Let $G=(V,E)$ be an undirected graph and $p \colon E \mapsto (0,1]$ defines weights of its edges.


Let's fix two connected vertices $v\_1, v\_2 \in V$.


Random graph $G'=(V,E')$ is obtained from $G$ by removing each edge $e \in E$ with probability $1-p(e)$.


What is the probability that connectivity between $v\_1$ and $v\_2$ is preserved in $G'$?



Probability of preserving connectivity between pair of vertices in weighted graph
Let L be the set of all simple paths in G from $v\_1$ to $v\_2$. By inclusion-exclusion, the probability that $v\_1$ and $v\_2$ are connected is 


$\sum\_{A \subseteq L} (-1)^{|A|-1} P(\cup A \subseteq E')$


where for any set S of edges,
$P(S \subseteq E') = \prod\_{e \in S} p(e)$.


Although explicit computation won't be feasible for large graphs, under appropriate conditions this might be used to get asymptotics.


