{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MJYH34DOZNL7XUHIPB5XCJAYFG","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":"5ce0eb052375d77fb84f1899ff64c039b08c8d185f0074ea2e06df10fb2587f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-04-20T17:52:57Z","title_canon_sha256":"5b468bea5e3f54b0859dd86fdc70f9376e200ba896aea93f88b87db05210a950"},"schema_version":"1.0","source":{"id":"1204.4688","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.4688","created_at":"2026-05-18T03:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1204.4688v3","created_at":"2026-05-18T03:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.4688","created_at":"2026-05-18T03:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"MJYH34DOZNL7","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MJYH34DOZNL7XUHI","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MJYH34DO","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:2b3e3094896a34b436454d0a3a6a3a47acb8b183ebe9885b76c956ea54243be1","target":"graph","created_at":"2026-05-18T03:08:14Z","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 a finite irreducible Markov chain with invariant distribution $\\pi$. We use the inner product induced by $\\pi$ and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection between the number of small eigenvalues of a graph's Laplacian and the expansion of small sets in that graph. We give a simplified proof which generalizes to the nonregular, directed case. This result implies an approximation algorithm for an \"analytic\" version of the Small-Set Expansion Prob","authors_text":"David Witmer, Ryan O'Donnell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-04-20T17:52:57Z","title":"Markov chain methods for small-set expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4688","kind":"arxiv","version":3},"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":"sha256:86239443b36774a715fc4bfb7b283d7ee5610e7e6451d6f985729b401ef53d45","target":"record","created_at":"2026-05-18T03:08:14Z","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":"5ce0eb052375d77fb84f1899ff64c039b08c8d185f0074ea2e06df10fb2587f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-04-20T17:52:57Z","title_canon_sha256":"5b468bea5e3f54b0859dd86fdc70f9376e200ba896aea93f88b87db05210a950"},"schema_version":"1.0","source":{"id":"1204.4688","kind":"arxiv","version":3}},"canonical_sha256":"62707df06ecb57fbd0e8787b71241829a25601fff73f8124338615658959c48b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62707df06ecb57fbd0e8787b71241829a25601fff73f8124338615658959c48b","first_computed_at":"2026-05-18T03:08:14.651684Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:14.651684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N2gnodL2rfJ+SzMhjJD+x0SabxXuFWXcx3vq0e4xKgkNO8M5a4nb5k0VUfwXJBcRzhvNuIKV6ZL9JLRSahY3CA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:14.652100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.4688","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86239443b36774a715fc4bfb7b283d7ee5610e7e6451d6f985729b401ef53d45","sha256:2b3e3094896a34b436454d0a3a6a3a47acb8b183ebe9885b76c956ea54243be1"],"state_sha256":"7fc3fecf45fbfcb1800db6d1cd13604a81304046f046dd4e5efe75b651f1441b"}