The emergence of collections of simple chemical entities that create self-sustaining reaction networks, embedding replication and catalysis, is cited as a potential mechanism for the appearance on the early Earth of systems that satisfy minimal definitions of life. In this work, a functional reaction network that creates and maintains a set of privileged replicator structures through auto- and cross-catalyzed reaction cycles is created from the pairwise combinations of four reagents. We show that the addition of individual preformed templates to this network, representing instructions to synthesize a specific replicator, induces changes in the output composition of the system that represent a network-level response. Further, we establish through sets of serial transfer experiments that the catalytic connections that exist between the four replicators in this network and the system-level behavior thereby encoded impose limits on the compositional variability that can be induced by repeated exposure to instructional inputs, in the form of preformed templates, to the system. The origin of this persistence is traced through kinetic simulations to the properties and inter-relationships between the critical ternary complexes formed by the auto- and crosscatalytic templates. These results demonstrate that in an environment where there is no continuous selection pressure, the network connectivity, described by the catalytic relationships and system-level interactions between the replicators, is persistent, thereby limiting the ability of this network to adapt and evolve.