Computable properties of huge reaction networks

[danielbarter]

Reaction networks have algebraic structure (reactions can be thought of as vectors in the space spanned by the species) which we may be able to use as a tool for understanding huge examples. So far there are two concepts which seem useful:

1. Deficiency: This is a non negative integer associated to any reaction network independent of rates. It counts the number of loops in the network which leave the number of each species fixed, but are non trivial at the level of complexes. Deficiency is closely related to the dynamics of the network with mass action kinetics, but it is hard to make quantifiable statements except in the deficiency zero case. It is explored thoroughly in the book Quantum techniques for Stochastic mechanics: https://math.ucr.edu/home/baez/stoch_stable.pdf

2. Concordance: This is a binary value associated to any reaction network independent of rates: A reaction network is either concordant or discordant. In https://arxiv.org/abs/1109.2923, it is proved that concordant systems cannot display many of the exotic phenomena which make reaction networks interesting. In his book Foundations of Chemical Reaction Network Theory https://www.springer.com/gp/book/9783030038571, Feinberg explains that most reaction networks are concordant and this is a reasonable mathematical explanation for the the dull behavior across vast regions of the reaction network landscape with occasional eruptions of exotic behavior.

Each of these quantities can be computed using linear algebra, so we can expect them to be computable even for large networks using GPUs. There are probably more things like deficiency and concordance which I am missing and will add here as I find