{"paper":{"title":"Maximal Steiner Trees in the Stochastic Mean-Field Model of Distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. Davidson, A. Ganesh","submitted_at":"2015-07-15T16:23:57Z","abstract_excerpt":"Consider the complete graph on $n$ vertices, with edge weights drawn independently from the exponential distribution with unit mean. Janson showed that the typical distance between two vertices scales as $\\log{n}/n$, whereas the diameter (maximum distance between any two vertices) scales as $3\\log{n}/n$. Bollob\\'{a}s et al. showed that, for any fixed k, the weight of the Steiner tree connecting $k$ typical vertices scales as $(k-1)\\log{n}/n$, which recovers Janson's result for $k=2$. We extend this result to show that the worst case $k$-Steiner tree, over all choices of $k$ vertices, has weigh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04282","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"}