{"paper":{"title":"On reversible cascades in scale-free and Erd\\H{o}s-R\\'enyi random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Ching-Lueh Chang","submitted_at":"2010-11-02T15:53:53Z","abstract_excerpt":"Consider the following cascading process on a simple undirected graph $G(V,E)$ with diameter $\\Delta$. In round zero, a set $S\\subseteq V$ of vertices, called the seeds, are active. In round $i+1,$ $i\\in\\mathbb{N},$ a non-isolated vertex is activated if at least a $\\rho\\in(\\,0,1\\,]$ fraction of its neighbors are active in round $i$; it is deactivated otherwise. For $k\\in\\mathbb{N},$ let $\\text{min-seed}^{(k)}(G,\\rho)$ be the minimum number of seeds needed to activate all vertices in or before round $k$. This paper derives upper bounds on $\\text{min-seed}^{(k)}(G,\\rho)$. In particular, if $G$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0653","kind":"arxiv","version":1},"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"}