28 research outputs found
Random tree growth by vertex splitting
We study a model of growing planar tree graphs where in each time step we
separate the tree into two components by splitting a vertex and then connect
the two pieces by inserting a new link between the daughter vertices. This
model generalises the preferential attachment model and Ford's -model
for phylogenetic trees. We develop a mean field theory for the vertex degree
distribution, prove that the mean field theory is exact in some special cases
and check that it agrees with numerical simulations in general. We calculate
various correlation functions and show that the intrinsic Hausdorff dimension
can vary from one to infinity, depending on the parameters of the model.Comment: 47 page
From trees to graphs: collapsing continuous-time branching processes
Continuous-time branching processes (CTBPs) are powerful tools in random
graph theory, but are not appropriate to describe real-world networks, since
they produce trees rather than (multi)graphs. In this paper we analyze
collapsed branching processes (CBPs), obtained by a collapsing procedure on
CTBPs, in order to define multigraphs where vertices have fixed out-degree
. A key example consists of preferential attachment models (PAMs), as
well as generalized PAMs where vertices are chosen according to their degree
and age. We identify the degree distribution of CBPs, showing that it is
closely related to the limiting distribution of the CTBP before collapsing. In
particular, this is the first time that CTBPs are used to investigate the
degree distribution of PAMs beyond the tree setting.Comment: 18 pages, 3 figure
Random recursive trees: A boundary theory approach
We show that an algorithmic construction of sequences of recursive trees
leads to a direct proof of the convergence of random recursive trees in an
associated Doob-Martin compactification; it also gives a representation of the
limit in terms of the input sequence of the algorithm. We further show that
this approach can be used to obtain strong limit theorems for various tree
functionals, such as path length or the Wiener index
Plane recursive trees, Stirling permutations and an urn model
We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Pólya urn