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.
Kernel of 
Relationship to Markov processes
References