Contagion and spreading processes

Understanding cascading processes on complex network topologies is paramount for modelling how diseases, information, fake news and other media spread. In this article, we extend the multi-type branching process method developed in Keating et al., (2022), which relies on networks having homogenous node properties, to a more general class of clustered networks. Using a model of socially inspired complex contagion we obtain results, not just for the average behaviour of the cascades but for full distributions of the cascade properties. We introduce a new method for the inversion of probability generating functions to recover their underlying probability distributions; this derivation naturally extends to higher dimensions. This inversion technique is used along with the multi-type branching process to obtain univariate and bivariate distributions of cascade properties. Finally, using clique-cover methods, we apply the methodology to synthetic and real-world networks and compare the theoretical distribution of cascade sizes with the results of extensive numerical simulations.

In recent years, research has increasingly focused on understanding how higher-order interactions shape dynamical processes, particularly in contagion dynamics like the SIS model and its extensions. Studies have highlighted the role of the microscopic structure of higher-order interactions in determining the onset of epidemic outbreaks.This work introduces a group-based framework for the Susceptible-Infected-Recovered (SIR) process, incorporating higher-order interactions across groups of arbitrary size. The model explicitly accounts for heterogeneity in degree distributions and inter-order overlap—shared connectivity among hyperedges of different orders. By capturing these dynamical correlations, the framework addresses gaps in existing models, enabling detailed exploration of how heterogeneity and overlap influence contagion dynamics.For systems with 2- and 3-body interactions, the framework yields analytical expressions for the epidemic threshold. The results show that increasing inter-order overlap makes the system more prone to epidemic outbreaks. Remarkably, high heterogeneity in 2-body degree distributions counteracts this sensitivity. This effect stems from the interplay between pairwise and 3-body interaction dynamics, which the model disentangles analytically.The study also examines the final epidemic size and the temporal evolution of infections. A striking finding is that higher heterogeneity in group interactions can trigger explosive outbreaks, a phenomenon well-documented in SIS dynamics but previously unobserved in SIR processes. Specifically, while pairwise interactions lead to smooth transitions, 3-body interactions exhibit abrupt and explosive growth as heterogeneity increases.These results underscore the importance of higher-order heterogeneity and dynamical correlations in shaping epidemic dynamics. By providing a versatile analytical framework, our model gives us new insights, deepening our understanding of complex contagion processes.

Models of spreading processes have been crucial for tackling a wide array of phenomena, from opinion formation to disease propagation. Such processes are arguably underpinned by fundamentally different mechanisms: in simple contagion, transmission occurs via contact with an infected neighbor, while in complex contagion it requires a level of consensus among neighbors. Despite huge advances in our understanding of spreading processes, our ability to assess the prevalence of underlying mechanisms from empirical data is limited. This is partly due to lack of data, but also due to the often non-trivial ways in which network topology interacts with the contagion dynamics on top. We propose a statistical inference framework for detecting the mechanisms underlying spreading phenomena.Our core assumption is that spreading mechanisms can be captured by matrices of infection and recovery rates that depend only on a node's degree and the number of infected neighbors. Focusing on a set of basic infection mechanisms, our main contributions are two-fold. First, we assess the dynamic range of different processes, i.e., the combined effect of network topology, parameters, sample size and initial conditions on our ability to identify mechanisms via statistical inference. Then, we infer which mechanisms explain the dynamics of three empirical datasets.Our results show that the detectability of spreading mechanisms is highly dependent on topology, infectivity, distribution of trials and sample size, including regimes where detection is highly unlikely. For example, for an SI model having many infected neighbors might lead to several infections and hinder detection, yet fewer infections mean that detection is possible. We also assess how the complete distribution of trials over time, including recovery events, can aid in the detectability of different processes, with the end goal of proposing estimators that account for regions of the dynamic range where inference is hindered.

Complex contagion models that involve contagion along higher-order structures, such assimplicial complexes and hypergraphs, yield new classes of mean-field models. Interestingly,the differential equations arising from many such models often exhibit a similar form, resultingin qualitatively comparable global bifurcation patterns. Motivated by this observation, weinvestigate a generalized mean-field-type model that provides a unified framework for analysinga range of different models. In particular, we derive analytical conditions for the emergence ofdifferent bifurcation regimes exhibited by three models of increasing complexity—ranging fromthree- and four-body interactions to two connected populations with both pairwise and three-body interactions. For the first two cases, we give a complete characterisation of all possibleoutcomes, along with the corresponding conditions on network and epidemic parameters. In thethird case, we demonstrate that multistability is possible despite only three-body interactions.Our results reveal that single population models with three-body interactions can only exhibitsimple transcritical transitions or bistability, whereas with four-body interactions multistabilitywith two distinct endemic steady states is possible. Surprisingly, the two-population modelexhibits multistability via symmetry breaking despite three-body interactions only. Our worksheds light on the relationship between equation structure and model behaviour and makes thefirst step towards elucidating mechanisms by which different system behaviours arise, and hownetwork and dynamic properties facilitate or hinder outcomes.

