Assume I have a set of $n$ nodes that I want to define their pairwise distances. I also assume that a node $i$ is characterised by an exponential distribution with parameter $\lambda _i$ in the following sense: I draw a sample $x_i$ for node $i$ and a sample $x_j$ for node $j$. I then define $d_{ij}=\min (x_i, x_j)$.
This is how I define the "distance" between all pair of nodes. I've made some calculations and came up with this:
$$\mathbb{P}(j-is-closest-to-i)=\prod_{k=1}^{n}(1+\frac{\lambda_i+\lambda_k}{\lambda_i+\lambda_j})^{-1}$$
The question is, are "j is closest to i" and "i is closest to j" independent in this model?
Thanks.