Smeared phase transitions in percolation on real complex networks

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 observed discrepancies may be due to the complex mesoscopic structure of real networks rather than to finite-size effects only. We support and illustrate this claim by studying real and synthetic networks through the lens of local order parameters, message passing, and local susceptibility. Our results not only shed light on the nature of the percolation transition in complex networks 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 discuss a shortcoming in state-of-the-art analytical approaches such as message passing, which can detect smeared transitions but not characterize their nature.