{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6RTWG4KP62IB2UYKAKA5P5MOA2","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":"99685f4bbe85c5d9997b3b1e656fdd4af829a48a6d5dfb42ef63c5462c2a99de","cross_cats_sorted":["cond-mat.stat-mech","cs.DM","cs.SI","physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-01-18T19:05:56Z","title_canon_sha256":"e6238f9c19487f7c7d4e9529ca1ceee3ed0614aeadcd575ed4ad409a6d1a7157"},"schema_version":"1.0","source":{"id":"1401.4580","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.4580","created_at":"2026-05-18T01:19:14Z"},{"alias_kind":"arxiv_version","alias_value":"1401.4580v4","created_at":"2026-05-18T01:19:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.4580","created_at":"2026-05-18T01:19:14Z"},{"alias_kind":"pith_short_12","alias_value":"6RTWG4KP62IB","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6RTWG4KP62IB2UYK","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6RTWG4KP","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:f34c9fd2518b61c18b06fadc614050e0dd5ad0812e0f6ac61143bd3d12673bc8","target":"graph","created_at":"2026-05-18T01:19:14Z","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":"Several expressions for the $j$-th component $\\left( x_{k}\\right)_{j}$ of the $k$-th eigenvector $x_{k}$ of a symmetric matrix $A$ belonging to eigenvalue $\\lambda_{k}$ and normalized as $x_{k}^{T}x_{k}=1$ are presented. In particular, the expression \\[ \\left( x_{k}\\right)_{j}^{2}=-\\frac{1}{c_{A}^{\\prime}\\left( \\lambda_{k}\\right) }\\det\\left( A_{\\backslash\\left\\{ j\\right\\} }-\\lambda_{k}I\\right) \\] where $c_{A}\\left( \\lambda\\right) =\\det\\left( A-\\lambda I\\right) $ is the characteristic polynomial of $A$, $c_{A}^{\\prime}\\left( \\lambda\\right) =\\frac{dc_{A}\\left( \\lambda\\right) }{d\\lambda}$ and $A_","authors_text":"Piet Van Mieghem","cross_cats":["cond-mat.stat-mech","cs.DM","cs.SI","physics.soc-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-01-18T19:05:56Z","title":"Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4580","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:c2f185d02da7dc6a9b7e288616485495f80526463c9cc3df1bfbd13eebc77627","target":"record","created_at":"2026-05-18T01:19:14Z","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":"99685f4bbe85c5d9997b3b1e656fdd4af829a48a6d5dfb42ef63c5462c2a99de","cross_cats_sorted":["cond-mat.stat-mech","cs.DM","cs.SI","physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-01-18T19:05:56Z","title_canon_sha256":"e6238f9c19487f7c7d4e9529ca1ceee3ed0614aeadcd575ed4ad409a6d1a7157"},"schema_version":"1.0","source":{"id":"1401.4580","kind":"arxiv","version":4}},"canonical_sha256":"f46763714ff6901d530a0281d7f58e06b79996d1fa55f881b712aa27705c86b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f46763714ff6901d530a0281d7f58e06b79996d1fa55f881b712aa27705c86b6","first_computed_at":"2026-05-18T01:19:14.499381Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:14.499381Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PS9hKOrcR8SCXKMvCASO9SviE9FpNlj1sa34/Y7J0l70AzTSFXrz7NkeldNsL5IqOjiioRffeSWA0/rAwkgCAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:14.500056Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.4580","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c2f185d02da7dc6a9b7e288616485495f80526463c9cc3df1bfbd13eebc77627","sha256:f34c9fd2518b61c18b06fadc614050e0dd5ad0812e0f6ac61143bd3d12673bc8"],"state_sha256":"351a0bda1488b3e7c1ab435da5b9b2d55c47b7004a203545c3dbdccf9c84ff74"}