We study a networked system of innovation processes, where each process is modeled as an urn with infinitely many colors-a classical framework for capturing the emergence of novelties. Extending this paradigm, we analyze a model of interacting urns, where the probability of generating or reusing elements in one process is influenced by the histories of others. This interaction is governed by two matrices that control innovation triggering and reinforcement dynamics across the system. The core contribution of this work is a detailed analysis of the second-order asymptotic behavior of the model. Building on these theoretical results, we develop statistical tools to infer the structure and strength of inter-process influence. The methodology is framed in a general setting, making it broadly applicable. We validate our approach with applications to two real-world datasets from Reddit discussions and Gutenberg text corpora.