{"paper":{"title":"Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Malcolm J. Williamson, Richard Arratia, Thomas M. Liggett","submitted_at":"2013-06-13T03:58:06Z","abstract_excerpt":"In discrete contexts such as the degree distribution for a graph, \\emph{scale-free} has traditionally been \\emph{defined} to be \\emph{power-law}. We propose a reasonable interpretation of \\emph{scale-free}, namely, invariance under the transformation of $p$-thinning, followed by conditioning on being positive.\n  For each $\\beta \\in (1,2)$, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$\\beta$, and different from the usual Yule--Simon power law-$\\beta$ that arises in preferential attachment models.\n  In addition to chara"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3017","kind":"arxiv","version":2},"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"}