Asynchronous dynamics of isomorphic Boolean networks

TL;DR

This work investigates how synchronous and asynchronous Boolean network dynamics relate under graph isomorphism. It proves that, for almost all $f$, the unlabelled asynchronous dynamics $\\mathcal{A}(f)$ uniquely determines the synchronous function $f$ up to isomorphism, but not vice versa; the latter can fail except in trivial cases. It then shows that, even when the synchronous dynamics are known only up to isomorphism, one can often reconstruct important asynchronous features or realize extremely large collections of small attractors by suitable relabellings, with high-probability behavior analyzed via probabilistic and combinatorial methods. The results illuminate how much structure in asynchronous behavior is fixed by the synchronous skeleton and reveal a dichotomy: strong identifiability in one direction versus dramatic combinatorial variability in the other, depending on fixed-point structure and derangements. Overall, the paper develops a toolkit (patterns, solidity, decreasing paths, and colorings) to connect graph-theoretic properties of the state space to dynamical outcomes, with implications for understanding gene networks and other Boolean systems under different updating schemes.

Abstract

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 $Ω(2^n)$ attractors.

Figures (16)

• Figure 1: Synchronous (left) and asynchronous (right) dynamics of the shift $f(x_1,x_2)=(x_2,x_1)$.
• Figure 2: Example of non-isomorphic Boolean networks with isomorphic asynchronous graphs.
• Figure 3: The three staples on $\mathbf{0} e_1$ for $n=4$.
• Figure 4: Illustration for Lemma \ref{['lem:2P_1']}.
• Figure 5: Illustration for Lemma \ref{['lem:no_2P_2']}.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 6