The susceptible-infectious-recovered (SIR) epidemic model and the bond percolation model are often treated as equivalent. However, this assumption is justified under certain conditions on transmission and recovery dynamics that are not necessarily realistic. In the general case, where recovery time can take multiple values, the isomorphism breaks down due to the correlated nature of epidemic propagation. For locally tree-like networks, the bond percolation scheme can still be used to accurately compute the epidemic threshold and the final epidemic size even though isomorphism cannot be assumed, but this agreement does not hold for general networks containing short cycles.To clarify the differences between the epidemic model and bond percolation, we construct a message passing formalism for the two models on tree-like and loopy networks. We note that this study is the first to formulate a message passing approach for the SIR epidemic model on general networks. Our approach highlights the conditions under which the two models are interchangeable and when one model can be a good approximation of the other.

Many network spreading models are challenged by the global structural correlations typical of real complex networks. We consider mitigating this challenge via a network property ideally suited to capturing spreading. This is the network correlation dimension, a property capturing how the number of reachable nodes scales with network distance. Applying this approach to susceptible-infected-recovered processes leads to a simple model which, for a range of synthetic and empirical networks, can outperform models of substantially higher complexity in predicting the early stages of spreading. The proposed dimensional spreading model also provides a basic reproduction number offering new information about final system state.

Recent advances in network science have highlighted the importance of non-pairwise approaches to modelling complex systems, particularly in social and biological networks where interactions often extend beyond dyadic relationships. Despite valuable insights obtained via inclusion of higher-order mechanisms, theoretical studies in this direction remain scarce.We extend classical birth-death processes (BDPs) to include higher-order interactions for modelling complex contagion dynamics. Our framework incorporates simplicial complexes, introducing an additional parameter to capture multi-node interactions, such as 2-simplices, and is easily extendable to higher-order simplicial complexes. The exact model for fully connected networks gives rise to a Markov Chain which can be specified in terms of the forward Kolmogorov equations.Building on Crawford's general BDP framework, we demonstrate that infection rates, both pairwise and higher-order, can be reconstructed from simulation observations. Furthermore, using simulations, we distinguish between infections via pairwise and complex contagion routes and observe that often pairwise infections dominate at the start of an outbreak. However, later in the epidemic, complex contagion tends to dominate. Simulations reveal the significant roles of higher-order terms in epidemic dynamics, offering new insights into complex contagion on networks. Following on from the approach in the previous paper, we explore whether the rates of complex contagion can be learnt on networks other than fully connected. Furthermore, we explore how the properties of higher-order networks manifest in the shape of these rates with the view of developing a classifier.

Superspreading events [1] have profoundly influenced the transmission dynamics of infectious disease outbreaks, in-cluding during the COVID-19 pandemic [2,3,4]. However, their sporadic nature makes prediction challenging. Theextensive genomic data collected during the pandemic presents an opportunity for retrospective analyses to identify thefactors driving these transmission patterns. This study evaluates a model integrating pairwise genetic and temporal dis-tances to infer transmission clusters associated with superspreading events via community detection algorithms. Syntheticdatasets with varying levels of stochasticity were utilised to assess the model’s performance in identifying superspreadingclusters and compare it to a recently published regression-based approach [5]. The results demonstrate that our proposedmodel has advantages in terms of simplicity, and under conditions of increasing epidemiological uncertainty and poorerquality data. Challenges such as low sampling density, high stochastic variability, and biased data can undermine cluster-ing performance and hinder the accurate identification of superspreading events. These findings underscore the criticalneed for comprehensive, high-resolution genomic and epidemiological data to enhance clustering accuracy and capturethe spatiotemporal transmission dynamics. We plan to apply this method with further refinement to datasets obtainedfrom Public Health Scotland [6] to characterise the spatiotemporal dynamics of superspreading and the factors drivingthis transmission pattern.