A Hebbian approach to complex network generation

TL;DR

This paper introduces a novel Hebbian-inspired method for generating complex networks, linking interaction types to emergent topologies and analyzing their statistical mechanics, including small-world effects and thermodynamics.

Contribution

It presents a new approach to model discrete systems with interacting components, revealing how interaction patterns influence network topology and thermodynamic properties.

Findings

Weighted random graphs with tunable correlations are analytically described.

Imitative couplings lead to natural small-world effects.

Thermodynamics are solved at replica symmetric level, revealing critical behavior.

Abstract

Through a redefinition of patterns in an Hopfield-like model, we introduce and develop an approach to model discrete systems made up of many, interacting components with inner degrees of freedom. Our approach clarifies the intrinsic connection between the kind of interactions among components and the emergent topology describing the system itself; also, it allows to effectively address the statistical mechanics on the resulting networks. Indeed, a wide class of analytically treatable, weighted random graphs with a tunable level of correlation can be recovered and controlled. We especially focus on the case of imitative couplings among components endowed with similar patterns (i.e. attributes), which, as we show, naturally and without any a-priori assumption, gives rise to small-world effects. We also solve the thermodynamics (at a replica symmetric level) by extending the double stochastic stability technique: free energy, self consistency relations and fluctuation analysis for a picture of criticality are obtained.