Multilayer Networks in a Nutshell

Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's constituents. During the last two decades, network science has provided many insights in natural, social, biological and technological systems. However, real systems are more often than not interconnected, with many interdependencies that are not properly captured by single layer networks. To account for this source of complexity, a more general framework, in which different networks evolve or interact with each other, is needed. These are known as multilayer networks. Here we provide an overview of the basic methodology used to describe multilayer systems as well as of some representative dynamical processes that take place on top of them. We round off the review with a summary of several applications in diverse fields of science.Comment: 16 pages and 3 figures. Submitted for publicatio

From degree-correlated to payoff-correlated activity for an optimal resolution of social dilemmas

An active participation of players in evolutionary games depends on several factors, ranging from personal stakes to the properties of the interaction network. Diverse activity patterns thus have to be taken into account when studying the evolution of cooperation in social dilemmas. Here we study the weak prisoner's dilemma game, where the activity of each player is determined in a probabilistic manner either by its degree or by its payoff. While degree-correlated activity introduces cascading failures of cooperation that are particularly severe on scale-free networks with frequently inactive hubs, payoff-correlated activity provides a more nuanced activity profile, which ultimately hinders systemic breakdowns of cooperation. To determine optimal conditions for the evolution of cooperation, we introduce an exponential decay to payoff-correlated activity that determines how fast the activity of a player returns to its default state. We show that there exists an intermediate decay rate, at which the resolution of the social dilemma is optimal. This can be explained by the emerging activity patterns of players, where the inactivity of hubs is compensated effectively by the increased activity of average-degree players, who through their collective influence in the network sustain a higher level of cooperation. The sudden drops in the fraction of cooperators observed with degree-correlated activity therefore vanish, and so does the need for the lengthy spatiotemporal reorganization of compact cooperative clusters. The absence of such asymmetric dynamic instabilities thus leads to an optimal resolution of social dilemmas, especially when the conditions for the evolution of cooperation are strongly adverse.Comment: 8 two-column pages, 6 figures; accepted for publication in Physical Review

Directionality reduces the impact of epidemics in multilayer networks

The study of how diseases spread has greatly benefited from advances in network modeling. Recently, a class of networks known as multilayer graphs has been shown to describe more accurately many real systems, making it possible to address more complex scenarios in epidemiology such as the interaction between different pathogens or multiple strains of the same disease. In this work, we study in depth a class of networks that have gone unnoticed up to now, despite of its relevance for spreading dynamics. Specifically, we focus on directed multilayer networks, characterized by the existence of directed links, either within the layers or across layers. Using the generating function approach and numerical simulations of a stochastic susceptible-infected-susceptible (SIS) model, we calculate the epidemic threshold for these networks for different degree distributions of the networks. Our results show that the main feature that determines the value of the epidemic threshold is the directionality of the links connecting different layers, regardless of the degree distribution chosen. Our findings are of utmost interest given the ubiquitous presence of directed multilayer networks and the widespread use of disease-like spreading processes in a broad range of phenomena such as diffusion processes in social and transportation systems.Comment: 20 pages including 7 figures. Submitted for publicatio

Prediction of scientific collaborations through multiplex interaction networks

Link prediction algorithms can help to understand the structure and dynamics of scientific collaborations and the evolution of Science. However, available algorithms based on similarity between nodes of collaboration networks are bounded by the limited amount of links present in these networks. In this work, we reduce the latter intrinsic limitation by generalizing the Adamic-Adar method to multiplex networks composed by an arbitrary number of layers, that encode diverse forms of scientific interactions. We show that the new metric outperforms other single-layered, similarity-based scores and that scientific credit, represented by citations, and common interests, measured by the usage of common keywords, can be predictive of new collaborations. Our work paves the way for a deeper understanding of the dynamics driving scientific collaborations, and provides a new algorithm for link prediction in multiplex networks that can be applied to a plethora of systems

A multilayer perspective for the analysis of urban transportation systems

Public urban mobility systems are composed by several transportation modes connected together. Most studies in urban mobility and planning often ignore the multi-layer nature of transportation systems considering only aggregated versions of this complex scenario. In this work we present a model for the representation of the transportation system of an entire city as a multiplex network. Using two different perspectives, one in which each line is a layer and one in which lines of the same transportation mode are grouped together, we study the interconnected structure of 9 different cities in Europe raging from small towns to mega-cities like London and Berlin highlighting their vulnerabilities and possible improvements. Finally, for the city of Zaragoza in Spain, we also consider data about service schedule and waiting times, which allow us to create a simple yet realistic model for urban mobility able to reproduce real-world facts and to test for network improvements

