3 research outputs found
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