Information Graphs: filtering and visualizing complex networks via q-state Potts model geometry

Abstract In this paper, we propose a novel framework for the post-hoc analysis and visualization of complex networks based on the statistical mechanics of the q-state Potts model. Our method, called Information Graph, leverages the estimation of the critical inverse temperature and information-geometric properties of the Potts model to identify the most and least informative nodes in a network. These estimates are used to weight the edges according to their contribution to intra and inter-community structure. By extracting the minimum and maximum information spanning trees, we isolate structurally relevant edges that respectively reinforce community cohesion and inter-community bridging. The union of these trees yields the Information Graph (IG), which offers a filtered representation of the original network by preserving semantically meaningful connections while removing redundancy. This process enhances both network modularity and coverage, providing an interpretable graph abstraction for downstream tasks. In practical terms, using a simple analogy with digital signal/image processing, Minimum Information Trees resemble a low-pass filtering process (smooth data) whereas Maximum Information Trees resemble a high-pass filtering process (emphasize edges and abrupt transitions). Potential applications include community-aware graph visualization, interpretation of classification outputs in relational data, simplification of biological and social networks, filtering of large-scale information graphs in machine learning pipelines, adaptive sampling for supervised classification and, eventually, the refinement of traditional community detection algorithms.