Mixed-order phase transition in a two-step contagion model with single infectious seed

TL;DR

This study investigates a generalized epidemic model showing a mixed-order phase transition, where the order parameter jumps discontinuously but critical behavior appears in outbreak size distribution, revealing complex critical phenomena.

Contribution

It uncovers the conditions under which a mixed-order transition occurs in a two-step contagion model with a single infectious seed, combining numerical simulations and finite-size scaling analysis.

Findings

Discontinuous jump in the order parameter without critical divergence.

Power-law distribution of outbreak sizes indicating critical behavior.

Diverging mean outbreak size consistent with percolation theory.

Abstract

A hybrid phase transition (HPT) that exhibits properties of continuous and discontinuous phase transitions at the same transition point has been observed in diverse complex systems. Previous studies of the HPTs on complex networks mainly focused on whether the order parameter is continuous or discontinuous. However, more careful and fundamental questions on the critical behaviors of the HPT such as how the divergences of the susceptibility and of the correlation size are affected by the discontinuity of the order parameter have been addressed. Here, we consider a generalized epidemic model that is known to exhibit a discontinuous transition as a spinodal transition. Performing extensive numerical simulations and using finite-size scaling analysis, we examine diverging behaviors of the susceptibility and the correlation size. We find that when there is one infectious node and under a certain condition, the order parameter can exhibit a discontinuous jump but does not exhibit any critical behavior before or after the jump. This feature differs from what we observed in HPTs in the percolation pruning process. However, critical behavior appears in the form of a power-law behavior of the outbreak size distribution. The mean outbreak size, corresponding to the susceptibility, diverge following the conventional percolation behavior. Thus a mixed-order transition occurs. The hyperscaling relation does not hold.