{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T08:50:27Z","timestamp":1782291027239,"version":"3.54.5"},"reference-count":40,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T00:00:00Z","timestamp":1762214400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as a hypothesis-testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on\n                    <jats:italic>n<\/jats:italic>\n                    vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100387_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">k equals o left parenthesis n right parenthesis<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>k<\/mml:mi>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:mi>o<\/mml:mi>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                        <\/mml:math>\n                        <jats:tex-math>$k=o(n)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    vertices are given spatial locations and connect following the\n                    <jats:italic>geometric<\/jats:italic>\n                    inhomogeneous random graph connection rule. The remaining\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100387_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">n minus k<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo>\u2212<\/mml:mo>\n                          <mml:mi>k<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$n-k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test based on counting normalized triangles to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100387_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">n right arrow normal infinity<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo stretchy=\"false\">\u2192<\/mml:mo>\n                          <mml:mi mathvariant=\"normal\">\u221e<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$n\\to\\infty$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and identifies large-degree vertices of the community with high probability.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10038","type":"journal-article","created":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T08:44:37Z","timestamp":1762245877000},"page":"484-500","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Localized geometry detection in scale-free random graphs"],"prefix":"10.1017","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8431-0636","authenticated-orcid":false,"given":"Gianmarco","family":"Bet","sequence":"first","affiliation":[{"name":"Universit\u00e0 degli Studi di Firenze"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4642-8507","authenticated-orcid":false,"given":"Riccardo","family":"Michielan","sequence":"additional","affiliation":[{"name":"Gran Sasso Science Institute"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Clara","family":"Stegehuis","sequence":"additional","affiliation":[{"name":"University of Twente"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,11,4]]},"reference":[{"key":"S0021900225100387_ref25","doi-asserted-by":"publisher","DOI":"10.1214\/21-EJP708"},{"key":"S0021900225100387_ref11","unstructured":"[11] Bresler, G. and Nagaraj, D. (2018). Optimal single sample tests for structured versus unstructured network data. Proc. Mach. Learn. Res. 75, 1657\u20131690."},{"key":"S0021900225100387_ref21","unstructured":"[21] Hajek, B. , Wu, Y. and Xu, J. (2015). Computational lower bounds for community detection on random graphs. Proc. Mach. Learn. Res. 40, 899\u2013928."},{"key":"S0021900225100387_ref33","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.69.026113"},{"key":"S0021900225100387_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/0378-8733(83)90021-7"},{"key":"S0021900225100387_ref5","doi-asserted-by":"publisher","DOI":"10.1145\/1839490.1839494"},{"key":"S0021900225100387_ref3","volume-title":"The Probabilistic Method","author":"Alon","year":"2016"},{"key":"S0021900225100387_ref2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199810\/12)13:3\/4<457::AID-RSA14>3.0.CO;2-W"},{"key":"S0021900225100387_ref24","unstructured":"[24] Jin, J. , Ke, Z. and Luo, S. (2018). 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Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}