Complex balance (${\sum }_{y^{\prime }\to y}{k}_{y^{\prime }\to y}{c}^{y^{\prime }}={\sum }_{y\to y^{\prime }}{k}_{y\to y^{\prime }}{c}^y$)

At a positive concentration vector $c$, the total incoming deterministic reaction flux equals the total outgoing flux at every reaction complex. Here $c^y={\prod }_i{c}_i^{y_i}$ is the law of mass action monomial. This is stronger than balancing net production of each chemical species, but does not require pairwise detailed balance between reverse chemical reactions. A directed cycle can have nonzero circulation while satisfying complex balance.