Smeared phase transitions in percolation on real complex networks

TL;DR

This paper investigates smeared phase transitions in percolation on real complex networks, revealing that inhomogeneities cause gradual transitions that challenge traditional detection methods and require new analytical tools.

Contribution

It introduces a local susceptibility measure for better detection of smeared transitions and critiques existing analytical methods for their limitations in characterizing these phenomena.

Findings

Smeared transitions are common in real networks due to inhomogeneities.

Sequential phase transitions can occur within correlated subsystems.

Current analytical tools struggle to fully characterize smeared transitions.

Abstract

Percolation on complex networks is used both as a model for dynamics on networks, such as network robustness or epidemic spreading, and as a benchmark for our models of networks, where our ability to predict percolation measures our ability to describe the networks themselves. In many applications, correctly identifying the phase transition of percolation on real-world networks is of critical importance. Unfortunately, this phase transition is obfuscated by the finite size of real systems, making it hard to distinguish finite size effects from the inaccuracy of a given approach that fails to capture important structural features. Here, we borrow the perspective of smeared phase transitions and argue that many observed discrepancies are due to the complex structure of real networks rather than to finite size effects only. In fact, several real networks often used as benchmarks feature a smeared phase transition where inhomogeneities in the topological distribution of the order parameter do not vanish in the thermodynamic limit. We find that these smeared transitions are sometimes better described as sequential phase transitions within correlated subsystems. Our results shed light not only on the nature of the percolation transition in complex systems, but also provide two important insights on the numerical and analytical tools we use to study them. First, we propose a measure of local susceptibility to better detect both clean and smeared phase transitions by looking at the topological variability of the order parameter. Second, we highlight a shortcoming in state-of-the-art analytical approaches such as message passing, which can detect smeared transitions but not characterize their nature.