A geometric entropy detecting the Erd\"os-R\'enyi phase transition

We propose a method to associate a differentiable Riemannian manifold to a generic many degrees of freedom discrete system which is not described by a Hamiltonian function. Then, in analogy with classical Statistical Mechanics, we introduce an entropy as the logarithm of the volume of the manifold. The geometric entropy so defined is able to detect a paradigmatic phase transition occurring in random graphs theory: the appearance of the `giant component' according to the Erd\"os-R\'enyi theorem.