{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:T7OHEF6AFL6MYEWBCFFDQJSLH4","short_pith_number":"pith:T7OHEF6A","canonical_record":{"source":{"id":"1812.11562","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-30T15:50:43Z","cross_cats_sorted":[],"title_canon_sha256":"073b05c4ffa209a660b265661c88d96b57356799b737bd8be04626bc17c9d8d7","abstract_canon_sha256":"591deb038cd515ebeccd0fd0c7d01ba26130800ead0371f7bae0023438f1f0f4"},"schema_version":"1.0"},"canonical_sha256":"9fdc7217c02afccc12c1114a38264b3f35077a68ee238467a0ee1bd4853850e7","source":{"kind":"arxiv","id":"1812.11562","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.11562","created_at":"2026-05-17T23:55:26Z"},{"alias_kind":"arxiv_version","alias_value":"1812.11562v2","created_at":"2026-05-17T23:55:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.11562","created_at":"2026-05-17T23:55:26Z"},{"alias_kind":"pith_short_12","alias_value":"T7OHEF6AFL6M","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"T7OHEF6AFL6MYEWB","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"T7OHEF6A","created_at":"2026-05-18T12:32:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:T7OHEF6AFL6MYEWBCFFDQJSLH4","target":"record","payload":{"canonical_record":{"source":{"id":"1812.11562","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-30T15:50:43Z","cross_cats_sorted":[],"title_canon_sha256":"073b05c4ffa209a660b265661c88d96b57356799b737bd8be04626bc17c9d8d7","abstract_canon_sha256":"591deb038cd515ebeccd0fd0c7d01ba26130800ead0371f7bae0023438f1f0f4"},"schema_version":"1.0"},"canonical_sha256":"9fdc7217c02afccc12c1114a38264b3f35077a68ee238467a0ee1bd4853850e7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:26.943046Z","signature_b64":"p3BbexgcHNQaFouS2aPl5agwY3kCea0PK24Ur86lgtl0iLkuNP+GMEY+1zZDqfcCy/Nj2NpQlYaSkvjzUSAPCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fdc7217c02afccc12c1114a38264b3f35077a68ee238467a0ee1bd4853850e7","last_reissued_at":"2026-05-17T23:55:26.942571Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:26.942571Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.11562","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:55:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V2zE1VAfcCFp3uet82P0rto+eTDY2DfIeXteherJb6sVJaAFq0lh1S4Prw3VQ2HnEXfYruUeH/DdYnGO7IJAAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:28:57.934960Z"},"content_sha256":"0ca5fdf9a90f106e32dc1df7bb59d6278dcc431e573a42ffa22c664ca80d68da","schema_version":"1.0","event_id":"sha256:0ca5fdf9a90f106e32dc1df7bb59d6278dcc431e573a42ffa22c664ca80d68da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:T7OHEF6AFL6MYEWBCFFDQJSLH4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Expanders - how to find them, and what to find in them","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Krivelevich","submitted_at":"2018-12-30T15:50:43Z","abstract_excerpt":"A graph $G=(V,E)$ is called an expander if every vertex subset $U$ of size up to $|V|/2$ has an external neighborhood whose size is comparable to $|U|$. Expanders have been a subject of intensive research for more than three decades and have become one of the central notions of modern graph theory.\n  We first discuss the above definition of an expander and its alternatives. Then we present examples of families of expanding graphs and state basic properties of expanders. Next, we introduce a way to argue that a given graph contains a large expanding subgraph. Finally we research properties of e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.11562","kind":"arxiv","version":2},"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":"2026-05-17T23:55:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LdHF6GAMqxlFctH7YqKQEVPwLj69QfkLk/wzNHNGavm+uUyixOcs3ES6/vi8rjCh3uxiltPpdy9KGe5r0yxKCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:28:57.935681Z"},"content_sha256":"db9e3363284f731789531c710f291bb75a6e7a6516cd622e2faca7cf2485a4ef","schema_version":"1.0","event_id":"sha256:db9e3363284f731789531c710f291bb75a6e7a6516cd622e2faca7cf2485a4ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4/bundle.json","state_url":"https://