{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:7NXHYKO3MP5ZIDPX2RA4UZORUI","short_pith_number":"pith:7NXHYKO3","schema_version":"1.0","canonical_sha256":"fb6e7c29db63fb940df7d441ca65d1a236fe99eb271c4ebdef418362f4c91704","source":{"kind":"arxiv","id":"1407.6248","version":3},"attestation_state":"computed","paper":{"title":"The phase transition in the multi-type binomial random graph $G(\\mathbf{n},P)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ang\\'elica Pach\\'on, Christoph Koch, Mihyun Kang","submitted_at":"2014-07-17T16:25:30Z","abstract_excerpt":"We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\\mathbf{n},P)$ every vertex has one of two types, the vector $\\mathbf{n}$ describes the number of vertices of each type, and any edge $\\{u,v\\}$ is present independently with a probability that is given by an entry of the probability matrix $P$ according to the types of $u$ and $v.$ We prove that in the weakly supercritical regime, i.e. if the distance to the critical point of the phase transition is given by an $\\va"},"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":"1407.6248","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-07-17T16:25:30Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0d4c4d7f703a1fd6a575826a2b773b81efa2e2df4319054e6c6477363a7d5ce2","abstract_canon_sha256":"251b59b8b6d62bca0400bcf957867c5f031759e56a08a1ba208db91a77fff3b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:22.703125Z","signature_b64":"RvhLSSD/doFhnZhutND76aE5qmJUQepBYLjuMU4OfNMelycHWVj9yX5rxLhaRIjaPu6iqptSSjmpxE/NkoWeBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb6e7c29db63fb940df7d441ca65d1a236fe99eb271c4ebdef418362f4c91704","last_reissued_at":"2026-05-18T01:35:22.702524Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:22.702524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The phase transition in the multi-type binomial random graph $G(\\mathbf{n},P)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ang\\'elica Pach\\'on, Christoph Koch, Mihyun Kang","submitted_at":"2014-07-17T16:25:30Z","abstract_excerpt":"We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\\mathbf{n},P)$ every vertex has one of two types, the vector $\\mathbf{n}$ describes the number of vertices of each type, and any edge $\\{u,v\\}$ is present independently with a probability that is given by an entry of the probability matrix $P$ according to the types of $u$ and $v.$ We prove that in the weakly supercritical regime, i.e. if the distance to the critical point of the phase transition is given by an $\\va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6248","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.6248","created_at":"2026-05-18T01:35:22.702628+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.6248v3","created_at":"2026-05-18T01:35:22.702628+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6248","created_at":"2026-05-18T01:35:22.702628+00:00"},{"alias_kind":"pith_short_12","alias_value":"7NXHYKO3MP5Z","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"7NXHYKO3MP5ZIDPX","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"7NXHYKO3","created_at":"2026-05-18T12:28:19.803747+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI","json":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI.json","graph_json":"https://pith.science/api/pith-number/7NXHYKO3MP5ZIDPX2RA4UZORUI/graph.json","events_json":"https://pith.science/api/pith-number/7NXHYKO3MP5ZIDPX2RA4UZORUI/events.json","paper":"https://pith.science/paper/7NXHYKO3"},"agent_actions":{"view_html":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI","download_json":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI.json","view_paper":"https://pith.science/paper/7NXHYKO3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.6248&json=true","fetch_graph":"https://pith.science/api/pith-number/7NXHYKO3MP5ZIDPX2RA4UZORUI/graph.json","fetch_events":"https://pith.science/api/pith-number/7NXHYKO3MP5ZIDPX2RA4UZORUI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI/action/storage_attestation","attest_author":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI/action/author_attestation","sign_citation":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI/action/citation_signature","submit_replication":"https://pith.science/pith/7NXHYKO3MP5ZIDPX2RA4UZORUI/action/replication_record"}},"created_at":"2026-05-18T01:35:22.702628+00:00","updated_at":"2026-05-18T01:35:22.702628+00:00"}