{"paper":{"title":"Global synchronization beyond dense graphs: the case of threshold graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.CO","math.OC"],"primary_cat":"math.DS","authors_text":"Hongjin Wu, Ulrik Brandes","submitted_at":"2025-11-16T15:28:07Z","abstract_excerpt":"Given a graph \\(G\\) with adjacency matrix \\(A\\), consider the homogeneous Kuramoto energy $E_G(\\boldsymbol{\\theta}):=\\frac{1}{2}\\sum_{1\\leq i,j\\leq n}A_{ij}\\bigl(1-\\cos(\\theta_i-\\theta_j)\\bigr)$. We call \\(G\\) \\emph{second-order globally synchronizing} if every second-order stationary point of \\(E_G\\) is fully synchronized. This property implies \\emph{global synchronization}, namely that, up to a measure-zero set of initial conditions, trajectories of the Kuramoto model converge to a fully synchronized state. A fundamental graph-theoretic question is to identify which graph structures have thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.12646","kind":"arxiv","version":6},"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/2511.12646/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"}