Consider a function $f$ computed by a boolean circuit $C$ with $n$ inputs of size $s(n) = \mathsf{poly}(n)$ over the basis $\{\mathsf{XOR},\mathsf{AND},\mathsf{NOT}\}$ (with indegree 2 for the $\mathsf{XOR},\mathsf{AND}$ gates).
A boolean circuit is layered if it can be arranged into $d$ layers ($d$ being the depth of the circuit) of gates such that any edge between two gates connects adjacent layers.
Given that $f$ has a boolean circuit of size $s$, what can we say about the size of a layered circuit computing $f$? There is a trivial upper bound: by adding dummy nodes to $C$ at each layer crossed by an edge, we get a layered circuit of size at most $O(s^2)$. But can we get better in general (e.g. $O(s\cdot \log s)$, or $O(s)$), or for interesting class of circuits?
Background. This question stems from recent results in cryptography which show how to securely compute layered boolean circuits of size $s$ with communication $o(s)$ (e.g. $s/\log s$ or $s/\log\log s)$; I'm trying to understand how restrictive this restriction to layered boolean circuits can be in practice, either for general circuits or for "natural" circuits. However, I've not found much about layered circuits in the literature; appropriate pointers would also be welcome.