Source-linked AI summary
Thresholds for epidemic spreading in networks
Claudio Castellano, Romualdo Pastor-Satorras
TL;DR
The paper examines epidemic thresholds on quenched power-law networks and challenges HMF expectations for SIS dynamics. It analyzes the spectral and hub mechanisms behind the result, finding a vanishing SIS threshold when the maximum degree diverges, while SIR retains a finite threshold on scale-rich networks; it conjectures that steady-state dynamics generally produce this distinction.
Problem
The paper addresses whether HMF theory correctly predicts epidemic thresholds for SIS and SIR processes on quenched networks with heterogeneous power-law connectivity.
Method
The authors combine spectral analysis with numerical simulations of SIS and SIR models on quenched configuration-model networks, including controlled maximum-degree experiments.
Results
SIS has a vanishing threshold whenever the maximum degree diverges, because the largest hub sustains activity and spreads infection, whereas SIR has a finite threshold on scale-rich networks consistent with HMF theory.
Takeaways & Limitations
On quenched scale-rich networks, epidemic thresholds may be governed by whether the model permits a steady state rather than by scale-free connectivity alone.
Takeaways & Limitations
For unrestricted maximum degree, sample-to-sample fluctuations can prevent a well-defined average epidemic threshold numerically unless γ < 3 and the degree cutoff exponent satisfies ω ≥2.
Abstract
from arXiv · showhide
We study the threshold of epidemic models in quenched networks with degree distribution given by a power-law. For the susceptible-infected-susceptible (SIS) model the activity threshold lambda_c vanishes in the large size limit on any network whose maximum degree k_max diverges with the system size, at odds with heterogeneous mean-field (HMF) theory. The vanishing of the threshold has not to do with the scale-free nature of the connectivity pattern and is instead originated by the largest hub in the system being active for any spreading rate lambda>1/sqrt{k_max} and playing the role of a self-sustained source that spreads the infection to the rest of the system. The susceptible-infected-removed (SIR) model displays instead agreement with HMF theory and a finite threshold for scale-rich networks. We conjecture that on quenched scale-rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state.