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：为了简化布尔网络的分析，通常考虑减少组件数量。一种常见的约简方法是通过变量替换来消除非自调控组件。本文展示了如何将此方法扩展到异步布尔网络动力学中，实现负自调控顶点的消除，并研究其对动力学和交互结构的影响。对于非自调控变量的消除，吸引子的保持通常仅能保证于不动点。本文给出了复杂吸引子保持的充分条件。常被考虑移除所谓的介导节点（即入度和出度均为1的顶点），且通常不影响吸引子景观。我们阐明并非总是如此，在某些情况下，即使交互结构的细微变化也可能导致不同的渐近行为。最后，利用本文引入的更通用消除方法的性质，为异步布尔网络吸引子数量上界（基于交互图正反馈顶点集基数）提供了一个替代证明。