  Stochastic chemical reaction networks (CRNs) are complex systems which
combine the features of concurrent transformation of multiple variables in each
elementary reaction event, and nonlinear relations between states and their
rates of change. Most general results concerning CRNs are limited to restricted
cases where a topological characteristic known as deficiency takes value 0 or
1. Here we derive equations of motion for fluctuation moments at all orders for
stochastic CRNs at general deficiency. We show, for the case of the mass-action
rate law, that the generator of the stochastic process acts on the hierarchy of
factorial moments with a finite representation. Whereas simulation of
high-order moments for many-particle systems is costly, this representation
reduces solution of moment hierarchies to a complexity comparable to solving a
heat equation. At steady states, moment hierarchies for finite CRNs interpolate
between low-order and high-order scaling regimes, which may be approximated
separately by distributions similar to those for deficiency-0 networks, and
connected through matched asymptotic expansions. In CRNs with multiple stable
or metastable steady states, boundedness of high-order moments provides the
starting condition for recursive solution downward to low-order moments,
reversing the order usually used to solve moment hierarchies. A basis for a
subset of network flows defined by having the same mean-regressing property as
the flows in deficiency-0 networks gives the leading contribution to low-order
moments in CRNs at general deficiency, in a $1/n$-expansion in large particle
numbers. Our results give a physical picture of the different informational
roles of mean-regressing and non-mean-regressing flows, and clarify the
dynamical meaning of deficiency not only for first-moment conditions but for
all orders in fluctuations.
