I derived the bond-percolation probability for a Bethe lattice and my result disagrees with a seemingly reputable reference by Albert & Barabasi (see below).
I would be happy if somebody could find my mistake, or, verify my result with an alternate reference (I tried to find one without success).
Problem Consider the Bethe graph (see below), where each node is connected to $3$ neighbors. From this lattice we construct a random graph, where each link exists with a given, fixed probability $p\in[0,1]$.
Our goal is to compute the percolation probability $P(p)$, that a given, fixed node is connected to infinity. By symmetry we can identify this arbitrary node with the origin.
(1) First we consider the white nodes (see the picture) and restrict to paths which only move away from the origin (i.e. from smaller to larger circles). Let $w$ be the probability, that one cannot reach infinity from such a white note. Then $w$ fulfills
$$w=(1-p+pw)^2.$$
This is because the path does either stop at the next link ($=1-p$) or it follows the link, but the next node does not lead to infinity ($=pw$). As there are two possible next links, $(1-p+pw)$ is squared. Solving this equation gives
$$w=\left(\frac{1}{p}-1\right)^2.$$
(2) Let $W$ be the probability, that one cannot reach infinity from the origin into any direction. Using the same idea as above, but keeping in mind that now there are three directions, we have
$$W=(1-p+pw)^3=w^{3/2}=\left(\frac{1}{p}-1\right)^3.$$
(3) Thus, the probability for reaching infinity is
$$P(p)=1-W=\frac{p^3-(1-p)^3}{p^3}.$$
I could not find any mistake, however, in this review article Albert & Barabasi claim that
$$P(p)=\frac{2p-1}{p^2},$$
(see Eq.(31)), referring again to Stauffer,Aharony, Introduction to percolation theory, London, 1992. However, in the book by Stauffer & Aharony I could not find this problem, but only a treatment of the site-percolation problem (where not the links, but the sites are occupied randomly) with a different result, while Albert & Barabasi explicitly consider bond-percolation (see 2nd paragraph of Sec. IV.C).
Now I wonder who is wrong, me or Albert & Barabasi ;)