I am interested in a formula that describes the probability that nodes i < j < k of a BA model of size n (that have been added, respectively, at time i < j < k) are such that there is a link from j to i and a link from k to i.
I managed to find a paper of Bollobás et al. deriving an asymptotic expression of the desired probability [1, Lemma 2, Eq. 2]. The calculation of the exact expression is however omitted. Moreover, there is step in the proof which I do not understand and which is needed to derive the exact expression: "Arguing as in the proof of (1) above, ...". My problem is that the expression in the expected value of interest has a multiplication and, as far as I can tell, requires most likely some cleverer massaging to get to a recursion similar to that appearing in the proof of (1). Any help is appreciated.
[1, Lemma 2, Eq. 2] B. BOLLOBÁS and O. RIORDAN. THE DIAMETER OF A SCALE-FREE RANDOM GRAPH https://www.math.cmu.edu/users/af1p/Teaching/INFONET/Papers/PowerLaw/swdiam.pdf
For what it is worth, I have figured it out myself. As claimed by the authors, it is indeed derived similarly to (1). Specifically, using the notation from the aforementioned paper, one first observes that
$\mathbb{E}(d_{t,i} I_{g_j=i} \mid G^{t-1}) = d_{t-1} I_{g_j=i} + \frac{d_{t-1}}{2t-1} I_{g_j=i} = (1 + \frac{1}{2t - 1}) d_{t-1,i} I_{g_j=i}$.
With this, it then follows that
$\mathbb{E}(d_{t,i} I_{g_j=i}) = \prod_{s=j}^t (1 + \frac{1}{2s - 1}) \mathbb{E}(d_{j,i} I_{g_j=i})$.