With modern molecular quantification methods, like, for instance, high throughput sequencing, biologists may perform multiple complex experiments and collect longitudinal data on RNA and DNA concentrations. Such data may be then used to infer cellular level interactions between the molecular entities of interest. One method which formalizes such inference is the stoichiometric algebraic statistical model (SASM) of [2] which allows to analyze the so-called conic (or single source) networks. Despite its intuitive appeal, up until now the SASM has been only heuristically studied on few simple examples. The current paper provides a more formal mathematical treatment of the SASM, expanding the original model to a wider class of reaction systems decomposable into multiple conic subnetworks. In particular, it is proved here that on such networks the SASM enjoys the so-called sparsistency property, that is, it asymptotically (with the number of observed network trajectories) discards the false interactions by setting their reaction rates to zero. For illustration, we apply the extended SASM to in silico data from a generic decomposable network as well as to biological data from an experimental search for a possible transcription factor for the heat shock protein 70 (Hsp70) in the zebrafish retina.