A geometric entropy detecting the Erdös-Rényi phase transition

Abstract

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 Erdös-Rényi theorem.

11 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1410.5459

References in corpus (8)

• Critical phenomena in complex networks
• The entropy of network ensembles
• The entropy of randomized network ensembles
• Entropy Rate of Diffusion Processes on Complex Networks
• Entropies of complex networks with hierarchically constrained topologies
• Asymptotic entropy and green speed for random walks on countable groups
• The emergence of a giant component in random subgraphs of pseudo-random graphs
• Random Walks on Complex Networks

Cited by in corpus (8)

• Network geometry with flavor: from complexity to quantum geometry
• Interdisciplinary and physics challenges of Network Theory
• Information geometric methods for complexity
• Geometric Algebra and Information Geometry for Quantum Computational Software
• Theoretical investigations of an information geometric approach to complexity
• Microcanonical entropy for classical systems
• Towards an Information Geometric characterization/classification of Complex Systems. I. Use of Generalized Entropies
• Catching homologies by geometric entropy