{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:AVTLAEN7YTJYN6H2XYODHNDZKZ","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":"aaeaa78ca950b4a7b833adf5632b5c2fb1aac9dadac4526f93374001a37a7a8e","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-11-19T01:26:59Z","title_canon_sha256":"1b00552931bea208c08feae3e83d302bb8ea5b777a587817d0154de8a7adbb6c"},"schema_version":"1.0","source":{"id":"1411.5087","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.5087","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"arxiv_version","alias_value":"1411.5087v4","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5087","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"pith_short_12","alias_value":"AVTLAEN7YTJY","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"AVTLAEN7YTJYN6H2","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"AVTLAEN7","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:122d0a5e164d28ace1851c99824f9e38424ddcb96bcc79f86a6d5700b5a1992b","target":"graph","created_at":"2026-05-18T01:25:16Z","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":"By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincar\\'e inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is ","authors_text":"Paul Horn, Shing-Tung Yau, Shuang Liu, Yong Lin","cross_cats":["math.CO","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-11-19T01:26:59Z","title":"Volume doubling, Poincar\\'e inequality and Guassian heat kernel estimate for nonnegative curvature graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5087","kind":"arxiv","version":4},"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":"sha256:8221fd852ce6e7f3b8ff84f600774b2ac94d4626481323e8d3570de0c4e07dc7","target":"record","created_at":"2026-05-18T01:25:16Z","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":"aaeaa78ca950b4a7b833adf5632b5c2fb1aac9dadac4526f93374001a37a7a8e","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-11-19T01:26:59Z","title_canon_sha256":"1b00552931bea208c08feae3e83d302bb8ea5b777a587817d0154de8a7adbb6c"},"schema_version":"1.0","source":{"id":"1411.5087","kind":"arxiv","version":4}},"canonical_sha256":"0566b011bfc4d386f8fabe1c33b479565f3a1d854c7e0137dc870dea1f4c3234","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0566b011bfc4d386f8fabe1c33b479565f3a1d854c7e0137dc870dea1f4c3234","first_computed_at":"2026-05-18T01:25:16.064548Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:16.064548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K8igurHmrqdV/nadc3u7lP10u77EQAHQaQntnvVUAJuy8WRSBvl5Lc8llIaFCMqazTfYNn6JydJtlXm3VMOSBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:16.065000Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.5087","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8221fd852ce6e7f3b8ff84f600774b2ac94d4626481323e8d3570de0c4e07dc7","sha256:122d0a5e164d28ace1851c99824f9e38424ddcb96bcc79f86a6d5700b5a1992b"],"state_sha256":"67695cff36ab99d8ea9f9be5e95021dd556c86f57c79bc71823f67873fba5d1a"}