{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:HO4OAQTRHM2K4DHZEHB5Y4XKXC","short_pith_number":"pith:HO4OAQTR","canonical_record":{"source":{"id":"1509.03568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-11T15:50:11Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"458d839420d8bdf15c74cdc0ead34c1c1cc455f685d874ca9e758c4a2e635764","abstract_canon_sha256":"85e40c4a07e1af5e41acb1cfe9894b7cfb13254e5b9525943aef58f701b0626a"},"schema_version":"1.0"},"canonical_sha256":"3bb8e042713b34ae0cf921c3dc72eab8b350c45e73e8a3e3fe8ddf4d9d9c198e","source":{"kind":"arxiv","id":"1509.03568","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03568","created_at":"2026-05-18T01:33:17Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03568v1","created_at":"2026-05-18T01:33:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03568","created_at":"2026-05-18T01:33:17Z"},{"alias_kind":"pith_short_12","alias_value":"HO4OAQTRHM2K","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HO4OAQTRHM2K4DHZ","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HO4OAQTR","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:HO4OAQTRHM2K4DHZEHB5Y4XKXC","target":"record","payload":{"canonical_record":{"source":{"id":"1509.03568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-11T15:50:11Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"458d839420d8bdf15c74cdc0ead34c1c1cc455f685d874ca9e758c4a2e635764","abstract_canon_sha256":"85e40c4a07e1af5e41acb1cfe9894b7cfb13254e5b9525943aef58f701b0626a"},"schema_version":"1.0"},"canonical_sha256":"3bb8e042713b34ae0cf921c3dc72eab8b350c45e73e8a3e3fe8ddf4d9d9c198e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:17.483812Z","signature_b64":"pITf9xGeNkzN0aSiXDBz1faxboKXzAmym0DXgt95zhyqd1iWxzSB6p3weUg4tgo3UXTQP7KfzqJrT83mvlgPCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bb8e042713b34ae0cf921c3dc72eab8b350c45e73e8a3e3fe8ddf4d9d9c198e","last_reissued_at":"2026-05-18T01:33:17.483103Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:17.483103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.03568","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":"2026-05-18T01:33:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8r9UmV7qWyRJFzQahYH0T8Wa+QP3Yu8RgZOuOf0ZJEVEoFcCva34ENGbsgdGynXL+aSrX3dKBefM5faE4/TNDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:04:38.780573Z"},"content_sha256":"760f4a417b5afc3d38228344266b2c05a7defcee5b83b3b8b8f6e4b0c1bb7b68","schema_version":"1.0","event_id":"sha256:760f4a417b5afc3d38228344266b2c05a7defcee5b83b3b8b8f6e4b0c1bb7b68"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:HO4OAQTRHM2K4DHZEHB5Y4XKXC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Connectivity and giant component in random distance graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Briana Oshiro, Joshua Flynn, Mary Radcliffe","submitted_at":"2015-09-11T15:50:11Z","abstract_excerpt":"Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric space elements. We here propose a model $G=G(X, f)$, in which $(X, d)$ is a metric space, $V(G)=X$, and $\\mathbb{P}(u\\sim v) = f(d(u, v))$, where $f$ is a decreasing function on the set of possible distances in $X$. We consider the case that $X$ is the $n\\times n \\times \\dots\\times n$ integer lattice in dimension $r$, with $d$ the $\\ell_1$ metric, and $f(d) = \\f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03568","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":"2026-05-18T01:33:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AJQyv0NGTQ5tzwA6kICPIpXeYrNYREM0+AjtcxKA9q5p1YPTmmY+o6BdoFv8y0PzK7S+dT3yeSIY8r4wEsPbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:04:38.780919Z"},"content_sha256":"95a181a9b100e3439b23d52c4aa93a6d14e50894fc726d693ee9fc900d6c3a7a","schema_version":"1.0","event_id":"sha256:95a181a9b100e3439b23d52c4aa93a6d14e50894fc726d693ee9fc900d6c3a7a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC/bundle.json","state_url":"https://pith.science/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC/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":"2026-06-23T14:04:38Z","links":{"resolver":"https://pith.science/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC","bundle":"https://pith.science/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC/bundle.json","state":"https://pith.science/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HO4OAQTRHM2K4DHZEHB5Y4XKXC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HO4OAQTRHM2K4DHZEHB5Y4XKXC","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":"85e40c4a07e1af5e41acb1cfe9894b7cfb13254e5b9525943aef58f701b0626a","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-11T15:50:11Z","title_canon_sha256":"458d839420d8bdf15c74cdc0ead34c1c1cc455f685d874ca9e758c4a2e635764"},"schema_version":"1.0","source":{"id":"1509.03568","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03568","created_at":"2026-05-18T01:33:17Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03568v1","created_at":"2026-05-18T01:33:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03568","created_at":"2026-05-18T01:33:17Z"},{"alias_kind":"pith_short_12","alias_value":"HO4OAQTRHM2K","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HO4OAQTRHM2K4DHZ","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HO4OAQTR","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:95a181a9b100e3439b23d52c4aa93a6d14e50894fc726d693ee9fc900d6c3a7a","target":"graph","created_at":"2026-05-18T01:33:17Z","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":"Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric space elements. We here propose a model $G=G(X, f)$, in which $(X, d)$ is a metric space, $V(G)=X$, and $\\mathbb{P}(u\\sim v) = f(d(u, v))$, where $f$ is a decreasing function on the set of possible distances in $X$. We consider the case that $X$ is the $n\\times n \\times \\dots\\times n$ integer lattice in dimension $r$, with $d$ the $\\ell_1$ metric, and $f(d) = \\f","authors_text":"Briana Oshiro, Joshua Flynn, Mary Radcliffe","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-11T15:50:11Z","title":"Connectivity and giant component in random distance graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03568","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":"sha256:760f4a417b5afc3d38228344266b2c05a7defcee5b83b3b8b8f6e4b0c1bb7b68","target":"record","created_at":"2026-05-18T01:33:17Z","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":"85e40c4a07e1af5e41acb1cfe9894b7cfb13254e5b9525943aef58f701b0626a","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-11T15:50:11Z","title_canon_sha256":"458d839420d8bdf15c74cdc0ead34c1c1cc455f685d874ca9e758c4a2e635764"},"schema_version":"1.0","source":{"id":"1509.03568","kind":"arxiv","version":1}},"canonical_sha256":"3bb8e042713b34ae0cf921c3dc72eab8b350c45e73e8a3e3fe8ddf4d9d9c198e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3bb8e042713b34ae0cf921c3dc72eab8b350c45e73e8a3e3fe8ddf4d9d9c198e","first_computed_at":"2026-05-18T01:33:17.483103Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:17.483103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pITf9xGeNkzN0aSiXDBz1faxboKXzAmym0DXgt95zhyqd1iWxzSB6p3weUg4tgo3UXTQP7KfzqJrT83mvlgPCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:17.483812Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.03568","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:760f4a417b5afc3d38228344266b2c05a7defcee5b83b3b8b8f6e4b0c1bb7b68","sha256:95a181a9b100e3439b23d52c4aa93a6d14e50894fc726d693ee9fc900d6c3a7a"],"state_sha256":"d77cd9809f2d891152447b716e4d86f1e79525b761bcfdb82296a6a726acf262"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u8xydmAmcwR8B0QJxwuBcObzfLIh6yZp6vdHnP7WSWnjCqqUnaps69c5kXDiU+J2Mkdo2j5VbcehbxhM8nMqCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T14:04:38.783013Z","bundle_sha256":"77e6f044b4be1efb3712fcd2012afece835861415ae47b70177b0fce17f827de"}}