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Networks are convenient mathematical models to represent the structure of complex systems, from cells to societies. In the past decade, multilayer network science -- the branch of the field dealing with units interacting in multiple distinct ways, simultaneously -- was demonstrated to be an effective modeling and analytical framework for a wide spectrum of empirical systems, from biopolymer networks (such as interactome and metabolomes) to neuronal networks (such as connectomes), from social networks to urban and transportation networks. In this Element, a decade after the publication of one of the most seminal papers on this topic, we review the most salient features of multilayer network science, covering both theoretical aspects and direct applications to real-world coupled/interdependent systems, from the point of view of multilayer structure, dynamics, and function. We discuss potential frontiers for this topic and the corresponding challenges in the field for the future., Comment: Published in Cambridge Elements. Cambridge University Press; 2022

During COVID-19, draconian countermeasures forbidding non-essential human activities have been adopted worldwide, providing an unprecedented setup for testing sustainability policies. We unravel causal relationships among 16 environmental conditions and human activity variables and argue that, despite a measurable decrease in NO2 concentration due to human activities, locking down a region is insufficient to significantly reduce emissions. Policy strategies more effective than lockdowns must be considered for pollution control and climate change mitigation., Comment: 16 pages, 2 figures

Revealing the structural features of a complex system from the observed collective dynamics is a fundamental problem in network science. In order to compute the various topological descriptors commonly used to characterize the structure of a complex system (e.g. the degree, the clustering coefficient), it is usually necessary to completely reconstruct the network of relations between the subsystems. Several methods are available to detect the existence of interactions between the nodes of a network. By observing some physical quantities through time, the structural relationships are inferred using various discriminating statistics (e.g. correlations, mutual information, etc.). In this setting, the uncertainty about the existence of the edges is reflected in the uncertainty about the topological descriptors. In this study, we propose a novel methodological framework to evaluate this uncertainty, replacing the topological descriptors, even at the level of a single node, with appropriate probability distributions, eluding the reconstruction phase. Our theoretical framework agrees with the numerical experiments performed on a large set of synthetic and real-world networks. Our results provide a grounded framework for the analysis and the interpretation of widely used topological descriptors, such as degree centrality, clustering and clusters, in scenarios where the existence of network connectivity is statistically inferred or when the probabilities of existence $\pi_{ij}$ of the edges are known. To this purpose we also provide a simple and mathematically grounded process to transform the discriminating statistics into the probabilities $\pi_{ij}$ ., Comment: 15 pages, 6 figures. Manuscript updated after peer-review. Appendices added