We introduce a unifying and generalizing framework for complex and detailed
balanced steady states in chemical reaction network theory. To this end, we
generalize the graph commonly used to represent a reaction network.
Specifically, we introduce a graph, called a reaction graph, that has one edge
for each reaction but potentially multiple nodes for each complex. A special
class of steady states, called node balanced steady states, is naturally
associated with such a reaction graph. We show that complex and detailed
balanced steady states are special cases of node balanced steady states by
choosing appropriate reaction graphs. Further, we show that node balanced
steady states have properties analogous to complex balanced steady states, such
as uniqueness and asymptotical stability in each stoichiometric compatibility
class. Moreover, we associate an integer, called the deficiency, to a reaction
graph that gives the number of independent relations in the reaction rate
constants that need to be satisfied for a positive node balanced steady state
to exist.
The set of reaction graphs (modulo isomorphism) is equipped with a partial
order that has the complex balanced reaction graph as minimal element. We
relate this order to the deficiency and to the set of reaction rate constants
for which a positive node balanced steady state exists
