{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:265NS4OSHXRZJEJB4OXBFHPSH6","short_pith_number":"pith:265NS4OS","canonical_record":{"source":{"id":"1401.7640","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-29T20:02:45Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"bd2afea8524af10d1b02cb18be11cfd9a0ffca4066c32cd130a7b8b118ded1f3","abstract_canon_sha256":"99c61e4e2985dc997627d79478d614c850a5276f9c094812a8640157ae568f18"},"schema_version":"1.0"},"canonical_sha256":"d7bad971d23de3949121e3ae129df23f9ffd6c2c9dcae3d683f85f7050328947","source":{"kind":"arxiv","id":"1401.7640","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.7640","created_at":"2026-05-18T00:45:04Z"},{"alias_kind":"arxiv_version","alias_value":"1401.7640v3","created_at":"2026-05-18T00:45:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.7640","created_at":"2026-05-18T00:45:04Z"},{"alias_kind":"pith_short_12","alias_value":"265NS4OSHXRZ","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"265NS4OSHXRZJEJB","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"265NS4OS","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:265NS4OSHXRZJEJB4OXBFHPSH6","target":"record","payload":{"canonical_record":{"source":{"id":"1401.7640","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-29T20:02:45Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"bd2afea8524af10d1b02cb18be11cfd9a0ffca4066c32cd130a7b8b118ded1f3","abstract_canon_sha256":"99c61e4e2985dc997627d79478d614c850a5276f9c094812a8640157ae568f18"},"schema_version":"1.0"},"canonical_sha256":"d7bad971d23de3949121e3ae129df23f9ffd6c2c9dcae3d683f85f7050328947","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:04.991166Z","signature_b64":"ixGcYFlr58ZOiSvkZwYDDRgpa9x8vk7zeHusq/JSYzUhjV6azDzrcUPKYTou6CIe+gXNvShab5egC8Brruf5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7bad971d23de3949121e3ae129df23f9ffd6c2c9dcae3d683f85f7050328947","last_reissued_at":"2026-05-18T00:45:04.990604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:04.990604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.7640","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:45:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e+UZDb1PISbHi6i99xZl85YXMTzQRywbonIsLaNWuV5EHfajnH9KKjtSAYIBInGFyDB6lheY5YaQpqywpV1rBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:08:17.423827Z"},"content_sha256":"75c3707b7f916c5111c1fc15c35663235610236632b84a84bde119d3ff675902","schema_version":"1.0","event_id":"sha256:75c3707b7f916c5111c1fc15c35663235610236632b84a84bde119d3ff675902"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:265NS4OSHXRZJEJB4OXBFHPSH6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modulus of families of walks on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CO","authors_text":"Faryad Darabi Sahneh, Max Goering, Nathan Albin, Pietro Poggi-Corradini","submitted_at":"2014-01-29T20:02:45Z","abstract_excerpt":"We introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7640","kind":"arxiv","version":3},"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":"2026-05-18T00:45:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6dPiya6hVDLJjToqK806rBcP8Esmy9Eya3wyF/l6z7oMwE6oVbU8hZWt9DjklOaalVjcAub9wF8lFjsxUymZDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:08:17.424570Z"},"content_sha256":"65ab4e65533cd83ab9cd381bc297e30f4f86e40eb154c1dd216bfdfdc1203aa6","schema_version":"1.0","event_id":"sha256:65ab4e65533cd83ab9cd381bc297e30f4f86e40eb154c1dd216bfdfdc1203aa6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/265NS4OSHXRZJEJB4OXBFHPSH6/bundle.json","state_url":"https://pith.science/pith/265NS4OSHXRZJEJB4OXBFHPSH6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/265NS4OSHXRZJEJB4OXBFHPSH6/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":"2026-06-09T19:08:17Z","links":{"resolver":"https://pith.science/pith/265NS4OSHXRZJEJB4OXBFHPSH6","bundle":"https://pith.science/pith/265NS4OSHXRZJEJB4OXBFHPSH6/bundle.json","state":"https://pith.science/pith/265NS4OSHXRZJEJB4OXBFHPSH6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/265NS4OSHXRZJEJB4OXBFHPSH6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:265NS4OSHXRZJEJB4OXBFHPSH6","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":"99c61e4e2985dc997627d79478d614c850a5276f9c094812a8640157ae568f18","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-29T20:02:45Z","title_canon_sha256":"bd2afea8524af10d1b02cb18be11cfd9a0ffca4066c32cd130a7b8b118ded1f3"},"schema_version":"1.0","source":{"id":"1401.7640","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.7640","created_at":"2026-05-18T00:45:04Z"},{"alias_kind":"arxiv_version","alias_value":"1401.7640v3","created_at":"2026-05-18T00:45:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.7640","created_at":"2026-05-18T00:45:04Z"},{"alias_kind":"pith_short_12","alias_value":"265NS4OSHXRZ","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"265NS4OSHXRZJEJB","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"265NS4OS","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:65ab4e65533cd83ab9cd381bc297e30f4f86e40eb154c1dd216bfdfdc1203aa6","target":"graph","created_at":"2026-05-18T00:45:04Z","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 introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide.","authors_text":"Faryad Darabi Sahneh, Max Goering, Nathan Albin, Pietro Poggi-Corradini","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-29T20:02:45Z","title":"Modulus of families of walks on graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7640","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":"sha256:75c3707b7f916c5111c1fc15c35663235610236632b84a84bde119d3ff675902","target":"record","created_at":"2026-05-18T00:45:04Z","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":"99c61e4e2985dc997627d79478d614c850a5276f9c094812a8640157ae568f18","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-29T20:02:45Z","title_canon_sha256":"bd2afea8524af10d1b02cb18be11cfd9a0ffca4066c32cd130a7b8b118ded1f3"},"schema_version":"1.0","source":{"id":"1401.7640","kind":"arxiv","version":3}},"canonical_sha256":"d7bad971d23de3949121e3ae129df23f9ffd6c2c9dcae3d683f85f7050328947","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d7bad971d23de3949121e3ae129df23f9ffd6c2c9dcae3d683f85f7050328947","first_computed_at":"2026-05-18T00:45:04.990604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:04.990604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ixGcYFlr58ZOiSvkZwYDDRgpa9x8vk7zeHusq/JSYzUhjV6azDzrcUPKYTou6CIe+gXNvShab5egC8Brruf5Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:04.991166Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.7640","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:75c3707b7f916c5111c1fc15c35663235610236632b84a84bde119d3ff675902","sha256:65ab4e65533cd83ab9cd381bc297e30f4f86e40eb154c1dd216bfdfdc1203aa6"],"state_sha256":"5d0c5ed71e41fc2ab183c50ac9dd9bc61b37fb4f02439485ecb0155cdab86d91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vbf9QN8y3eDANUZsoiRzsRyG+V1Bb30OqsvZFggveeXefiu9RLauoCqLIDxTSGt2wTg1G+tyVuCBtMOWMRLlCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T19:08:17.429034Z","bundle_sha256":"a00c9ed1c271e43e39a2f40fd997dcb9e53f29a487a0dca777a0c5c42f79d3bc"}}