Fast algorithm to find pair of triangles with a common edge in a complete graph
There's a straightforward algorithm.  For each edge $(u,v)$, compute the multiset of products of weights of paths of length 2 from $u$ to $v$:

$$S_{u,v} = \{\text{wt}(u,t) \cdot \text{wt}(t,w) : (u,t) \in E \land (t,w) \in E\}.$$

These multisets can be computed in at most $O(|V| \cdot |E|)$ time.  In practice it suffices to enumerate all edges $(u,v)$, then enumerate all edges $(u,t)$ incident on $u$, then check whether $(t,w)$ is an edge.

Next, for each edge $(u,v)$ and each pair of values $x,x' \in S_{u,v}$, plot the point $(\text{wt}(u,v) \cdot x, \text{wt}(u,v) \cdot x')$ on the scatterplot.  This can be done in at most $O(|V|^2 \cdot |E|)$ time.

It might not be easy to do asymptotically significantly better than this.  For a complete graph, the scatter plot can have $\Theta(|V|^4)$ points, so there is no hope for an algorithm whose running time is faster than $O(|V|^4)$ on complete graphs; and this algorithm takes $O(|V|^4)$ time on complete graphs, so it is asymptotically optimal for such graphs.  For sparse graphs, this algorithm takes $O(|V|^3)$ time, and there might be some hope to get the running time a little faster for sparse graphs.

If it's still too slow, rather than creating the entire scatterplot, you could sample the points on the scatterplot and plot that.  One way to do that is to compute the multisets $S_{u,v}$, then for each edge $(u,v)$ sample a bunch of pairs of values $x,x' \in S_{u,v}$ and plot the points $(\text{wt}(u,v) \cdot x, \text{wt}(u,v) \cdot x')$ on the scatterplot for each such pair.  If there are a lot of points on the scatterplot, plotting a random sample of the points should also be sufficient for visualization purposes and should give you a good sense of the "shape" of the scatterplot.