{"paper":{"title":"Graphon branching processes and fractional isomorphism","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Anna Margarethe Limbach, Eng Keat Hng, Jan Hladk\\'y","submitted_at":"2024-08-05T14:48:42Z","abstract_excerpt":"In their study of the giant component in inhomogeneous random graphs, Bollob\\'as, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that the tree structures underlying two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic, a notion introduced by Greb\\'ik and Rocha.\n  A different class of branching processes was introduced by Hladk\\'y, Nachmias, and Tran in relation to uniform spanning trees in finite graphs approximating a given connected graphon. We prove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.02528","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.02528/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}