Reachability queries on uncertain graphs We have an uncertain graph $G$ where each edge $(u,v)$ exists with a probability $p_{(u,v)} \in (0, 1]$. We want to assign a score in $[0, 1]$ to each pair of vertices $u$ and $v$ which represents the likelihood that they lie in the same connected component.

If the only possible scores were discrete $0$ and $1$, then this problem reduces to the connected components listing or graph reachability problem. Is there any work on uncertain graphs that can solve this problem?
I edited your question to make it self-contained.

The literature on uncertain graphs is really rich for this problem. What you are looking for is "reachability in uncertain graphs", which has been studied extensively in database community [1].

Basically, you have an edge between two vertices $u$ and $v$ with probability $p_{uv} \in (0,1]$. You need to find whether two given vertices are reachable, therefore, lie in the same connected component or disjoint set. I suggest you take a look at the following paper as starting point, which describes the basics and can lead you to more works that have been done on this problem:

[1] Jin, Ruoming, et al. "Distance-constraint reachability computation in uncertain graphs." Proceedings of the VLDB Endowment 4.9 (2011): 551-562.