{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:47Z","timestamp":1775527607609,"version":"3.50.1"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T00:00:00Z","timestamp":1762128000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The generalised random graph contains\n                    <jats:italic>n<\/jats:italic>\n                    vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments, and study the point process\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225100351_inline1.png\"\/>\n                        <jats:tex-math>$\\mathcal{C}_n$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225100351_inline2.png\"\/>\n                        <jats:tex-math>$\\{3,4,\\ldots\\}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , which counts how many cycles of the respective length are present in the graph. We establish convergence of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225100351_inline3.png\"\/>\n                        <jats:tex-math>$\\mathcal{C}_n$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225100351_inline4.png\"\/>\n                        <jats:tex-math>$\\mathcal{C}_n$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is evaluated on a bounded set\n                    <jats:italic>A<\/jats:italic>\n                    , we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets\n                    <jats:italic>A<\/jats:italic>\n                    at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and longest cycles when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung\u2013Lu model and the Norros\u2013Reittu model.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10035","type":"journal-article","created":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T10:51:48Z","timestamp":1762167108000},"page":"433-457","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Poisson approximation for cycles in the generalised random graph"],"prefix":"10.1017","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-7269-3342","authenticated-orcid":false,"given":"Matthias","family":"Lienau","sequence":"first","affiliation":[{"name":"Hamburg University of Technology"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,11,3]]},"reference":[{"key":"S0021900225100351_ref16","doi-asserted-by":"publisher","DOI":"10.1214\/07-AAP490"},{"key":"S0021900225100351_ref14","volume-title":"Random Graphs and Complex Networks","author":"van der Hofstad","year":"2017"},{"key":"S0021900225100351_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2020.108733"},{"key":"S0021900225100351_ref24","doi-asserted-by":"publisher","DOI":"10.1214\/11-PS182"},{"key":"S0021900225100351_ref17","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20297"},{"key":"S0021900225100351_ref10","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.252631999"},{"key":"S0021900225100351_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/18-AIHP926"},{"key":"S0021900225100351_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v15-817"},{"key":"S0021900225100351_ref11","doi-asserted-by":"publisher","DOI":"10.1142\/9789812567680_0002"},{"key":"S0021900225100351_ref22","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2017.2704880"},{"key":"S0021900225100351_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-008-9594-z"},{"key":"S0021900225100351_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/20-AAP1580"},{"key":"S0021900225100351_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-61871-1"},{"key":"S0021900225100351_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-006-9168-x"},{"key":"S0021900225100351_ref13","doi-asserted-by":"crossref","unstructured":"[13] Gao, P. , van der Hofstad, R. , Southwell, A. and Stegehuis, C. (2020). Counting triangles in power-law uniform random graphs. Electron. J. Combin. 27, P3.19.","DOI":"10.37236\/9239"},{"key":"S0021900225100351_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-019-02248-w"},{"key":"S0021900225100351_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-76829-4_5"},{"key":"S0021900225100351_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOP680"},{"key":"S0021900225100351_ref23","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1143936140"},{"key":"S0021900225100351_ref2","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176991491"},{"key":"S0021900225100351_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-41598-7"},{"key":"S0021900225100351_ref26","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.1.033034"},{"key":"S0021900225100351_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"S0021900225100351_ref5","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2005\/06\/P06005"},{"key":"S0021900225100351_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-93635-8"},{"key":"S0021900225100351_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20168"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100351","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:56Z","timestamp":1775525156000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100351\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,3]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100351"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10035","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,3]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}