{"paper":{"title":"Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","cs.DM","cs.SI","physics.soc-ph"],"primary_cat":"math.SP","authors_text":"Piet Van Mieghem","submitted_at":"2014-01-18T19:05:56Z","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_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4580","kind":"arxiv","version":4},"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"}