Smeared phase transitions in percolation on real complex networks

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.

Large deviations of random walks on random graphs

“…Whether this transition is a "genuine" DPT that becomes sharp in the thermodynamic limit is not clear at this point. For N < ∞, the transition is rounded or smeared [42] because the URW has a finite probability to reach hairs from the bulk, so it is not strictly absorbing (see Appendix A). For the transition to become sharp, one needs to show that this return probability vanishes as N → ∞, which happens if the bulk is a complete graph of size N or, more generally, if the degrees of the nodes in the bulk grow uniformly in N , none of which applies to ER graphs in the sparse limit.…”

Heterogeneity in outcomes of repeated instances of percolation experiments

“…Along with the full pdfs we also show their degree-based deconvolutions defined by Eqs. (13) and (14). Band edges appear far less sharp than for the Gnutella network, in part as contributions from different degrees strongly overlap.…”

“…The main bands correspond to the contribution to π τ (g) from nodes of different degrees k. The location of the sharp upper cut-offs of the main bands can be predicted from Eqs. (13), (14), as 0 ≤g ν ≤ 1. Upon insertion of the upper boundg ν = 1, one obtains an upper bound for the support of the contribution of degree-k nodes to π τ (g), viz.…”

“…(5), (6), and using the solution to evaluate Eqs. (14) and (13). Panels correspond to results for different numbers of instances of the percolation process.…”

Scaling relations and finite-size scaling in gravitationally correlated lattice percolation models

“…Network percolation has been playing an important role as a simplified model to understand spreading processes of message, disease, matter and dynamic processes in complex systems [12][13][14][15][16][17][18][19][20]. It has been attracting more and more attention from physics and other research communities.…”