{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:5MXIJ2SKIV54GO6IQEA4WBVTOQ","short_pith_number":"pith:5MXIJ2SK","schema_version":"1.0","canonical_sha256":"eb2e84ea4a457bc33bc88101cb06b37408d720f6d568aa176e963f1b2a9db1e2","source":{"kind":"arxiv","id":"1611.05728","version":1},"attestation_state":"computed","paper":{"title":"Component structure of the configuration model: barely supercritical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Malwina Luczak, Remco van der Hofstad, Svante Janson","submitted_at":"2016-11-17T15:11:00Z","abstract_excerpt":"We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $\\nu_n={\\mathbb E} [D_n(D_n-1)]/{\\mathbb E}[D_n]$ and, assuming that $\\nu_n \\to 1$ as $n \\to \\infty$, we write $\\varepsilon_n=\\nu_n-1$. We call the setting where $\\varepsilon_n n^{1/3}/({\\mathbb E}[D_n^3])^{2/3} \\to \\infty$ the {\\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \\to \\infty$.\n  Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \\rho_n {\\mathbb E} D_n (1+o"},"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":"1611.05728","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-17T15:11:00Z","cross_cats_sorted":[],"title_canon_sha256":"bbf893079b7545de26c3c9a2e4cebc3e334d20b7019e23755efa36b273d18872","abstract_canon_sha256":"d9e77e6194cc910554e724ce375dcd24817f1a062546f90bd216056c444a3e2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:02.776862Z","signature_b64":"PwlYW6Y7Y5Jnu55/US1lHA988/spI3sZGpjxp+svumaBZr9f6nkCAUNqpC2swtOsUmUFmvrlJPH169DZfeWOBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb2e84ea4a457bc33bc88101cb06b37408d720f6d568aa176e963f1b2a9db1e2","last_reissued_at":"2026-05-18T00:58:02.776329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:02.776329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Component structure of the configuration model: barely supercritical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Malwina Luczak, Remco van der Hofstad, Svante Janson","submitted_at":"2016-11-17T15:11:00Z","abstract_excerpt":"We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $\\nu_n={\\mathbb E} [D_n(D_n-1)]/{\\mathbb E}[D_n]$ and, assuming that $\\nu_n \\to 1$ as $n \\to \\infty$, we write $\\varepsilon_n=\\nu_n-1$. We call the setting where $\\varepsilon_n n^{1/3}/({\\mathbb E}[D_n^3])^{2/3} \\to \\infty$ the {\\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \\to \\infty$.\n  Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \\rho_n {\\mathbb E} D_n (1+o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05728","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1611.05728","created_at":"2026-05-18T00:58:02.776419+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.05728v1","created_at":"2026-05-18T00:58:02.776419+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05728","created_at":"2026-05-18T00:58:02.776419+00:00"},{"alias_kind":"pith_short_12","alias_value":"5MXIJ2SKIV54","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5MXIJ2SKIV54GO6I","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5MXIJ2SK","created_at":"2026-05-18T12:30:01.593930+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/5MXIJ2SKIV54GO6IQEA4WBVTOQ","json":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ.json","graph_json":"https://pith.science/api/pith-number/5MXIJ2SKIV54GO6IQEA4WBVTOQ/graph.json","events_json":"https://pith.science/api/pith-number/5MXIJ2SKIV54GO6IQEA4WBVTOQ/events.json","paper":"https://pith.science/paper/5MXIJ2SK"},"agent_actions":{"view_html":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ","download_json":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ.json","view_paper":"https://pith.science/paper/5MXIJ2SK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.05728&json=true","fetch_graph":"https://pith.science/api/pith-number/5MXIJ2SKIV54GO6IQEA4WBVTOQ/graph.json","fetch_events":"https://pith.science/api/pith-number/5MXIJ2SKIV54GO6IQEA4WBVTOQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ/action/storage_attestation","attest_author":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ/action/author_attestation","sign_citation":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ/action/citation_signature","submit_replication":"https://pith.science/pith/5MXIJ2SKIV54GO6IQEA4WBVTOQ/action/replication_record"}},"created_at":"2026-05-18T00:58:02.776419+00:00","updated_at":"2026-05-18T00:58:02.776419+00:00"}