Asynchronous dynamics of isomorphic Boolean networks

A Boolean network is a function $f:\{0,1\}^n\to\{0,1\}^n$ from which several dynamics can be derived, depending on the context. The most classical ones are the synchronous and asynchronous dynamics. Both are digraphs on $\{0,1\}^n$, but the synchronous dynamics (which is identified with $f$) has an arc from $x$ to $f(x)$ while the asynchronous dynamics $\mathcal{A}(f)$ has an arc from $x$ to $x+e_i$ whenever $x_i\neq f_i(x)$. Clearly, $f$ and $\mathcal{A}(f)$ share the same information, but what can be said on these objects up to isomorphism? We prove that if $\mathcal{A}(f)$ is only known up to isomorphism then, with high probability, $f$ can be fully reconstructed up to isomorphism. We then show that the converse direction is far from being true. In particular, if $f$ is only known up to isomorphism, very little can be said on the attractors of $\mathcal{A}(f)$. For instance, if $f$ has $p$ fixed points, then $\mathcal{A}(f)$ has at least $\max(1,p)$ attractors, and we prove that this trivial lower bound is tight: there always exists $h\sim f$ such that $\mathcal{A}(h)$ has exactly $\max(1,p)$ attractors. But $\mathcal{A}(f)$ may often have much more attractors since we prove that, with high probability, there exists $h\sim f$ such that $\mathcal{A}(h)$ has $\Omega(2^n)$ attractors.