Short version: Consider the SI model of infectious disease spread on a random graph $G$ with a given degree sequence. Let $j$ be a vertex and let $i$ and $k$ be two of its neighbors. If $G$ contains no short cycles, then, conditioning on $j$ being susceptible at time $t$, it is reasonable to suspect that the states of $i$ and $k$ at time $t$ are close to being independent, since they only interact "at a long distance", as it were. Under what assumptions on the degree distribution of $G$ can this suspicion be made precise?
Long version, with details and definitions: (The derivation of the relevant quantities is a bit lengthy. The calculations given here follow Section 17.10 of M.E.J. Newman's Networks: An Introduction.) In the SI model of infectious diseases, members of a population can be in one of two states, "susceptible" and "infected". Once a member of the population is infected, it remains infected forever.
We consider this model on a random graph $G$ with a given degree sequence (more formally, a sequence of random graphs $G_n$ with a given asymptotic degree sequence). Each infected vertex $v$ infects its susceptible neighbors independently with rate $\beta$--that is, the probability that $v$ infects a susceptible neighbor $w$ at some point in the interval $[t, t + dt]$ is $\beta dt$.
Let $s_i(t)$ denote the indicator function for the event that $i$ is susceptible at time $t$, and let $x_i$ denote the indicator function for the event that $i$ is infected at time $t$. How does $\mathbb{E} s_i(t)$ change with $t$? Well, the expected number of infections that $i$ receives in a time interval $dt$ equals the number of infected neighbors that "fire" during the interval $dt$. Then we have $$\frac{d\mathbb{E} s_i(t)}{dt} = -\beta\sum_{j = 1}^n A_{ij} \mathbb{E} \bigl(s_i(t) x_j(t)\bigr),$$ where $A_{ij}$ denotes the corresponding entry of the adjacency matrix. The expression on the right-hand side does not tell us much, because we don't know the values of the $\mathbb{E} \bigl(s_i(t) x_j(t)\bigr)$. However, with some more thought, one can determine that (suppressing the dependence on $t$) $$\frac{d\mathbb{E} (s_i x_j)}{dt} = \beta\sum_{k \neq i} A_{jk} \mathbb{E} (s_i s_j x_k) - \beta\sum_{\ell \neq j} A_{i \ell} \mathbb{E} (x_{\ell} s_i x_j) - \beta \mathbb{E} (s_i x_j). \tag*{(1)}$$
My question is about computing, e.g., $$\mathbb{E} (s_i s_j x_k) = \mathbb{P}(s_i = 1, s_j = 1) \mathbb{P}(x_k = 1 \mid s_i = 1, s_j = 1). \tag*{(2)}$$ Here, the "moment closure" or "pair approximation" method says that if $G$ is close to being a tree near $i$, then, conditioning on $j$ being susceptible, the states of $i$ and $k$ are close to being independent, because the only interactions between them occur at a long range. (Note that $k$ is a neighbor of $j$, not of $i$.) Since random graphs with a given degree distribution are often locally tree-like (e.g., random $d$-regular graphs), this seems like a very plausible approximation. Then we have, e.g., $$\mathbb{P}(x_k = 1 \mid s_i = 1, s_j = 1) = \frac{\mathbb{E} (s_j x_k)}{\mathbb{E} s_j},$$ which reduces $(2)$ to a combination of one- and two-variable expectations. This, in turn, makes the system of differential equations $(1)$ solvable, albeit complicated.
Can this idea of "near independence" be made precise? If so, how? Under what conditions on the degree distribution of $G$ do we have, for example, that $$\mathbb{E}(s_i x_k) = \mathbb{E} s_i \mathbb{E} x_k + o(1),$$ and why?