Complex networks are pervasive in various fields such as chemistry, biology, and sociology. In chemistry, first-order reaction networks are represented by a set of first-order differential equations, which can be constructed from the underlying energy landscape. However, as the number of nodes increases, it becomes more challenging to understand complex kinetics across different timescales. Hence, how to construct an interpretable, coarse-graining scheme that preserves the underlying timescales of overall reactions is of crucial importance. Here, we develop a scheme to capture the underlying hierarchical subsets of nodes, and a series of coarse-grained (reduced-dimensional) rate equations between the subsets as a function of time resolution from the original reaction network. Each of the coarse-grained representations guarantees to preserve the underlying slow characteristic timescales in the original network. The crux is the construction of a lumping scheme incorporating a similarity measure in deciphering the underlying timescale hierarchy, which does not rely on the assumption of equilibrium. As an illustrative example, we apply the scheme to four-state Markovian models and Claisen rearrangement of allyl vinyl ether (AVE), and demonstrate that the reduced-dimensional representation accurately reproduces not only the slowest but also the faster timescales of overall reactions although other reduction schemes based on equilibrium assumption well reproduce the slowest timescale but fail to reproduce the second-to-fourth slowest timescales with the same accuracy. Our scheme can be applied not only to the reaction networks but also to networks in other fields, which helps us encompass their hierarchical structures of the complex kinetics over timescales.

中文翻译：

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一阶反应网络中时间尺度层次结构的包含表示


复杂网络普遍存在于化学、生物学和社会学等各个领域。在化学中，一阶反应网络由一组一阶微分方程表示，可以从底层的能量景观构建。然而，随着节点数量的增加，理解不同时间尺度的复杂动力学变得更具挑战性。因此，如何构建一个可解释的、粗粒度的方案来保留总体反应的基本时间尺度至关重要。在这里，我们开发了一种方案来捕获节点的底层分层子集，以及子集之间的一系列粗粒度（降维）速率方程，作为原始反应网络时间分辨率的函数。每个粗粒度表示都保证保留原始网络中底层的慢特征时间尺度。关键是构建一个集总方案，该方案在破译底层时间尺度层次结构时结合了相似性度量，该方案不依赖于均衡假设。作为一个说明性的例子，我们将该方案应用于四态马尔可夫模型和烯丙基乙烯基醚（AVE）的克莱森重排，并证明降维表示不仅准确地再现了整个反应的最慢的时间尺度，而且还准确地再现了整个反应的更快的时间尺度，尽管其他反应基于平衡假设的缩减方案可以很好地重现最慢的时间尺度，但无法以相同的精度重现第二到第四个最慢的时间尺度。我们的方案不仅可以应用于反应网络，还可以应用于其他领域的网络，这有助于我们在时间尺度上涵盖复杂动力学的层次结构。