{"paper":{"title":"A Hilbert space approach to effective resistance metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MG"],"primary_cat":"math.FA","authors_text":"Erin P. J. Pearse, Palle E. T. Jorgensen","submitted_at":"2009-06-14T13:50:09Z","abstract_excerpt":"A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\\mathcal E$ produces a Hilbert space structure (which we call the energy space ${\\mathcal H}_{\\mathcal E}$) on the space of functions of finite energy.\n  We use the reproducing kernel $\\{v_x\\}$ constructed in \\cite{DGG} to analyze the effective resistance $R$, which is a natural metric for such a network. It is known that when $(G,c)$ supports nonconstant harmonic functions of finite energy, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2535","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"}