Johnson Samuel, Jones Nick S
Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom;
Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom.
Proc Natl Acad Sci U S A. 2017 May 30;114(22):5618-5623. doi: 10.1073/pnas.1613786114. Epub 2017 May 16.
Many natural, complex systems are remarkably stable thanks to an absence of feedback acting on their elements. When described as networks these exhibit few or no cycles, and associated matrices have small leading eigenvalues. It has been suggested that this architecture can confer advantages to the system as a whole, such as "qualitative stability," but this observation does not in itself explain how a loopless structure might arise. We show here that the number of feedback loops in a network, as well as the eigenvalues of associated matrices, is determined by a structural property called trophic coherence, a measure of how neatly nodes fall into distinct levels. Our theory correctly classifies a variety of networks-including those derived from genes, metabolites, species, neurons, words, computers, and trading nations-into two distinct regimes of high and low feedback and provides a null model to gauge the significance of related magnitudes. Because trophic coherence suppresses feedback, whereas an absence of feedback alone does not lead to coherence, our work suggests that the reasons for "looplessness" in nature should be sought in coherence-inducing mechanisms.
许多自然的复杂系统由于其组成元素不存在反馈作用而具有显著的稳定性。当被描述为网络时，这些系统几乎没有或根本没有循环，并且相关矩阵的主特征值较小。有人认为这种架构可以为整个系统带来优势，比如“定性稳定性”，但这一观察结果本身并不能解释无环结构是如何产生的。我们在此表明，网络中反馈环的数量以及相关矩阵的特征值，是由一种称为营养连贯性的结构属性决定的，营养连贯性是衡量节点如何整齐地落入不同层次的一种度量。我们的理论正确地将各种网络——包括那些源自基因、代谢物、物种、神经元、单词、计算机和贸易国家的网络——分为高反馈和低反馈两种不同的状态，并提供了一个零模型来衡量相关量的重要性。由于营养连贯性抑制反馈，而仅缺乏反馈并不会导致连贯性，我们的研究表明，自然界中“无环性”的原因应该从诱导连贯性的机制中去寻找。
