abstract:
One of the simplest models for spatial growth and competition is the
Richardson model. The original version describes the growth of an
infectious phenomenon on Z^d, but the mechanism can also be extended to
comprise two phenomena, making it a model for competition on Z^d. In this
talk, continuum counterparts of both the one-type and the two-type
Richardson model are defined. These models describe growth and competition
respectively on R^d instead of Z^d.
The main result for the continuum one-type model is a shape theorem where
the rotational invariance with R^d allows for a stronger conclusion than
in the discrete case. For the two-type continuum model, the question at
issue is whether the infection types can grow to occupy infinite parts of
R^d simultaneously and it is conjectured that this is possible if and only
if the infection types have the same intensity. Existing results are
described in the talk along with a number of open problems.
abstract:
In this talk I will present results on a random graph with N
nodes, where node j has degree D_j and \{D_j\}_{j=1}^N are
i.i.d. with Prob(D_j <= x)=F(x). Our main assumption is
that 1-F(x)= x^{-\tau+1}L(x) for some \tau>1, and where L is slowly
varying at infinity.
The graph model is a variant of the so-called configuration model.
The minimal number of edges between two arbitrary connected nodes,
also known as the graph distance or the hopcount, is investigated
when N\rightarrow \infty. We prove that for \tau>3 the graph distance grows
like \log_{\nu}N, where the base of the logarithm equals
\nu=E[D_j(D_j -1)]/\E[D_j]>1. This confirms the
heuristic argument of Newman, Strogatz and Watts.
In addition we characterize the asymptotics of the
random fluctuations around \log_{\nu}{N}.
For \tau \in (2,3) and under some additional technical assumption,
we prove that the graph distance grows
like 2 \log\log N |\log(\tau-2)|.
Again we are able to characterize the asymptotics of the
random fluctuations around this mean.
Finally for \tau\in (1,2), the graph distance is concentrated on the values
2 and 3, as N\to \infty.
For \tau>3, the talk is based on a paper written jointly with Remco van der
Hofstad and Piet Van Mieghem. The other cases are based on two papers together
with Remco van der Hofstad and Dmitri Znamenski. A survey article, that
treats all three
regions for \tau can be downloaded from the website:
http://ssor.twi.tudelft.nl/~gerardh/