199 research outputs found
The maximal degree in random recursive graphs with random weights
We study a generalisation of the random recursive tree (RRT) model and its
multigraph counterpart, the uniform directed acyclic graph (DAG). Here,
vertices are equipped with a random vertex-weight representing initial
inhomogeneities in the network, so that a new vertex connects to one of the old
vertices with a probability that is proportional to their vertex-weight. We
first identify the asymptotic degree distribution of a uniformly chosen vertex
for a general vertex-weight distribution. For the maximal degree, we
distinguish several classes that lead to different behaviour: For bounded
vertex-weights we obtain results for the maximal degree that are similar to
those observed for RRTs and DAGs. If the vertex-weights have unbounded support,
then the maximal degree has to satisfy the right balance between having a high
vertex-weight and being born early. For vertex-weights in the Gumbel maximum
domain of attraction the first order behaviour of the maximal degree is
deterministic, while for those in the Fr\'echet maximum domain of attraction
are random to leading order.Comment: 33 page
Longest path distance in random circuits
We study distance properties of a general class of random directed acyclic
graphs (DAGs). In a DAG, many natural notions of distance are possible, for
there exists multiple paths between pairs of nodes. The distance of interest
for circuits is the maximum length of a path between two nodes. We give laws of
large numbers for the typical depth (distance to the root) and the minimum
depth in a random DAG. This completes the study of natural distances in random
DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also
obtain large deviation bounds for the minimum of a branching random walk with
constant branching, which can be seen as a simplified version of our main
result.Comment: 21 pages, 2 figure
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) and triangular urns, the martingale tail sum for the
number of balls of a given color admits both a Gaussian central limit theorem
as well as a law of the iterated logarithm. The laws of the iterated logarithm
are new even in the standard model when only one ball is drawn from the urn in
each step (except for the classical Polya urn model). Finally, we prove that
the martingale limits exhibit densities (bounded under suitable assumptions)
and exponentially decaying tails. Applications are given in the context of node
degrees in random linear recursive trees and random circuits.Comment: 17 page
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