4,087 research outputs found
First-passage percolation on Cartesian power graphs
We consider first-passage percolation on the class of "high-dimensional"
graphs that can be written as an iterated Cartesian product of some base graph as the number of factors tends to
infinity. We propose a natural asymptotic lower bound on the first-passage time
between and as , the number of
factors, tends to infinity, which we call the critical time . Our
main result characterizes when this lower bound is sharp as
. As a corollary, we are able to determine the limit of the
so-called diagonal time-constant in as for
a large class of distributions of passage times.Comment: 30 pages, 1 figur
An improved energy argument for the Hegselmann-Krause model
We show that the freezing time of the -dimensional Hegselmann-Krause model
is where is the number of agents. This improves the best known
upper bound whenever
- β¦