{"paper":{"title":"Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","q-bio.PE"],"primary_cat":"math.PR","authors_text":"P. L. Krapivsky, Yoshiaki Itoh","submitted_at":"2012-06-16T22:46:00Z","abstract_excerpt":"We study a class of directed random graphs. In these graphs, the interval [0,x] is the vertex set, and from each y\\in [0,x], directed links are drawn to points in the interval (y,x] which are chosen uniformly with density one. We analyze the length of the longest directed path starting from the origin. In the large x limit, we employ traveling wave techniques to extract the asymptotic behavior of this quantity. We also study the size of a cascade tree composed of vertices which can be reached via directed paths starting at the origin."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3711","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"}