A simple solution to the k-core problem
2007 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 30, no 1-2, 50-62 p.Article in journal (Refereed) Published
We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty and other conditions that imply that with high probability the k-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer, and Wormald (J Combinator Theory 67 (1996), 111-151) on the existence and size of a k-core in G(n,p) and G(n,m), see also Molloy (Random Struct Algor 27 (2005), 124-135) and Cooper (Random Struct Algor 25 (2004), 353-375). Our method is based on the properties of empirical distributions of independent random variables and leads to simple proofs.
Place, publisher, year, edition, pages
2007. Vol. 30, no 1-2, 50-62 p.
cores, random graphs, balls and bins, death process, empirical distributions, law of large numbers
IdentifiersURN: urn:nbn:se:uu:diva-22671DOI: 10.1002/rsa.20147ISI: 000243477700004OAI: oai:DiVA.org:uu-22671DiVA: diva2:50444