{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:EWYRD7JTZKC45VQECDKVI7LVOS","short_pith_number":"pith:EWYRD7JT","canonical_record":{"source":{"id":"1609.00290","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-01T15:51:56Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"10b127ebd5c22668fd2e8ef8be4ea38709d4d86aa7010e7225d085430165de4d","abstract_canon_sha256":"8e513553276a4ac1d6ead51f36c27554386403b3effd03f67541ec37caa9d865"},"schema_version":"1.0"},"canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","source":{"kind":"arxiv","id":"1609.00290","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00290","created_at":"2026-05-18T01:06:25Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00290v1","created_at":"2026-05-18T01:06:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00290","created_at":"2026-05-18T01:06:25Z"},{"alias_kind":"pith_short_12","alias_value":"EWYRD7JTZKC4","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"EWYRD7JTZKC45VQE","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"EWYRD7JT","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:EWYRD7JTZKC45VQECDKVI7LVOS","target":"record","payload":{"canonical_record":{"source":{"id":"1609.00290","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-01T15:51:56Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"10b127ebd5c22668fd2e8ef8be4ea38709d4d86aa7010e7225d085430165de4d","abstract_canon_sha256":"8e513553276a4ac1d6ead51f36c27554386403b3effd03f67541ec37caa9d865"},"schema_version":"1.0"},"canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:25.210612Z","signature_b64":"u8KzHcupRH+OR6BoRNmPPPmI5a+uApTvyv/DNElxmrukxo8kiXp0aiGy+BoN+NDyKLbZUejCEGeXMVOwqDPiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","last_reissued_at":"2026-05-18T01:06:25.210047Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:25.210047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.00290","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T2+757qImAuSY26HLBxlFleQZruFJMR4SkqYZNuS8aDCcmWGUlwBzA/KA6+moSo3wPMxNCJZxztYMPciG0YMBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T01:02:01.691989Z"},"content_sha256":"afd61a1e8141670362d3759bafe4a97e1f79bbc7c268e2e780224f13fb88e28c","schema_version":"1.0","event_id":"sha256:afd61a1e8141670362d3759bafe4a97e1f79bbc7c268e2e780224f13fb88e28c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:EWYRD7JTZKC45VQECDKVI7LVOS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting strongly connected $(k_1,k_2)$-directed cores","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Boris Pittel","submitted_at":"2016-09-01T15:51:56Z","abstract_excerpt":"Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\\min\\{k_1,k_2\\}\\ge 2$ and $M/N>\\max\\{k_1,k_2\\}$, $M=\\Theta(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00290","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+47gYk3Gc2wUkQj0YoIAFKi0SAGha489rMrMvCdzKJnwq/hglZDBTn3I8J2eS/YJXwPhSQXK2yg9660lCMWRBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T01:02:01.692353Z"},"content_sha256":"bfba4a80e3e780a55a94f16d9e782c84ce935612685394b43ce9641c5e856158","schema_version":"1.0","event_id":"sha256:bfba4a80e3e780a55a94f16d9e782c84ce935612685394b43ce9641c5e856158"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/bundle.json","state_url":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EWYRD7JTZKC45VQECDKVI7LVOS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T01:02:01Z","links":{"resolver":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS","bundle":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/bundle.json","state":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EWYRD7JTZKC45VQECDKVI7LVOS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:EWYRD7JTZKC45VQECDKVI7LVOS","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":"8e513553276a4ac1d6ead51f36c27554386403b3effd03f67541ec37caa9d865","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-01T15:51:56Z","title_canon_sha256":"10b127ebd5c22668fd2e8ef8be4ea38709d4d86aa7010e7225d085430165de4d"},"schema_version":"1.0","source":{"id":"1609.00290","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00290","created_at":"2026-05-18T01:06:25Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00290v1","created_at":"2026-05-18T01:06:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00290","created_at":"2026-05-18T01:06:25Z"},{"alias_kind":"pith_short_12","alias_value":"EWYRD7JTZKC4","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"EWYRD7JTZKC45VQE","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"EWYRD7JT","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:bfba4a80e3e780a55a94f16d9e782c84ce935612685394b43ce9641c5e856158","target":"graph","created_at":"2026-05-18T01:06:25Z","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":"Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\\min\\{k_1,k_2\\}\\ge 2$ and $M/N>\\max\\{k_1,k_2\\}$, $M=\\Theta(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core e","authors_text":"Boris Pittel","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-01T15:51:56Z","title":"Counting strongly connected $(k_1,k_2)$-directed cores"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00290","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:afd61a1e8141670362d3759bafe4a97e1f79bbc7c268e2e780224f13fb88e28c","target":"record","created_at":"2026-05-18T01:06:25Z","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":"8e513553276a4ac1d6ead51f36c27554386403b3effd03f67541ec37caa9d865","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-01T15:51:56Z","title_canon_sha256":"10b127ebd5c22668fd2e8ef8be4ea38709d4d86aa7010e7225d085430165de4d"},"schema_version":"1.0","source":{"id":"1609.00290","kind":"arxiv","version":1}},"canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","first_computed_at":"2026-05-18T01:06:25.210047Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:25.210047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u8KzHcupRH+OR6BoRNmPPPmI5a+uApTvyv/DNElxmrukxo8kiXp0aiGy+BoN+NDyKLbZUejCEGeXMVOwqDPiDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:25.210612Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.00290","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afd61a1e8141670362d3759bafe4a97e1f79bbc7c268e2e780224f13fb88e28c","sha256:bfba4a80e3e780a55a94f16d9e782c84ce935612685394b43ce9641c5e856158"],"state_sha256":"0e0927d2ec0823aed798ea8385a50c8694c95d839209da21667f67795af579a5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AjI+tfjF9s5sWRZxKBqlsPsWcwFLXTLbZsE34K3Fe9sWbYV8kz6BWFajoSiFbr5XS2dbKyxU2M5SBTAHh9bwCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T01:02:01.694387Z","bundle_sha256":"f3817badb7e00d38290d9583c6dc4edf664c13bd4fc29f2597e411fa35a218a9"}}