384 research outputs found
The probability that a random multigraph is simple
Consider a random multigraph G* with given vertex degrees d_1,...,d_n,
contructed by the configuration model. We show that, asymptotically for a
sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending
to infinity, the probability that the multigraph is simple stays away from 0 if
and only if \sum d_i^2=O(\sum d_i). This was previously known only under extra
assumtions on the maximum degree. We also give an asymptotic formula for this
probability, extending previous results by several authors.Comment: 24 page
Preferential attachment without vertex growth: emergence of the giant component
We study the following preferential attachment variant of the classical
Erdos-Renyi random graph process. Starting with an empty graph on n vertices,
new edges are added one-by-one, and each time an edge is chosen with
probability roughly proportional to the product of the current degrees of its
endpoints (note that the vertex set is fixed). We determine the asymptotic size
of the giant component in the supercritical phase, confirming a conjecture of
Pittel from 2010. Our proof uses a simple method: we condition on the vertex
degrees (of a multigraph variant), and use known results for the configuration
model.Comment: 20 page
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