用禁止子网络表示的布尔网络的不动点定理
Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks
摘要：
我们感兴趣的是布尔网络中的不动点，{ \em，即}从$\{0,1\}^n$到其自身的函数$f$。定义了$f$的子网络作为$f$对$\{0,1\}^n$的子立方体的限制，刻画了一类满足以下性质的布尔网络$\mathcal{F}$：$f$的每个子网络有唯一不动点当且仅当$f$在$\mathcal{F}$中没有子网络.这一刻画推广了Shih和Dong的不动点定理，该定理断言：如果对于$\{0,1\}^n$中的每个$x$，在邻接矩阵为$f$在点$x$处的离散雅可比矩阵的有向图中不存在有向圈，则$f$有唯一不动点.然后，用$\mathcal{C}^+$表示（分别$\mathcal{C}^-$）交互图为正的网络（分别为负）循环，证明了$\mathcal{F}$的非扩张网络正是$\mathcal{C}^+\cup \mathcal{C}^-$的网络，并对这类非扩张网络给出了上述禁止子网络定理的“二分法”：$f$的每个子网络最多有（分别为.至少）一个固定点当且仅当$f$在$\mathcal{C}^+$中没有子网（分别为$\mathcal{C}^-$）子网。最后，证明了如果$f$是一个合取网络，则$f$的每个子网至多有一个不动点当且仅当$f$在$\mathcal{C}^+$中没有子网.
英文摘要：
We are interested in fixed points in Boolean networks, {\em i.e.} functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the subcubes of $\{0,1\}^n$, and we characterizes a class $\mathcal{F}$ of Boolean networks satisfying the following property: Every subnetwork of $f$ has a unique fixed point if and only if $f$ has no subnetwork in $\mathcal{F}$. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that if for every $x$ in $\{0,1\}^n$ there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. Then, denoting by $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of $\mathcal{F}$ are exactly the networks of $\mathcal{C}^+\cup \mathcal{C}^-$; and for the class of non-expansive networks we get a "dichotomization" of the previous forbidden subnetwork theorem: Every subnetwork of $f$ has at most (resp. at least) one fixed point if and only if $f$ has no subnetworks in $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) subnetwork. Finally, we prove that if $f$ is a conjunctive network then every subnetwork of $f$ has at most one fixed point if and only if $f$ has no subnetwork in $\mathcal{C}^+$.
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