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The ability for people and organizations to connect in the digital age has allowed the growth of networks that cover an increasing proportion of human interactions. The research community investigating networks asks a range of questions such as which participants are most central, and which community label to apply to each member. This paper deals with the question on how to label nodes based on the features (attributes) they contain, and then how to model the changes in the label assignments based on the influence they produce and receive in their networked neighborhood. The methodological approach applies the simple graph convolutional neural network in a novel setting. Primarily that it can be used not only for label classification, but also for modeling the spread of the influence of nodes in the neighborhoods based on the length of the walks considered. This is done by noticing a common feature in the formulations in methods that describe information diffusion which rely upon adjacency matrix powers and that of graph neural networks. Examples are provided to demonstrate the ability for this model to aggregate feature information from nodes based on a parameter regulating the range of node influence which can simulate a process of exchanges in a manner which bypasses computationally intensive stochastic simulations.