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  Craciun (2009) 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 presentation will  provide 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 will be shown  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,  the extended SASM is applied 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.