{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:G6Y23UG3A2C4GPVETMY43GVYIO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"748fef792a8634d68022773939f9bd449cd6874af5bd0ec5e04ca2b1b08c5d32","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-16T10:24:28Z","title_canon_sha256":"8a4fec551e854be3557d3416aff7cd88d90a98645421827f7eef57f22a2a0f2e"},"schema_version":"1.0","source":{"id":"1807.05772","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.05772","created_at":"2026-05-18T00:10:42Z"},{"alias_kind":"arxiv_version","alias_value":"1807.05772v1","created_at":"2026-05-18T00:10:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.05772","created_at":"2026-05-18T00:10:42Z"},{"alias_kind":"pith_short_12","alias_value":"G6Y23UG3A2C4","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"G6Y23UG3A2C4GPVE","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"G6Y23UG3","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:15847cf2164496d3268429716e13683822bb64671b9ffc9dceb27a0862b67880","target":"graph","created_at":"2026-05-18T00:10:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of \\emph{patchworks} to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows us to tackle a wide range of problems. We obtain the asymptotic number and the limiting distribution of the number of subgraphs which are isomorphic to a graph from a given set of graphs. The","authors_text":"Bernhard Gittenberger, Dani\\`ele Gardy, \\'Elie de Panafieu, Gwendal Collet, Vlady Ravelomanana","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-16T10:24:28Z","title":"Threshold functions for small subgraphs in simple graphs and multigraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.05772","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1570a82d21c317f93e2f44a3a483f370577e29cc8aa0562ed07fac106fd46412","target":"record","created_at":"2026-05-18T00:10:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"748fef792a8634d68022773939f9bd449cd6874af5bd0ec5e04ca2b1b08c5d32","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-16T10:24:28Z","title_canon_sha256":"8a4fec551e854be3557d3416aff7cd88d90a98645421827f7eef57f22a2a0f2e"},"schema_version":"1.0","source":{"id":"1807.05772","kind":"arxiv","version":1}},"canonical_sha256":"37b1add0db0685c33ea49b31cd9ab8438ebd6663fb329b39e87bba6438aec3a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37b1add0db0685c33ea49b31cd9ab8438ebd6663fb329b39e87bba6438aec3a2","first_computed_at":"2026-05-18T00:10:42.584835Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:42.584835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rP+SYESUZgTqOQfTZRUObN3cRg/vkBmd7xDrqmu+KouSR0CuKlrRCPfl0m1LBi1/3LYoCDOvZlnCTd463CttDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:42.585522Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.05772","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1570a82d21c317f93e2f44a3a483f370577e29cc8aa0562ed07fac106fd46412","sha256:15847cf2164496d3268429716e13683822bb64671b9ffc9dceb27a0862b67880"],"state_sha256":"312d7baf338f536f786ad872525849d750f8d0992c32fcf44dc0299ef2e056df"}