Criticality of time series for spreading phenomena on networks: the visibility graph
approach

Criticality can be understood as a state of a large system of interacting agents lingering
at the edge between order and disorder, in which nontrivial emerging phenomena are
ubiquitous. Formerly born in the realm of phase transitions in condensed matter physics at
the thermodynamic equilibrium, critically has been expanded to a broad set of
non-equilibrium and natural systems. Evidences indicate that criticality also plays a
central role in biological systems as, for example, neuron firing dynamics on brain
networks and animal collective motion.

In this talk, we examine critical time series of the order parameter in spreading dynamics
undergoing active-to-inactive phase transitions on heterogeneous networks. Different
activation mechanisms near the critical point are investigated. Using the visibility graph
(VG) method, we find that a disassortative degree correlation in the VG signals
criticality, while assortative correlation indicates off-critical dynamics. This VG
signature is confirmed for collective activation phenomena, as observed in homogeneous
networks. Similarly, for localized activation driven by a densely connected hub set,
identified via maximum k-core decomposition, the VG method reliably detects critical time
series. However, when activation is driven by sparsely distributed hubs, criticality is
obscured and only discernible in very large systems. In cases of strong structural
localization due to rare regions, the VG exhibits an assortative degree correlation,
characteristic of off-critical series. Thus, while macroscopic time series effectively
indicate criticality in collective or maximum k-core activation, spatial localization can
delay these signatures or, in extreme cases, produce false negatives for time series
criticality.