Recently a simple model was introduced [Phys. Rev. Lett. 106 058701 (2011)], which attempts to describe the process of emergence, expansion and decline of scientific ideas. It is an agent-based model, which takes into account the agents' memory. Possible interactions between the agents are defined by the interaction network. We try to develop an analytical description of the dynamics, in the case of two competing ideas, considering different topologies of the interaction networks (including homogeneous networks, such as the chain graph, as well as heterogeneous networks, e.g. Barabási-Albert graphs). We examine the pace of an innovative idea expansion, as well as the time after which a new idea replaces the old one. Our analytical results agree with the results of the simulations.

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