On conditional configuration graphs with random distribution of vertex degrees

Abstract

We consider a configuration graph with N vertices. The degrees of the vertices are drawn independently from a discrete power-law distribution with positive parameter τ . They are equal to the number of each vertex’s numbered semiedges. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. Research in the last years showed that configuration power-law random graphs with τ ∈ (1, 2) are deemed to be a good implementation of Internet topology. Such graphs could be used also for modeling forest fires as well as banking system defaults. But in these cases usually τ > 2. Parameter τ may depend on N and even be random. In the paper we consider configuration random graphs under the condition that the sum of vertex degrees is equal to n. Random graph dynamics as N → ∞ is assumed to take place in a random environment, where τ is a random variable following uniform distribution on the interval [a, b], 0 < a < b < ∞. We obtained the limit distributions of the maximum vertex degree and the number of vertices with a given degree as N, n → ∞.