We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance, "formal balance," and the new notion of "cycle balance" are all defined in terms of the underlying graph. This fact allows elementary graph-theoretic (non-algebraic) proofs of a previous result (detailed balance = complex balance + formal balance), our main result (detailed balance = complex balance + cycle balance), and a corresponding result in the setting of continuous-time Markov chains.

中文翻译：

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详细余额 $$=$$ 复杂余额 $$+$$ 循环余额：反应网络和马尔可夫链的图论证明

我们进一步阐明了可逆化学反应网络的详细平衡和复杂平衡之间的关系。我们的结果适用于任意动力学，也适用于边界平衡。详细平衡、复杂平衡、“形式平衡”和“循环平衡”的新概念都是根据底层图来定义的。这一事实允许对先前结果（详细平衡 = 复杂平衡 + 形式平衡）、我们的主要结果（详细平衡 = 复杂平衡 + 循环平衡）和设置中的相应结果进行基本图论（非代数）证明连续时间马尔可夫链。