同构布尔网络的非同步动态
Asynchronous dynamics of isomorphic Boolean networks
AI总结：
研究同构布尔网络的非同步动态与同步动态的关系，证明在同构下非同步动态可重建同步动态，但反之则无法确定吸引子数量。
摘要：
布尔网络是一种函数 $f:\{0,1\}^n\to\{0,1\}^n$，可以根据上下文推导出几种动态。最经典的是同步动态和非同步动态。两者都是在 $\{0,1\}^n$ 上的有向图，但同步动态（与 $f$ 一致）从 $x$ 到 $f(x)$ 有一条边，而非同步动态 $\mathcal{A}(f)$ 在 $x$ 到 $x+e_i$ 有一条边，当 $x_i\neq f_i(x)$。显然，$f$ 和 $\mathcal{A}(f)$ 有相同的信息，但这些对象在同构下能说些什么？我们证明，如果 $\mathcal{A}(f)$ 只在同构下已知，那么以高概率，$f$ 可以在同构下完全重建。然后我们显示，反向方向远非成立。特别是，如果 $f$ 只在同构下已知，关于 $\mathcal{A}(f)$ 的吸引子几乎无法得出结论。例如，如果 $f$ 有 $p$ 个固定点，那么 $\mathcal{A}(f)$ 至少有 $\max(1,p)$ 个吸引子，并且我们证明这个下界是紧的：总存在 $h\sim f$ 使得 $\mathcal{A}(h)$ 恰好有 $\max(1,p)$ 个吸引子。但 $\mathcal{A}(f)$ 可能经常有更多的吸引子，因为我们证明，以高概率存在 $h\sim f$ 使得 $\mathcal{A}(h)$ 有 $Ω(2^n)$ 个吸引子。
英文摘要：
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.
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