To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph.

翻译：为简化布尔网络的分析，常考虑减少组件数量。一种常用的简化方法是通过变量替换剔除无自调控作用的组件。本研究展示了如何将该方法扩展至布尔网络异步动力学中具有负自调控作用的顶点消除，并探讨其对动力学行为与交互结构的影响。对于非自调控变量的消除，通常仅能保证不动点吸引子的保留。本文给出了复杂吸引子保留的充分条件。常被考虑的所谓中介节点（即入度与出度均为1的顶点）移除通常不影响吸引子景观。我们明确指出情况并非总是如此，某些情境下交互结构的细微变化便可能导致不同的渐近行为。最后，利用本文提出的更普适消除方法的性质，为异步布尔网络吸引子数量上限提供了另一种证明方法，该上限以交互图正反馈顶点集基数的形式表达。