Characterising the role of human behaviour in the effectiveness of contact-tracing applications

Albeit numerous countries relied on contact-tracing (CT) applications as an epidemic control measure against the COVID-19 pandemic, the debate around their effectiveness is still open. Most studies indicate that very high levels of adoption are required to stop disease progression, placing the main interest of policymakers in promoting app adherence. However, other factors of human behaviour, like delays in adherence or heterogeneous compliance, are often disregarded. To characterise the impact of human behaviour on the effectiveness of CT apps we propose a multilayer network model reflecting the co-evolution of an epidemic outbreak and the app adoption dynamics over a synthetic population generated from survey data. The model was initialised to produce epidemic outbreaks resembling the first wave of the COVID-19 pandemic and was used to explore the impact of different changes in behavioural features in peak incidence and maximal prevalence. The results corroborate the relevance of the number of users for the effectiveness of CT apps but also highlight the need for early adoption and, at least, moderate levels of compliance, which are factors often not considered by most policymakers. The insight obtained was used to identify a bottleneck in the implementation of several apps, such as the Spanish CT app, where we hypothesise that a simplification of the reporting system could result in increased effectiveness through a rise in the levels of compliance.Comment: 25 pages including all figures and S

Digital cities and the spread of COVID-19: characterizing the impact of non-pharmaceutical interventions in five cities in Spain

Mathematical modeling has been fundamental to achieving near real-time accurate forecasts of the spread of COVID-19. Similarly, the design of non-pharmaceutical interventions has played a key role in the application of policies to contain the spread. However, there is less work done regarding quantitative approaches to characterize the impact of each intervention, which can greatly vary depending on the culture, region, and specific circumstances of the population under consideration. In this work, we develop a high-resolution, data-driven agent-based model of the spread of COVID-19 among the population in five Spanish cities. These populations synthesize multiple data sources that summarize the main interaction environments leading to potential contacts. We simulate the spreading of COVID-19 in these cities and study the effect of several non-pharmaceutical interventions. We illustrate the potential of our approach through a case study and derive the impact of the most relevant interventions through scenarios where they are suppressed. Our framework constitutes a first tool to simulate different intervention scenarios for decision-making.Comment: Main text with 5 figures and 1 table, and Supplementary Materia

Assessing the Risk of Spatial Spreading of Diseases in Hospitals

In recent years, the transmission of healthcare-associated infections (HAIs) has led to substantial economic loss, extensive damage, and many preventable deaths. With the increasing availability of data, mathematical models of pathogen spreading in healthcare settings are becoming more detailed and realistic. Here, we make use of spatial and temporal information that has been obtained from healthcare workers (HCWs) in three hospitals in Canada and generate data-driven networks that allow us to realistically simulate the spreading of an airborne respiratory pathogen in such settings. By exploring in depth the dynamics of HAIs on the generated networks, we quantify the infection risk associated with both the spatial units of the hospitals and HCWs categorized by their occupations. Our findings show that the "inpatient care" and "public area" are the riskiest categories of units and "nurse" is the occupation at a greater risk of getting infected. Our results provide valuable insights that can prove important for measuring risks associated with HAIs and for strengthening prevention and control measures with the potential to reduce transmission of infections in hospital settings

Human mobility networks and persistence of rapidly mutating pathogens

Rapidly mutating pathogens may be able to persist in the population and reach an endemic equilibrium by escaping hosts' acquired immunity. For such diseases, multiple biological, environmental and population-level mechanisms determine the dynamics of the outbreak, including pathogen's epidemiological traits (e.g. transmissibility, infectious period and duration of immunity), seasonality, interaction with other circulating strains and hosts' mixing and spatial fragmentation. Here, we study a susceptible-infected-recovered-susceptible model on a metapopulation where individuals are distributed in subpopulations connected via a network of mobility flows. Through extensive numerical simulations, we explore the phase space of pathogen's persistence and map the dynamical regimes of the pathogen following emergence. Our results show that spatial fragmentation and mobility play a key role in the persistence of the disease whose maximum is reached at intermediate mobility values. We describe the occurrence of different phenomena including local extinction and emergence of epidemic waves, and assess the conditions for large scale spreading. Findings are highlighted in reference to previous works and to real scenarios. Our work uncovers the crucial role of hosts' mobility on the ecological dynamics of rapidly mutating pathogens, opening the path for further studies on disease ecology in the presence of a complex and heterogeneous environment.Comment: 29 pages, 7 figures. Submitted for publicatio