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A reaction network together with a choice of rate constants uniquely givesrise to a system of differential equations, according to the law of mass-actionkinetics. On the other hand, different networks can generate the same dynamicalsystem under mass-action kinetics. Therefore, the problem of identifying "the"underlying network of a dynamical system is not well-posed, in general. Here weshow that the problem of identifying an underlying weakly reversible deficiencyzero network is well-posed, in the sense that the solution is unique wheneverit exists. This can be very useful in applications because from the perspectiveof 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 areremarkably stable for any choice of rate constants: they have a unique positivesteady state within each invariant polyhedron, and cannot give rise tooscillations or chaotic dynamics. We also prove that both of our hypotheses(i.e., weak reversibility and deficiency zero) are necessary for uniqueness.

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems