Renormalization group analysis of the small-world network model

TL;DR

This paper applies real-space renormalization group analysis to the small-world network model, revealing a continuous phase transition and deriving exact critical exponents and scaling laws, supported by numerical simulations.

Contribution

It introduces an exact renormalization group transformation for the small-world network model and derives the critical behavior analytically.

Findings

Identifies a continuous phase transition with a divergent correlation length.

Calculates the exact critical exponent for the model.

Derives the scaling form for degrees of separation between nodes.

Abstract

We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a normal continuous phase transition with a divergent correlation length as the degree of randomness tends to zero. We propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. We use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of "degrees of separation" between two nodes on the network as a function of the three independent variables. We confirm our results by extensive numerical simulation.