{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QHHAOYBZUTS2ECJZWKARUT2ATP","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":"4b5d8e76a9e34aedcb72b0efa4755d0dfaeb40e2d8bede356171fadedeaf231c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-10T20:25:16Z","title_canon_sha256":"1308b030e1e579ea1acb2aa69ab36b36e6b8927ba98ceb5b39cd308d65e72f40"},"schema_version":"1.0","source":{"id":"1105.2049","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.2049","created_at":"2026-05-18T04:22:21Z"},{"alias_kind":"arxiv_version","alias_value":"1105.2049v1","created_at":"2026-05-18T04:22:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.2049","created_at":"2026-05-18T04:22:21Z"},{"alias_kind":"pith_short_12","alias_value":"QHHAOYBZUTS2","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QHHAOYBZUTS2ECJZ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QHHAOYBZ","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:39a91bb2ab4f5aeec0a32066c71776ce63484d683b782e2261bc2e4238eacaa9","target":"graph","created_at":"2026-05-18T04:22:21Z","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":"We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen and of Lyons and Peres with the sufficient criterion of Soardi. We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos.","authors_text":"Johannes Carmesin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-10T20:25:16Z","title":"A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2049","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:fe5cffc72506245b54bd7c75e0df8ef4b94aa75c06a791ad13458ddf5bbf835c","target":"record","created_at":"2026-05-18T04:22:21Z","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":"4b5d8e76a9e34aedcb72b0efa4755d0dfaeb40e2d8bede356171fadedeaf231c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-10T20:25:16Z","title_canon_sha256":"1308b030e1e579ea1acb2aa69ab36b36e6b8927ba98ceb5b39cd308d65e72f40"},"schema_version":"1.0","source":{"id":"1105.2049","kind":"arxiv","version":1}},"canonical_sha256":"81ce076039a4e5a20939b2811a4f409bcd236193f3fa409972781ce8dc1b1f3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81ce076039a4e5a20939b2811a4f409bcd236193f3fa409972781ce8dc1b1f3d","first_computed_at":"2026-05-18T04:22:21.589842Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:22:21.589842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+F6lGP8uTAU0C63GwdGfgiarKXqKeKhq/821aHd/o1xbPMWq2Gtz9WkKzRrc1BLKNDlB0yej/pFhxdyPUP7LDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:22:21.590407Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.2049","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe5cffc72506245b54bd7c75e0df8ef4b94aa75c06a791ad13458ddf5bbf835c","sha256:39a91bb2ab4f5aeec0a32066c71776ce63484d683b782e2261bc2e4238eacaa9"],"state_sha256":"82241ec31e66119018cbba6524c02d571d46a06cdaf2771aa43f345171ee5759"}