{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2U3WQLIK47ON46Z5IC5X3MQE3T","short_pith_number":"pith:2U3WQLIK","canonical_record":{"source":{"id":"1507.01196","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-07-05T11:22:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5b01e6e8bd08257d3230a2ba5dbec73ca2db7e6946081fd4a8c278c6073ee216","abstract_canon_sha256":"fa81273466f5ec8651627292ceb2cf2bbce905334af81a816cce4fbbab6e5130"},"schema_version":"1.0"},"canonical_sha256":"d537682d0ae7dcde7b3d40bb7db204dce85d9c2bd21e452072548e6e41799751","source":{"kind":"arxiv","id":"1507.01196","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.01196","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"arxiv_version","alias_value":"1507.01196v1","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.01196","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"pith_short_12","alias_value":"2U3WQLIK47ON","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2U3WQLIK47ON46Z5","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2U3WQLIK","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2U3WQLIK47ON46Z5IC5X3MQE3T","target":"record","payload":{"canonical_record":{"source":{"id":"1507.01196","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-07-05T11:22:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5b01e6e8bd08257d3230a2ba5dbec73ca2db7e6946081fd4a8c278c6073ee216","abstract_canon_sha256":"fa81273466f5ec8651627292ceb2cf2bbce905334af81a816cce4fbbab6e5130"},"schema_version":"1.0"},"canonical_sha256":"d537682d0ae7dcde7b3d40bb7db204dce85d9c2bd21e452072548e6e41799751","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:17.766253Z","signature_b64":"y5+2QcRa0n4B41+rfAoxYchmHlzTcc4GjVhhnI5zpEPjCeeIRZueRWPVxFAcAoNkl+ACnTI2jrH6EtP+HugCCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d537682d0ae7dcde7b3d40bb7db204dce85d9c2bd21e452072548e6e41799751","last_reissued_at":"2026-05-18T01:37:17.765551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:17.765551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.01196","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":"2026-05-18T01:37:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VC0HwFxDYraVDWchKf3UKA2dS7gPauA1O+A/D/XA2kAleW8p9hzOAljtumvecGDOy17qi8n6P/b6fQBIEQK9Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:35:32.547442Z"},"content_sha256":"e26a20f50f1e3093059098d8a509fd38565e314b83fe7bdbe5506d81f3e6bc85","schema_version":"1.0","event_id":"sha256:e26a20f50f1e3093059098d8a509fd38565e314b83fe7bdbe5506d81f3e6bc85"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2U3WQLIK47ON46Z5IC5X3MQE3T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit Expanding Expanders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Asaf Valadarsky, Michael Dinitz, Michael Schapira","submitted_at":"2015-07-05T11:22:45Z","abstract_excerpt":"Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite family is not enough: we need expanders which are \"close\" to each other. We study the following question: Construct an an infinite sequence of expanders $G_0,G_1,\\dots$, such that for every two consecutive graphs $G_i$ and $G_{i+1}$, $G_{i+1}$ can be obtained from $G_i$ by adding a single vertex and inserting/removing a small number of edges, which we call th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01196","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":"2026-05-18T01:37:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7lQW+FZH7ZsascFtjmVt7ixj0pGs78fKYYtp/WPjx69g6GTS+4Xfce/Gvvj7cTpUFBZZKwVDVefFXq+77Q3NCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:35:32.547772Z"},"content_sha256":"ad54047dde4b76d367226f3a77b930d1a9da655bcbcf95f39ba169b945add6df","schema_version":"1.0","event_id":"sha256:ad54047dde4b76d367226f3a77b930d1a9da655bcbcf95f39ba169b945add6df"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2U3WQLIK47ON46Z5IC5X3MQE3T/bundle.json","state_url":"https://pith.science/pith/2U3WQLIK47ON46Z5IC5X3MQE3T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2U3WQLIK47ON46Z5IC5X3MQE3T/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":"2026-06-08T22:35:32Z","links":{"resolver":"https://pith.science/pith/2U3WQLIK47ON46Z5IC5X3MQE3T","bundle":"https://pith.science/pith/2U3WQLIK47ON46Z5IC5X3MQE3T/bundle.json","state":"https://pith.science/pith/2U3WQLIK47ON46Z5IC5X3MQE3T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2U3WQLIK47ON46Z5IC5X3MQE3T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2U3WQLIK47ON46Z5IC5X3MQE3T","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":"fa81273466f5ec8651627292ceb2cf2bbce905334af81a816cce4fbbab6e5130","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-07-05T11:22:45Z","title_canon_sha256":"5b01e6e8bd08257d3230a2ba5dbec73ca2db7e6946081fd4a8c278c6073ee216"},"schema_version":"1.0","source":{"id":"1507.01196","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.01196","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"arxiv_version","alias_value":"1507.01196v1","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.01196","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"pith_short_12","alias_value":"2U3WQLIK47ON","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2U3WQLIK47ON46Z5","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2U3WQLIK","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:ad54047dde4b76d367226f3a77b930d1a9da655bcbcf95f39ba169b945add6df","target":"graph","created_at":"2026-05-18T01:37:17Z","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":"Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite family is not enough: we need expanders which are \"close\" to each other. We study the following question: Construct an an infinite sequence of expanders $G_0,G_1,\\dots$, such that for every two consecutive graphs $G_i$ and $G_{i+1}$, $G_{i+1}$ can be obtained from $G_i$ by adding a single vertex and inserting/removing a small number of edges, which we call th","authors_text":"Asaf Valadarsky, Michael Dinitz, Michael Schapira","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-07-05T11:22:45Z","title":"Explicit Expanding Expanders"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01196","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:e26a20f50f1e3093059098d8a509fd38565e314b83fe7bdbe5506d81f3e6bc85","target":"record","created_at":"2026-05-18T01:37:17Z","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":"fa81273466f5ec8651627292ceb2cf2bbce905334af81a816cce4fbbab6e5130","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-07-05T11:22:45Z","title_canon_sha256":"5b01e6e8bd08257d3230a2ba5dbec73ca2db7e6946081fd4a8c278c6073ee216"},"schema_version":"1.0","source":{"id":"1507.01196","kind":"arxiv","version":1}},"canonical_sha256":"d537682d0ae7dcde7b3d40bb7db204dce85d9c2bd21e452072548e6e41799751","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d537682d0ae7dcde7b3d40bb7db204dce85d9c2bd21e452072548e6e41799751","first_computed_at":"2026-05-18T01:37:17.765551Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:17.765551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y5+2QcRa0n4B41+rfAoxYchmHlzTcc4GjVhhnI5zpEPjCeeIRZueRWPVxFAcAoNkl+ACnTI2jrH6EtP+HugCCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:17.766253Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.01196","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e26a20f50f1e3093059098d8a509fd38565e314b83fe7bdbe5506d81f3e6bc85","sha256:ad54047dde4b76d367226f3a77b930d1a9da655bcbcf95f39ba169b945add6df"],"state_sha256":"5d3dd9f60624b495fc9a639703378d721ddc924ecdcb20b68d02d7bdc9fc2ebb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f7qfdlUDf8U002UvEJeOYt/Bgqa6K2asyodEexVSDbY6CBCDCGRyBoIqU1m8wP9ANYG0oP7YiQmtrUQXXabGCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T22:35:32.549605Z","bundle_sha256":"ea0d5159a59551983f9e5f092d41fa6f95e40cd7c429cc6fd70b483c5a0aa8cb"}}