Anomalous finite-size scaling in higher-order processes with absorbing states

TL;DR

This paper analyzes higher-order birth-death processes with absorbing states using large-deviation theory, revealing complex finite-size-scaling behaviors and the importance of next-to-leading terms for understanding fluctuations at criticality.

Contribution

It provides a general expression for stationary distributions in higher-order processes and uncovers novel finite-size-scaling phenomena with implications for complex contagion models.

Findings

Variance-to-mean ratio diverges at criticality for certain q-values

Maximal variability occurs at q=2 in the $q$-SIS model

Next-to-leading terms are crucial for capturing singularities in systems with absorbing states

Abstract

We study standard and higher-order birth-death processes on fully connected networks, within the perspective of large-deviation theory (also referred to as Wentzel-Kramers-Brillouin (WKB) method in some contexts). We obtain a general expression for the leading and next-to-leading terms of the stationary probability distribution of the fraction of "active" sites, as a function of parameters and network size $N$. We reproduce several results from the literature and, in particular, we derive all the moments of the stationary distribution for the $q$-susceptible-infected-susceptible ($q-SIS$) model, i.e., a high-order epidemic model requiring of $q$ active ("infected") sites to activate an additional one. We uncover a very rich scenario for the fluctuations of the fraction of active sites, with non-trivial finite-size-scaling properties. In particular, we show that the variance-to-mean ratio diverges at criticality for $[1 \leq q\leq 3]$, with a maximal variability at $q=2$, confirming that complex-contagion processes can exhibit peculiar scaling features including wild variability and that the leading-order in a large-deviation approach does not suffice to describe them: next-to-leading terms are essential to capture the intrinsic singularity at the origin of systems with absorbing states.