Oscillations in two-stage contagion on networks: contagion structure, turnover and connectivity
Abstract
Two-stage contagion describes processes, such as the adoption of an opinion or of a technological novelty, in which an individual must be exposed to an adopter twice before adopting. With demographic turnover these processes can generate waves of adoption and abandonment. We study these oscillations in a two-stage contagion model with turnover through a reservoir of inactive individuals and a small rate of spontaneous activation, and report four main findings. (i) In the mean-field description the oscillations are born at a supercritical Hopf bifurcation and require a predominantly two-stage contagion (first-contact adoption probability Q≲0.17) and a slow turnover; spontaneous activation acts as a perturbation that destroys them. (ii) In finite populations demographic noise sustains quasi-cycles also outside this region, with a spectrum and variance predicted by a linear noise approximation away from the Hopf bifurcation. (iii) Reducing the connectivity of the network damps the oscillation, because naive nodes see fewer active neighbours than in a well-mixed population, and correlations measured in simulations confirm this damping mechanism; a pair approximation predicts that the sustained cycle disappears below a critical mean degree kc, which decreases as Q and the turnover rate decrease, and a finite-size test with up to 3.2×105 nodes confirms this threshold in simulations. (iv) At low connectivity the amplitude no longer depends on Q, so that sparsity erases the signature of the two-stage mechanism, whereas a faster turnover reduces the relative amplitude at every connectivity. The irregular dynamics observed in simulations are noise-driven quasi-cycles, not chaos.
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BibTeX
@misc{munoz2026oscillations,
title = {{Oscillations in two-stage contagion on networks: contagion structure, turnover and connectivity}},
author = {Francisco J. Muñoz and Juan Carlos Nuño and Luca Meacci and Mario Primicerio},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.12920},
eprint = {2611.12920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.12920},
}
RIS
TY - UNPB
TI - Oscillations in two-stage contagion on networks: contagion structure, turnover and connectivity
AU - Francisco J. Muñoz
AU - Juan Carlos Nuño
AU - Luca Meacci
AU - Mario Primicerio
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.12920
UR - https://arxiv.org/abs/2611.12920
AB - Two-stage contagion describes processes, such as the adoption of an opinion or of a technological novelty, in which an individual must be exposed to an adopter twice before adopting. With demographic turnover these processes can generate waves of adoption and abandonment. We study these oscillations in a two-stage contagion model with turnover through a reservoir of inactive individuals and a small rate of spontaneous activation, and report four main findings. (i) In the mean-field description the oscillations are born at a supercritical Hopf bifurcation and require a predominantly two-stage contagion (first-contact adoption probability Q≲0.17) and a slow turnover; spontaneous activation acts as a perturbation that destroys them. (ii) In finite populations demographic noise sustains quasi-cycles also outside this region, with a spectrum and variance predicted by a linear noise approximation away from the Hopf bifurcation. (iii) Reducing the connectivity of the network damps the oscillation, because naive nodes see fewer active neighbours than in a well-mixed population, and correlations measured in simulations confirm this damping mechanism; a pair approximation predicts that the sustained cycle disappears below a critical mean degree kc, which decreases as Q and the turnover rate decrease, and a finite-size test with up to 3.2×10^5 nodes confirms this threshold in simulations. (iv) At low connectivity the amplitude no longer depends on Q, so that sparsity erases the signature of the two-stage mechanism, whereas a faster turnover reduces the relative amplitude at every connectivity. The irregular dynamics observed in simulations are noise-driven quasi-cycles, not chaos.
ER -