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Large multipartite quantum systems tend to rapidly reach extraordinary levels of complexity as their number of constituents and entanglement links grow. Here we use complex network theory to study a class of continuous variables quantum states that present both multipartite entanglement and non-Gaussian statistics. In particular, the states are built from an initial imprinted cluster state created via Gaussian entangling operations according to a complex network structure. To go beyond states that can be easily simulated via classical computers we engender non-Gaussian statistics via multiple photon subtraction operations. We then use typical networks measures, the degree and clustering, to characterize the emergent complex network of photon-number correlations after photon subtractions. We show that, in contrast to regular clusters, in the case of imprinted complex network structures the emergent correlations are strongly affected by photon subtraction. On the one hand, we unveil that photon subtraction universally increases the average photon-number correlations, regardless of the imprinted network structure. On the other hand, we show that the shape of the distributions in the emergent networks after subtraction is greatly influenced by the structure of the imprinted network, as witnessed by their higher-moments. Thus for the field of network theory, we introduce a new class of networks to study. At the same time for the field of continuous variable quantum states, this work presents a new set of practical tools to benchmark systems of increasing complexity.