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The psychological research on the formation of collective memories has explored how individuals come to hold similar memories following conversational interactions in social networks. These collective memories are dependent on both individual-level cognitive mechanisms as well as on the social influence exerted during people's conversations. Building on this work, we investigate the impact of the network structure of conversational interactions on the formation of collective memories in 16-member networks. We manipulate two parameters of conversational networks that were previously found to impact information diffusion in networks: network reachability and network clustering. We find that clustering significantly impacts the emerging collective memory, but reachability does not. Additionally, a participant's influence on the content of the network's collective memory depends on both their topological position in the network and on how early their conversations occur in the conversational network.