pith.science/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4/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":"2026-05-25T20:28:57Z","links":{"resolver":"https://pith.science/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4","bundle":"https://pith.science/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4/bundle.json","state":"https://pith.science/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T7OHEF6AFL6MYEWBCFFDQJSLH4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:T7OHEF6AFL6MYEWBCFFDQJSLH4","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":"591deb038cd515ebeccd0fd0c7d01ba26130800ead0371f7bae0023438f1f0f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-30T15:50:43Z","title_canon_sha256":"073b05c4ffa209a660b265661c88d96b57356799b737bd8be04626bc17c9d8d7"},"schema_version":"1.0","source":{"id":"1812.11562","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.11562","created_at":"2026-05-17T23:55:26Z"},{"alias_kind":"arxiv_version","alias_value":"1812.11562v2","created_at":"2026-05-17T23:55:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.11562","created_at":"2026-05-17T23:55:26Z"},{"alias_kind":"pith_short_12","alias_value":"T7OHEF6AFL6M","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"T7OHEF6AFL6MYEWB","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"T7OHEF6A","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:db9e3363284f731789531c710f291bb75a6e7a6516cd622e2faca7cf2485a4ef","target":"graph","created_at":"2026-05-17T23:55:26Z","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":"A graph $G=(V,E)$ is called an expander if every vertex subset $U$ of size up to $|V|/2$ has an external neighborhood whose size is comparable to $|U|$. Expanders have been a subject of intensive research for more than three decades and have become one of the central notions of modern graph theory.\n  We first discuss the above definition of an expander and its alternatives. Then we present examples of families of expanding graphs and state basic properties of expanders. Next, we introduce a way to argue that a given graph contains a large expanding subgraph. Finally we research properties of e","authors_text":"Michael Krivelevich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-30T15:50:43Z","title":"Expanders - how to find them, and what to find in them"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.11562","kind":"arxiv","version":2},"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":"sha256:0ca5fdf9a90f106e32dc1df7bb59d6278dcc431e573a42ffa22c664ca80d68da","target":"record","created_at":"2026-05-17T23:55:26Z","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":"591deb038cd515ebeccd0fd0c7d01ba26130800ead0371f7bae0023438f1f0f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-30T15:50:43Z","title_canon_sha256":"073b05c4ffa209a660b265661c88d96b57356799b737bd8be04626bc17c9d8d7"},"schema_version":"1.0","source":{"id":"1812.11562","kind":"arxiv","version":2}},"canonical_sha256":"9fdc7217c02afccc12c1114a38264b3f35077a68ee238467a0ee1bd4853850e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9fdc7217c02afccc12c1114a38264b3f35077a68ee238467a0ee1bd4853850e7","first_computed_at":"2026-05-17T23:55:26.942571Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:26.942571Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p3BbexgcHNQaFouS2aPl5agwY3kCea0PK24Ur86lgtl0iLkuNP+GMEY+1zZDqfcCy/Nj2NpQlYaSkvjzUSAPCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:26.943046Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.11562","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ca5fdf9a90f106e32dc1df7bb59d6278dcc431e573a42ffa22c664ca80d68da","sha256:db9e3363284f731789531c710f291bb75a6e7a6516cd622e2faca7cf2485a4ef"],"state_sha256":"ea33ce0ce3c4dd5ef5374dc6c7fb88f5d63c71576dd7d40e8e36205130a1b1e2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sse/UxeZKl/6JZpH2rLyPjv6To+3DqPE+dXTPiUrvlo94c6i6PFg/opPp/FRis9UtFp3fknPK9HCqZcUyzucDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T20:28:57.939203Z","bundle_sha256":"1bcd7895967a89323e4d1bd6e9466fa3dd2b86be3b81f0008704afad55b25f78"}}