{"paper":{"title":"Spreading Processes and Large Components in Ordered, Directed Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.SI"],"primary_cat":"math.CO","authors_text":"Malik Magdon-Ismail, Paul Horn","submitted_at":"2012-09-10T18:43:56Z","abstract_excerpt":"Order the vertices of a directed random graph \\math{v_1,...,v_n}; edge \\math{(v_i,v_j)} for \\math{i<j} exists independently with probability \\math{p}. This random graph model is related to certain spreading processes on networks. We consider the component reachable from \\math{v_1} and prove existence of a sharp threshold \\math{p^*=\\log n/n} at which this reachable component transitions from \\math{o(n)} to \\math{\\Omega(n)}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2088","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}