9
votes
0answers
198 views

First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...
5
votes
1answer
231 views

Probability of two vertices to be connected in G(n,p)

A question I asked at math.SE without elliciting an answer. Let $G(n,p)$ be an Erdős–Rényi graph on $n$ vertices. Is there an explicit expression for the probability $P_{n,p}(u,v)$ that two fixed ...
4
votes
0answers
219 views

Expected number of components with multiple cycles in a subgraph of a square lattice

Short version Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the ...