This article investigates in a systematic way the properties of the classical continuous mean-field theory governed by the generalized Poisson-Boltzmann-Emden equationOrigins of the theory are traced back. Past studies (freezing theories, electrostatic and self-gravitating systems) are relocated in a broader framework. New results concerning the thermodynamic limit, phase transitions, metastability, and the shape of density profiles are provided. In particular, the question of ground states (in relationship to condensation and wetting phenomena) is illustrated by numerous explicit solutions.
Abstract. General models of network navigation must contain a deterministic or drift component, encouraging the agent to follow routes of least cost, as well as a random or diffusive component, enabling free wandering. This paper proposes a thermodynamic formalism involving two path functionals, namely an energy functional governing the drift and an entropy functional governing the diffusion. A freely adjustable parameter, the temperature, arbitrates between the conflicting objectives of minimising travel costs and maximising spatial exploration. The theory is illustrated on various graphs and various temperatures. The resulting optimal paths, together with presumably new associated edges and nodes centrality indices, are analytically and numerically investigated.
General clustering deals with weighted objects and fuzzy memberships. We investigate the group-or object-aggregation-invariance properties possessed by the relevant functionals (effective number of groups or objects, centroids, dispersion, mutual object-group information, etc.). The classical squared Euclidean case can be generalized to non-Euclidean distances, as well as to non-linear transformations of the memberships, yielding the c-means clustering algorithm as well as two presumably new procedures, the convex and pairwise convex clustering. Cluster stability and aggregation-invariance of the optimal memberships associated to the various clustering schemes are examined as well.
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