Comment on "Abrupt transition in the structural formation of interconnected networks", F. Radicchi and A. Arenas, Nature Phys. 9, 717 (2013)

TL;DR

This paper critiques a previous study claiming an abrupt structural transition in interconnected networks, demonstrating that the observed singularity is a trivial mathematical consequence rather than a genuine phase transition.

Contribution

It clarifies that the previously reported abrupt transition is a mathematical artifact, not a real structural change in interconnected networks.

Findings

The singularity is due to reducibility of the supra-Laplacian operator.

No evidence of a genuine first-order phase transition.

The observed phenomena are mathematically trivial.

Abstract

A recent paper [F. Radicchi and A. Arenas, Nature Phys. 9, 717 (2013)] presented the finding of an abrupt transition in the structure of interconnected networks. This transition was said to be generic and to occur even in networks of finite size. Furthermore, it was remarked that this singular behaviour could be understood in the spirit of a first-order phase transition. We show here that the generic singularity found in that paper is a trivial consequence of the reducibility of the "supra-Laplacian" operator studied. The singular changes observed are therefore not related to any collective abrupt structural transformation in the interconnected networks.