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
