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Each node of a tree has degree selected from a finite predefined set of non-negative integers and starting from any node, all nodes at the same graph distance from it have the same degree. We show the existence of the critical threshold $p_f(\\theta) \\in (0,1)$ such that with high probability, (i) if $p > p_f(\\theta)$ then the periodic tree becomes fully active, while (ii) if $p < p_f(\\theta)$ then a periodic tree does not become ful"},"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":"1311.7449","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-29T00:20:51Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"ef0cfff13c0339113a48d99d36fc97b4237e0045628d3a810bea7036b3d09fa0","abstract_canon_sha256":"4327a27bb79c5a0eb6ea40982731f84a9ba1cccb93dff2ab8a284cace34ef229"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:56.194638Z","signature_b64":"Oa+uf5JD8zMW553p5qmImAy0rIxHSuPtyyYyyMLksbIr++Y1gcJ2hs9U1LmCsiWY9SSs82mjWl02REokYiEdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d3664808c263f49e2d29ac7fe28168852a0f6aa211d9761e77321125231814b","last_reissued_at":"2026-05-18T03:05:56.194099Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:56.194099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bootstrap Percolation on Periodic Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.PR","authors_text":"Iraj Saniee, Milan Bradonji\\'c","submitted_at":"2013-11-29T00:20:51Z","abstract_excerpt":"We study bootstrap percolation with the threshold parameter $\\theta \\geq 2$ and the initial probability $p$ on infinite periodic trees that are defined as follows. 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