Vezzani A, Muñoz MA, Burioni R. Anomalous finite-size scaling in higher-order processes with absorbing states.
Phys Rev E 2023;
107:014105. [PMID:
36797930 DOI:
10.1103/physreve.107.014105]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 12/21/2022] [Indexed: 01/09/2023]
Abstract
Here we study standard and higher-order birth-death processes on fully connected networks, within the perspective of large-deviation theory [also referred to as the 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 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 nontrivial finite-size-scaling properties. In particular, we show that the variance-to-mean ratio diverges at criticality for [1≤q≤3], with a maximal variability at q=2, confirming that complex-contagion processes can exhibit peculiar scaling features including wild variability. Moreover, 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. Some possible extensions of this work are also discussed.
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