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Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set.\n  We study this process on the binomial random graph $G:=G(n,p)$ with $p: = \\frac{d}{n}$ and $1 \\ll d \\ll \\left(\\frac{n \\log \\log n}{\\log^2 n}\\right)^{\\frac{r-1}{r}}$. Assuming $r > 1$ to be a constant that does not depend on $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1602.01751","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-04T17:10:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5ea0e0b3dd2261886c7276599205324f12ca5b4ad24c40bc646df6124910c524","abstract_canon_sha256":"86ec8edf9d0e81bfd467f58492b5dd3b213e097306a51b6eb680a94dd74aa180"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:18.196420Z","signature_b64":"MbTrDgNf1rs5CmEDhaOaWBJONLsdgchFgRe9Gy9oMS1MiE8w10LjuRNlQh2zn++QgqRJtMB7r82d5Fzp617eBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca2bdf80c11fca1241a4e70f9d13aa0b5d63e53dae40cc2406b4f147c8467dec","last_reissued_at":"2026-05-18T01:21:18.195934Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:18.195934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Contagious Sets in Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Daniel Reichman, Michael Krivelevich, Uriel Feige","submitted_at":"2016-02-04T17:10:06Z","abstract_excerpt":"We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A \\emph{contagious set} is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set.\n  We study this process on the binomial random graph $G:=G(n,p)$ with $p: = \\frac{d}{n}$ and $1 \\ll d \\ll \\left(\\frac{n \\log \\log n}{\\log^2 n}\\right)^{\\frac{r-1}{r}}$. 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