12 research outputs found
Fringe trees, Crump-Mode-Jagers branching processes and -ary search trees
This survey studies asymptotics of random fringe trees and extended fringe
trees in random trees that can be constructed as family trees of a
Crump-Mode-Jagers branching process, stopped at a suitable time. This includes
random recursive trees, preferential attachment trees, fragmentation trees,
binary search trees and (more generally) -ary search trees, as well as some
other classes of random trees.
We begin with general results, mainly due to Aldous (1991) and Jagers and
Nerman (1984). The general results are applied to fringe trees and extended
fringe trees for several particular types of random trees, where the theory is
developed in detail. In particular, we consider fringe trees of -ary search
trees in detail; this seems to be new.
Various applications are given, including degree distribution, protected
nodes and maximal clades for various types of random trees. Again, we emphasise
results for -ary search trees, and give for example new results on protected
nodes in -ary search trees.
A separate section surveys results on height, saturation level, typical depth
and total path length, due to Devroye (1986), Biggins (1995, 1997) and others.
This survey contains well-known basic results together with some additional
general results as well as many new examples and applications for various
classes of random trees
Fringe trees for random trees with given vertex degrees
We prove asymptotic normality for the number of fringe subtrees isomorphic to
any given tree in uniformly random trees with given vertex degrees. As
applications, we also prove corresponding results for random labelled trees
with given vertex degrees, for random simply generated trees (or conditioned
Galton--Watson trees), and for additive functionals.
The key tool for our work is an extension to the multivariate setting of a
theorem by Gao and Wormald (2004), which provides a way to show asymptotic
normality by analysing the behaviour of sufficiently high factorial moments.Comment: 41 page
Normal limit laws for vertex degrees in randomly grown hooking networks and bipolar networks
We consider two types of random networks grown in blocks. Hooking networks
are grown from a set of graphs as blocks, each with a labelled vertex called a
hook. At each step in the growth of the network, a vertex called a latch is
chosen from the hooking network and a copy of one of the blocks is attached by
fusing its hook with the latch. Bipolar networks are grown from a set of
directed graphs as blocks, each with a single source and a single sink. At each
step in the growth of the network, an arc is chosen and is replaced with a copy
of one of the blocks. Using P\'olya urns, we prove normal limit laws for the
degree distributions of both networks. We extend previous results by allowing
for more than one block in the growth of the networks and by studying
arbitrarily large degrees.Comment: 28 pages, 6 figure
Degrees in random -ary hooking networks
The theme in this paper is a composition of random graphs and P\'olya urns.
The random graphs are generated through a small structure called the seed. Via
P\'olya urns, we study the asymptotic degree structure in a random -ary
hooking network and identify strong laws. We further upgrade the result to
second-order asymptotics in the form of multivariate Gaussian limit laws. We
give a few concrete examples and explore some properties with a full
representation of the Gaussian limit in each case. The asymptotic covariance
matrix associated with the P\'olya urn is obtained by a new method that
originated in this paper and is reported in [25].Comment: 21 pages, 5 figure
Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees
An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger
Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes
Our main result is to prove almost-sure convergence of a stochasticapproximation algorithm defined on the space of measures on a noncompact space. Our motivation is to apply this result to measure-valued Pólya processes (MVPPs, also known as infinitely-many Pólya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochasticapproximation methods to measure-valued Markov processes with a noncompact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs; this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding "weights"to the different colors of the infinitelymany- color urn. We also exhibit a link between non-"balanced"MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the nonbalanced case. Finally, we show how our result can be applied to designing stochasticapproximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on noncompact spaces