The blue nodes represent the number $b = b_3b_2b_1b_0$, and the green nodes represent the number $a = a_3a_2a_1a_0$. The yellow nodes are the output bits after multiplying $a \cdot b$. If $a\cdot b$ would not normally fit in a 4-bit word, then the nodes only represent the lower bits of that multiplication. Would working with this graph make integer factorization easier?
For example.
Here's the graph supposing that all $c_i$ bits are $1$. And we've moved up to 5-bit multiplication:
The other red nodes are immediately deduced to be $1$. Now what's the minimal set of remaining nodes such that if we knew their bit settings, then we can immediately deduce the rest of the graph?
You can be absolutely certain that everybody who works in factorization knows how multiplication works, and how to wire up a straightforward binary multiplier. In spite of this, factorization is still considered a hard problem -- which ought to answer your question about whether drawing such a network makes factorization easier.
Essentially what you're proposing is to rephrase the factorization problem as a boolean-circuit SAT problem, but SAT is generally believed to be even harder than integer factorization (in the sense that SAT is NP-complete whereas factorization probably isn't). So that wouldn't generally be expected to represent progress.