The contribution exposes and illustrates a general, flexible formalism, together with an associated iterative procedure, aimed at determining soft memberships of marked nodes in a weighted network. Gathering together spatial entities which are both spatially close and similar regarding their features is an issue relevant in image segmentation, spatial clustering, and data analysis in general. Unoriented weighted networks are specified by an ``exchange matrix", determining the probability to select a pair of neighbors. We present a family of membership-dependent free energies, whose local minimization specifies soft clusterings. The free energy additively combines a mutual information, as well as various energy terms, concave or convex in the memberships: within-group inertia, generalized cuts (extending weighted Ncut and modularity), and membership discontinuities (generalizing Dirichlet forms). The framework is closely related to discrete Markov models, random walks, label propagation and spatial autocorrelation (Moran's I), and can express the Mumford-Shah approach. Four small datasets illustrate the theory.