A Balog-Szemeredi-Gowers-type question A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds
$$
|B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \},
$$
where the standard notation for the product set $B.B$ is used. Now suppose that instead of $B.B$ we have a partial productset $B\stackrel{G}{.}B$ along the edges of a graph $G$ of edge density $\epsilon$ (meaning that $b_1b_2 \in B\stackrel{G}{.}B$ if only if $b_1, b_2 \in B$ are adjacent in $G$).
Is it true that if $B < \sqrt{p}$ a similar estimate
$$
 |B\stackrel{G}{.}B + B\stackrel{G}{.}B + B\stackrel{G}{.}B - B\stackrel{G}{.}B - B\stackrel{G}{.}B - B\stackrel{G}{.}B| \gg_{\epsilon} |B|^2 
$$
holds?
UPD
Sorry for sloppy notation. So what it means. $B.B = \{b_ib_j |b_i, b_j \in B \}$, 
$B.B-B.B = \{b_ib_j - b_kb_l |b_i, b_j, b_k, b_k \in B\}$, and $B.B+B.B$ etc. being defined the same way. 
 A: This is a sketch of a solution in a certain case.
Let $P$ be the set of pairs $(b,b')$ such that $|N_G(b)\cap N_G(b')|\approx_\epsilon |B|$ up to epsilons. Now let $Q=\{(b-b')/(c-c')\colon (b,b'),(c,c')\in P\}$. We need $\approx_\epsilon |B|^2$ such pairs (off the top of my head I'm not sure if this is possible, but I think it's a standard Cauchy-Schwarz argument).
Consider the special case $Q=\mathbb{F}_p$. Since the number of pairs $(b,b')$ is $\approx |B|^2$, we have
$$
|B|^4\approx \sum_{\lambda\in Q}E_+(B,\lambda\cdot B).
$$
Hence there is a $\lambda\in Q$ such that $E_+(B,\lambda\cdot B)\lesssim |B|^4/p$.
If $S$ and $T$ are subsets of $B$, then $E_+(S,\lambda\cdot T)\lesssim |B|^4/p$ since we can't get more quadruples by taking subsets. If $\lambda=(b-b')/(c-c')$, let $S$ be the neighborhood of $c-c'$ and let $T$ be the neighborhood of $b-b'$. Then
$$
|(c-c')\cdot S+(b-b')\cdot T|=|S+\lambda\cdot T|\geq \frac{|S|^2|T|^2}{E_+(S,\lambda T)}\gtrsim_\epsilon p.
$$
Now because $(c,x),(c',y)\in G$ for all $x,y\in S$ (and similarly for $b,b',T$), we have
$$
|B\cdot_G B-B\cdot_G B+B\cdot_G B-B\cdot_G B|\geq |cS-c'S+bT-b'T|\gtrsim p.
$$
