  A reaction network together with a choice of rate constants uniquely gives
rise to a system of differential equations, according to the law of mass-action
kinetics. On the other hand, different networks can generate the same dynamical
system under mass-action kinetics. Therefore, the problem of identifying "the"
underlying network of a dynamical system is not well-posed, in general. Here we
show that the problem of identifying an underlying weakly reversible deficiency
zero network is well-posed, in the sense that the solution is unique whenever
it exists. This can be very useful in applications because from the perspective
of both dynamics and network structure, a weakly reversibly deficiency zero
($\textit{WR}_\textit{0}$) realization is the simplest possible one. Moreover,
while mass-action systems can exhibit practically any dynamical behavior,
including multistability, oscillations, and chaos, $WR_0$ systems are
remarkably stable for any choice of rate constants: they have a unique positive
steady state within each invariant polyhedron, and cannot give rise to
oscillations or chaotic dynamics. We also prove that both of our hypotheses
(i.e., weak reversibility and deficiency zero) are necessary for uniqueness.
