Let $G_{0}=(V,E)$ be a connected graph.
Let $G_{1}=(V,E')$ be the probalistic subgraph of $G_{0}$ such that for each $e_{i}\in E$ the probability of $e\in E'$ is $p_{i}$ (each edge can get different probability)
I need to find a way to describe the probability of $G_{1}$ to stay connected, meaning that we might lose some of $G_{0}$ edges but still stay connected.
I have found only papers that describe the $G(n,p)$ model and the $G(n,m)$ model or related issues, all of them describe an equal p probability for each edge.
I have tried to build a function of the vertices degree, something like $\Sigma_{i=0}^{n}\Pi_{j\in N(V_{i})}P_{j}$ where $n$ is the number of vertices and $N(v_{i})$ represents $v_{i}$ neighbors but I have duplications.
I am thinking about the probability of finding a tree in $G_{1}$ but I can't find any material.
Does anyone have any idea if there is a relevant material that is related to my problem?
Let me make sure I've got you correctly. $G_0$ is fixed, with $G_0=(V,E)$. For each $e_i\in E$, you flip a biased coin with probability of Heads $p_i$. If it comes Heads, keep edge $e_i$, otherwise erase, right?
With this, the answer seems to heavily depend on $G_0$. For instance, consider $G_0$ to be a cycle. Then, you are allowed to erase at most one edge, once you erase, say, two edges, the graph immediately becomes disconnected.
There are examples, even with deletion of one edge makes the graph disconnected. For instance, consider two vertex-disjoint connected subgraphs $G_0'$ and $G_0''$, and let $G_0$ be such that, $G_0$ includes $G_0'$, $G_0''$, and only one extra edge between an arbitrary vertex from $G_0'$ and $G_0''$. Immediately see that, once you erase this edge, the graph becomes simply disconnected.
Even more dramatic examples exist. Take a cycle, and erase one edge only. Now, you have a (connected) path. Observe that for this object to remain connected, you should not erase any edges.