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Abstract：
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.
摘要：
复杂网络在化学等各个领域都很普遍，生物学和社会学。在化学方面,一阶反应网络由一组一阶微分方程表示，可以从潜在的能源景观中构建出来。然而,随着节点数量的增加，理解不同时间尺度上的复杂动力学变得更具挑战性。因此，如何构造一个可解释的，保留整体反应的基本时间尺度的粗粒度方案至关重要。这里,我们开发了一个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，作为原始反应网络的时间分辨率的函数。每个粗粒度表示保证在原始网络中保留潜在的缓慢特征时间尺度。关键是构建一个包含相似性度量的集总方案，以破译基本的时间尺度层次结构，这不依赖于均衡的假设。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，并证明了降维表示不仅准确地再现了整体反应的最慢时间尺度，而且还准确地再现了整体反应的较快时间尺度，尽管基于平衡假设的其他简化方案很好地再现了最慢的时间尺度，但无法以相同的精度再现第二至第四最慢的时间尺度。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。这有助于我们涵盖时间尺度上复杂动力学的层次结构。
