{"paper":{"title":"Diffusion and consensus on weakly connected directed graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.SI"],"primary_cat":"math.CO","authors_text":"E. Kummel, J.J.P. Veerman","submitted_at":"2018-07-25T20:45:02Z","abstract_excerpt":"Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\\dot x=-{\\cal L}x$ for consensus and $\\dot p=-p{\\cal L}$ for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors $\\{\\bar \\gamma_i\\}_{i=1}^k$ of the left null-space of ${\\cal L}$ and a basis of column vectors $\\{\\gamma_i\\}_{i=1}^k$ of the right null-space of ${\\cal L}$ in terms of the partition of $G$ into strongly connected components. This allows for complete characteriza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.09846","kind":"arxiv","version":1},"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"}