Renormalization group analysis of the small-world network model

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 critical point with a divergent characteristic 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. (C) 1999 Published by Elsevier Science B.V. All rights reserved.