Strong localization blurs criticality of time series for spreading phenomena on networks

TL;DR

This paper investigates how localization affects the ability of time series analysis, via visibility graphs, to detect criticality in spreading phenomena on networks, revealing limitations in localized cases.

Contribution

It demonstrates that localization can obscure critical signatures in time series, especially in sparse or highly localized network structures, challenging existing detection methods.

Findings

VG degree correlation signals criticality in collective activation.

Localization can delay or hide criticality signatures in time series.

Strong localization may cause false negatives in criticality detection.

Abstract

We analyze critical time series of the order parameter generated with active to inactive phase transitions of spreading dynamics running on the top of heterogeneous networks. Different activation mechanisms that govern the dynamics near the critical point were investigated. The time series were analyzed using the visibility graph (VG) method where a disassortative degree correlation of the VG is a signature of criticality. In contrast, assortative correlation is associated with offcritical dynamics. The signature of criticality given by the VG is confirmed for collective activation phenomena, as in the case of homogeneous networks. Similarly, for a localized activation driven by a densely connected set of hubs, identified by a maximum k-core decomposition, critical times series were also successfully identified by the VG method. However, in the case of activation driven by sparsely distributed hubs, the time series criticality is blurred, being observable only for huge systems. In the case of strong structural localization induced by the presence of rare regions, an assortative VG degree correlation, typical of off-critical series, is observed. We conclude that while macroscopic times series remain good proxies for the analysis of criticality for collective or maximum k-core activation, systems under spatial localization can postpone the signatures of or, in case of extreme localization, lead to false negatives for criticality of time series.