Unstable supercritical discontinuous percolation transitions

TL;DR

This paper investigates the nature of discontinuous percolation transitions in random networks, revealing complex behaviors such as unstable jumps and the non-coincidence of the largest jump with the percolation threshold, especially in genuinely discontinuous models.

Contribution

It uncovers the rich supercritical behavior of genuinely discontinuous percolation transitions, including unstable jumps and parameter-dependent phenomena, expanding understanding beyond classical models.

Findings

Largest jump coincides with percolation threshold in continuous models

Genuinely discontinuous models show non-coincidence and unstable transitions

Supercritical regime exhibits complex, rich behaviors

Abstract

The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a `Devil's staircase' of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erdos-Renyi percolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erdos-Renyi and other processes exhibiting global continuity, including models exhibiting an `explosive' transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions, which is a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.