论文标题
签名交互图上的nilpotent动力学和托马斯规则的弱对话
Nilpotent dynamics on signed interaction graphs and weak converses of Thomas' rules
论文作者
论文摘要
具有$ n $组件的有限动态系统是函数$ f：x \ to x $其中$ x = x_1 \ times \ dots \ times \ times x_n $是$ n $整数间隔的产品。这种系统$ f $的结构由签名的Digraph $ g $表示，称为交互图：有$ n $的顶点，每个组件一个，签名的弧形描述了它们之间的正面和负面影响。有限的动力系统是基因网络的通常模型。在这种情况下，通常假定$ f $是{\ em-gund的}，也就是说，每个$ x_i $的大小最多是$ g $ in $ g $ plus of的$ i $ in。假设$ g $已连接并且$ f $是限制度的，我们证明了以下内容：如果$ g $不是周期，则$ f^{n+1} $可能是常数。在这种情况下，$ f $描述了一种非常简单的动态：$ n+1 $迭代中唯一固定点的全局收敛。这表明，在与程度结合的情况下，$ f $描述了一个复杂的动力学{\ em不能}的事实是从其交互图中推导的。然后，我们将上述结果广泛概括为直接的后果，将其他限制仅从交互图中推论出来，因为托马斯规则的以下弱对话：如果$ g $连接并具有正（负）周期，则$ f $可能具有两个（否）固定点。
A finite dynamical system with $n$ components is a function $f:X\to X$ where $X=X_1\times\dots\times X_n$ is a product of $n$ finite intervals of integers. The structure of such a system $f$ is represented by a signed digraph $G$, called interaction graph: there are $n$ vertices, one per component, and the signed arcs describe the positive and negative influences between them. Finite dynamical systems are usual models for gene networks. In this context, it is often assumed that $f$ is {\em degree-bounded}, that is, the size of each $X_i$ is at most the out-degree of $i$ in $G$ plus one. Assuming that $G$ is connected and that $f$ is degree-bounded, we prove the following: if $G$ is not a cycle, then $f^{n+1}$ may be a constant. In that case, $f$ describes a very simple dynamics: a global convergence toward a unique fixed point in $n+1$ iterations. This shows that, in the degree-bounded case, the fact that $f$ describes a complex dynamics {\em cannot} be deduced from its interaction graph. We then widely generalize the above result, obtaining, as immediate consequences, other limits on what can be deduced from the interaction graph only, as the following weak converses of Thomas' rules: if $G$ is connected and has a positive (negative) cycle, then $f$ may have two (no) fixed points.
