Message-Passing Methods for Complex Contagions

Message-passing methods provide a powerful approach for calculating the expected size of cascades either on random networks (e.g., drawn from a configuration-model ensemble or its generalizations) asymptotically as the number $N$ of nodes becomes infinite or on specific finite-size networks. We review the message-passing approach and show how to derive it for configuration-model networks using the methods of (Dhar et al., 1997) and (Gleeson, 2008). Using this approach, we explain for such networks how to determine an analytical expression for a "cascade condition", which determines whether a global cascade will occur. We extend this approach to the message-passing methods for specific finite-size networks (Shrestha and Moore, 2014; Lokhov et al., 2015), and we derive a generalized cascade condition. Throughout this chapter, we illustrate these ideas using the Watts threshold model.Comment: 14 pages, 3 figure

Spreading of infection on temporal networks: an edge-centered perspective

We discuss a continuous-time extension of the contact-based (CB) model, as proposed in [Koher et al. Phys. Rev. X 9, 031017 (2019)], for infections with permanent immunity on temporal networks. At the core of our methodology is a fundamental change to an edge-centered perspective, which allows for an accurate model on temporal networks, where the underlying time-aggregated graph has a tree structure. From the continuous-time CB model, we derive the infection propagator for the low prevalence limit and propose a novel spectral criterion to estimate the epidemic threshold. In addition, we explore the relation between the continuous-time CB model and the previously proposed edge-based compartmental model, as well as the message-passing framework