doi.org/10.1145/3796701.3815922","ama":"DโArchivio N, Almahmoud H, Natale E, Mallmann-Trenn F. Order statistics in population protocols via simple dynamics. In: Proceedings of the Annual ACM Symposium on Principles of Distributed Computing. Association for Computing Machinery; 2026:425-436. doi:10.1145/3796701.3815922","mla":"DโArchivio, Niccolรฒ, et al. โOrder Statistics in Population Protocols via Simple Dynamics.โ Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Association for Computing Machinery, 2026, pp. 425โ36, doi:10.1145/3796701.3815922."},"doi":"10.1145/3796701.3815922"}