Recent studies have examined the relationship between the intensity of extreme rainfall and temperature. Two main reasons justify this interest. First, the moisture-holding capacity of the atmosphere is governed by the Clausius–Clapeyron (CC) equation. Second, the temperature dependence of extreme-intensity rainfalls should follow a similar relationship assuming relative humidity remains constant and extreme rainfalls are driven by the actual water content of the atmosphere. The relationship between extreme rainfall intensity and air temperature (Pextr–Ta) was assessed by analyzing maximum daily rainfall intensities for durations ranging from 5 min to 12 h for more than 100 meteorological stations across Canada. Different factors that could influence this relationship have been analyzed. It appears that the duration and the climatic region have a strong influence on this relationship. For short durations, the Pextr–Ta relationship is close to the CC scaling for coastal regions while a super-CC scaling followed by an upper limit is observed for inland regions. As the duration increases, the slope of the relationship Pextr–Ta decreases for all regions. The shape of the Pextr–Ta curve is not sensitive to the percentile or season. Complementary analyses have been carried out to understand the departures from the expected Clausius–Clapeyron scaling. The relationship between dewpoint temperature and extreme rainfall intensity shows that the relative humidity is a limiting factor for inland regions, but not for coastal regions. Using hourly rainfall series, an event-based analysis is proposed in order to understand other deviations (super-CC, sub-CC, and monotonic decrease). The analyses suggest that the observed scaling is primarily due to the rainfall event dynamic.

Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.

Network growth processes can be understood as generative models of the structure and history of complex networks. This point of view naturally leads to the problem of network archaeology: reconstructing all the past states of a network from its structure-a difficult permutation inference problem. In this paper, we introduce a Bayesian formulation of network archaeology, with a generalization of preferential attachment as our generative mechanism. We develop a sequential Monte Carlo algorithm to evaluate the posterior averages of this model, as well as an efficient heuristic that uncovers a history well correlated with the true one, in polynomial time. We use these methods to identify and characterize a phase transition in the quality of the reconstructed history, when they are applied to artificial networks generated by the model itself. Despite the existence of a no-recovery phase, we find that nontrivial inference is possible in a large portion of the parameter space as well as on empirical data.

We present a degree-based theoretical framework to study the susceptible-infected-susceptible (SIS) dynamics on time-varying (rewired) configuration model networks. Using this framework on a given degree distribution, we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the SIS process. This analysis is suitable for the characterization of the phase transition and leads to three main contributions. (i) We obtain a self-consistent expression for the absorbing-state threshold, able to capture both collective and hub activation. (ii) We recover the predictions of a number of existing approaches as limiting cases of our analysis, providing thereby a unifying point of view for the SIS dynamics on random networks. (iii) We obtain bounds for the critical exponents of a number of quantities in the stationary state. This allows us to reinterpret the concept of hub-dominated phase transition. Within our framework, it appears as a heterogeneous critical phenomenon : observables for different degree classes have a different scaling with the infection rate. This phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold.

Forecasting the evolution of contagion dynamics is still an open problem to which mechanistic models only offer a partial answer. To remain mathematically or computationally tractable, these models must rely on simplifying assumptions, thereby limiting the quantitative accuracy of their predictions and the complexity of the dynamics they can model. Here, we propose a complementary approach based on deep learning where effective local mechanisms governing a dynamic on a network are learned from time series data. Our graph neural network architecture makes very few assumptions about the dynamics, and we demonstrate its accuracy using different contagion dynamics of increasing complexity. By allowing simulations on arbitrary network structures, our approach makes it possible to explore the properties of the learned dynamics beyond the training data. Finally, we illustrate the applicability of our approach using real data of the COVID-19 outbreak in Spain. Our results demonstrate how deep learning offers a new and complementary perspective to build effective models of contagion dynamics on networks.