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

Juliane T. Moraes1 and Silvio C. Ferreira1,2

  • 1Departamento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, Minas Gerais, Brazil
  • 2National Institute of Science and Technology for Complex Systems, 22290-180, Rio de Janeiro, Brazil

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Phys. Rev. E 111, 044302 – Published 8 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.044302

Abstract

We analyze the 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 off-critical 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 very large 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 the time series.

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Phys. Rev. E 111, 044302– Published 8 April, 2025
  • Received 23 November 2024
  • Accepted 14 March 2025
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