A mixture of molecules can be regarded as a network if all the molecular components participate in some kind of interaction with other molecules--either physical or functional interactions. Template-assisted ligation reactions that direct replication processes can serve as the functional elements that connect two members of a chemical network. In such a process, the template does not necessarily catalyze its own formation, but rather the formation of another molecule, which in turn can operate as a template for reactions within the network medium. It was postulated that even networks made up of small numbers of molecules possess a wealth of molecular information sufficient to perform rather complex behavior. To probe this assumption, we have constructed virtual arrays consisting of three replicating molecules, in which dimer templates are capable of catalyzing reactants to form additional templates. By using realistic parameters from peptides or DNA replication experiments, we simulate the construction of various functional motifs within the networks. Specifically, we have designed and implemented each of the three-element Boolean logic gates, and show how these networks are assembled from four basic "building blocks". We also show how the catalytic pathways can be wired together to perform more complex arithmetic units and network motifs, such as the half adder and half subtractor computational modules, and the coherent feed-forward loop network motifs under different sets of parameters. As in previous studies of chemical networks, some of the systems described display behavior that would be difficult to predict without the numerical simulations. Furthermore, the simulations reveal trends and characteristics that should be useful as "recipes" for future design of experimental functional motifs and for potential integration into modular circuits and molecular computation devices.