Consider a chemical reaction network endowed with mass-action kinetics and using complex-centered indexing of the reactions and rate constants , for . Then the kinetic or Kirchhoff matrix of the system is defined as
where is the rate constant corresponding to the reaction (and if ).
The kinetic matrix of a system keeps track of the connections in the reaction graph of a network, as well as the corresponding reaction weights associated with mass-action kinetics. For example, for off-diagonal entries , it is clear that for all and if and only if . In this sense, a kinetics matrix can be said to indicate the structure of a chemical reaction network underlying a mass action system. It is also clear that where and are the vectors of all ones and zeroes, respectively. This implies, among other things, that a kinetics matrix has a non-trivial kernel.
The kinetics matrix differs from the weighted Laplacian matrix of a weighted directed graph only by the sign of its entries.
Relationship to Markov processes