Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs

We study how graph topology can condition a categorical distribution associated with node-indexed attributes. We introduce an algebraic–categorical framework based on the point of view of a node, represented by an under category that captures its node-centered structural perspective. Finite indexed projections of these viewpoints are aggregated into a graph-level monoidal object, separating the abstract topological representation from its concrete matrix realizations. We then introduce a distributional realization in which a supplied categorical distribution induces weighted graph matrices, yielding node-level and graph-level point-of-view (POV) distributions for both fixed distributions and sequences of distributions. For a connected graph and a strictly positive fixed distribution, we characterize the common limiting POV distribution by the normalized left Perron eigenvector of the induced weighted matrix. For complete graphs, we further prove recovery of the limiting supplied distribution for convergent distribution sequences, providing a sanity-check boundary when topology carries no distinguishing information. We complement the theoretical development with empirical studies that examine POV across several settings, including recovery behavior, distribution-conditioned graph propagation, topology conditioning of externally supplied node distributions, and representations of evolving policy distributions in reinforcement learning. Together, the framework provides a general mechanism for incorporating graph structure into supplied distributional information through node-centered viewpoints and path-based representations.