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A Random Tree Model Associated with Random Graphs

Summary: A Random Tree Model Associated
with Random Graphs
David Aldous*
Department of Statistics, University of California, Berkeley CA 94720
Grow a tree on n vertices by starting with no edges and successivelyadding an edge chosen
uniformly from the set of possible edges whose addition would not create a cycle. This
process is closely related to the classical random graph process. We describe the asymptotic
structure of the tree, as seen locally from a given vertex. In particular, we give an explicit
expression for the asymptotic degree distribution. Our results an be applied to study the
random minimum-weight spanning tree question, when the edge-weight distribution is
allowed to vary almost arbitrarily with n.
The construction indicated in the abstract, and stated more formally in Section 2,
yields a certain random tree F,,on n vertices. It is easy to calculate that, for the
star graph t, centered at vertex 1,
P(F" = I,) = 2"-'(n - 1)!/(2n - 2)!
and so for n 1.4, F,, is not the uniform random labeled tree (for which the
probability is 1ln"-2). Other explicit calculations are harder. The substance of
this paper is Theorem 1 below, which gives information about certain aspects of


Source: Aldous, David J. - Department of Statistics, University of California at Berkeley


Collections: Mathematics