Complex networks are critical in many scientific, technological, and societal contexts due to their ability to represent and analyze intricate systems with interdependent components. Often, after labeling the nodes of a network with a community detection algorithm, its modular organization emerges, allowing a better understanding of the underlying structure by uncovering hidden relationships. In this paper, we introduce a novel information-geometric framework for the filtering and decomposition of networks whose nodes have been labeled. Our approach considers the labeled network as the outcome of a Markov random field modeled by a q-state Potts model. According to information geometry, the first and second order Fisher information matrices are related to the metric and curvature tensor of the parametric space of a statistical model. By computing an approximation to the local shape operator, the proposed methodology is able to identify low and high information nodes, allowing the decomposition of the labeled network in two complementary subgraphs. Hence, we call this method as the LO-HI decomposition. Experimental results with several kinds of networks show that the high information subgraph is often related to edges and boundaries, while the low information subgraph is a smoother version of the network, in the sense that the modular structure is improved.