Unifying Network Connectivity: From Geodesics to Random Walks via the Random Cluster Model

Connectivity is a central concept in network science, capturing how interactions propagate through indirect paths. While numerous connectivity metrics exist, such as shortest paths, effective resistance and minimum cut, each emphasizes different structural aspects, and their relationships have remained fragmented. In this work, we show that these classical measures emerge as limiting cases of a broader statistical physics framework: the random cluster (RC) model. By tuning its parameters, the RC model interpolates between contrasting notions of connectivity, from efficient geodesic paths to redundant parallel routes, offering a principled synthesis of series and parallel communication. This perspective provides both theoretical insight into the notion of connectivity and practical tools for learning tasks on networks. In particular, RC connectivity naturally encodes the kinetics of growing paths, enhancing predictive performance in dynamical settings such as epidemic spreading and neurodynamics. By bridging structural, dynamic, and learning-based views, RC connectivity lays the groundwork for a more general and interpretable theory of networked systems.