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Suppose we have a tree $T = (V,E)$, in which each nodes $v_i \in V$ has a probability $p_i$ to vanish. Let $v_0\in V$, we define random variable $\boldsymbol{X} = \boldsymbol{X}(T, v_0)$ stands for the size (number of nodes) of connected component which includes $v_0$. To illustrate the problem more precisely, note that we could write a formula about MGF of $\boldsymbol{X}$ like (regarding $v_0$ as the root of $T$) $$ E[\exp(t \cdot \boldsymbol{X}(T,v_0))] = p_0 + (1-p_0)\exp(t)\prod_{\text{subtree}\ T_i \ \text{for} \ v_0 \text{'s child}\ v_i} E[\exp(t \cdot \boldsymbol{X}(T_i,v_i))].$$ Since this problem looks very natural, I tend to believe that some works have be done for this. However it is hard for me to search them out (I have used Google Scholar to try it but failed). Could you give me some references on this problem?

PS: feel free to edit this post if you want, it would help ^_^

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  • $\begingroup$ This sounds like vertex percolation on $T$. Have you tried searching for that? By the way, I recommend MathSciNet over google scholar. Classic work on percolation deals with geometric graphs, such as lattices $\Bbb{Z}^d$, but your question is very natural and I would be surprised if it has not been studied. $\endgroup$
    – Victor Protsak
    Commented Apr 6, 2017 at 14:53
  • $\begingroup$ To @VictorProtsak, your keyword is very helpful. I've searched by using keywords "tree percolation" and found something interesting. However the result which is most similar to my question is about the LOCAL FINITE (so INFINITY) tree model, while I'm interested in the FINITE case. $\endgroup$
    – Lwins
    Commented Apr 6, 2017 at 15:19

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