The increasing availability of temporal network data is calling for more research on extracting and characterizing mesoscopic structures in temporal networks and on relating such structure to specific functions or properties of the system. An outstanding challenge is the extension of the results achieved for static networks to time-varying networks, where the topological structure of the system and the temporal activity patterns of its components are intertwined. Here we investigate the use of a latent factor decomposition technique, non-negative tensor factorization, to extract the community-activity structure of temporal networks. The method is intrinsically temporal and allows to simultaneously identify communities and to track their activity over time. We represent the time-varying adjacency matrix of a temporal network as a three-way tensor and approximate this tensor as a sum of terms that can be interpreted as communities of nodes with an associated activity time series. We summarize known computational techniques for tensor decomposition and discuss some quality metrics that can be used to tune the complexity of the factorized representation. We subsequently apply tensor factorization to a temporal network for which a ground truth is available for both the community structure and the temporal activity patterns. The data we use describe the social interactions of students in a school, the associations between students and school classes, and the spatio-temporal trajectories of students over time. We show that non-negative tensor factorization is capable of recovering the class structure with high accuracy. In particular, the extracted tensor components can be validated either as known school classes, or in terms of correlated activity patterns, i.e., of spatial and temporal coincidences that are determined by the known school activity schedule.

As the first decentralized digital currency introduced in 2009 together with the blockchain, Bitcoin offers new opportunities both for developed and developing countries. Bitcoin peer-to-peer transactions are independent of the banking system, facilitating foreign exchanges with low transaction fees, such as remittances, and offering a high degree of anonymity. These opportunities together with other key factors led the Bitcoin to become extremely popular and caused its price to skyrocket during 2017 (Henry et al. in J Digit Bank 2(4):311–337, 2018). However, while the Bitcoin blockchain attracts a lot of attention, it remains difficult to investigate where this attention comes from, due to the pseudo-anonymity of the system, and consequently to appreciate its social impact. Here we make an attempt to characterize the adoption of the Bitcoin blockchain by country. In the first part of the work we show that information about the number of Bitcoin software client downloads, the IP addresses that act as relays for the transactions, and the Internet searches about Bitcoin provide together a coherent picture of the system evolution in different countries. Using these quantities as a proxy for user adoption, we identify several socio-economic indexes such as the GDP per capita, freedom of trade and the Internet penetration as key variables correlated with the degree of user adoption. In the second part of the work, we build a network of Bitcoin transactions between countries using the IP addresses of nodes relaying transactions and we develop an augmented version of the gravity model of trade in order to identify socio-economic factors linked to the flow of Bitcoin between countries. In a nutshell our study provides a new insight on Bitcoin adoption by country and on the potential socio-economic drivers of the international Bitcoin flow.

After more than 1 year into the COVID-19 pandemic, governments worldwide still face the challenge of adopting non-pharmaceutical interventions to mitigate the risks posed by the emergence of new SARS-CoV-2 variants and the lack of a worldwide equitable vaccine allocation. Thus, it becomes crucial to identify the drivers of mobility responses to mitigation efforts during different restriction regimes, for planning interventions that are both economically and socially sustainable while effective in controlling an outbreak. Here, using anonymous and privacy-enhanced cell phone data from Italy, we investigate the determinants of spatial variations of reductions in mobility and co-location in response to the adoption and the lift of restrictions, considering both provinces and city neighbourhoods. In large urban areas, our analysis uncovers the desertification of historic city centres, which persisted after the end of the lockdown. Such centre-periphery gradient was mainly associated with differences in educational attainment. At the province level, the local structure of the labour market mainly explained the variations in mobility responses, together with other demographic factors, such as the population’s age and sex composition. In the future, targeted interventions should take into account how the ability to comply with restrictions varies across geographical areas and socio-demographic groups.

Despite the availability of effective vaccines against SARS-CoV-2, non-pharmaceutical interventions remain an important part of the effort to reduce viral circulation caused by emerging variants with the capability of evading vaccine-induced immunity. With the aim of striking a balance between effective mitigation and long-term sustainability, several governments worldwide have adopted systems of tiered interventions, of increasing stringency, that are calibrated according to periodic risk assessments. A key challenge remains in quantifying temporal changes in adherence to interventions, which can decrease over time due to pandemic fatigue, under such kind of multilevel strategies. Here, we examine whether there was a reduction in adherence to tiered restrictions that were imposed in Italy from November 2020 through May 2021, and in particular we assess whether temporal trends in adherence depended on the intensity of the restrictions adopted. We analyzed daily changes in movements and in residential time, combining mobility data with the restriction tier enforced in the Italian regions. Through mixed-effects regression models, we identified a general trend of reduction in adherence and an additional effect of faster waning associated with the most stringent tier. We estimated both effects being of the same order of magnitude, suggesting that adherence decreased twice as fast during the strictest tier as in the least stringent one. Our results provide a quantitative measure of behavioral responses to tiered interventions—a metric of pandemic fatigue—that can be integrated into mathematical models to evaluate future epidemic scenarios.

Italy is currently experiencing the largest COVID-19 outbreak in Europe so far, with more than 100,000 confirmed cases. Following the identification of the first infections, on February 21, 2020, national authorities have put in place an increasing number of restrictions aimed at containing the outbreak and delaying the epidemic peak. Since March 12, the whole country is under lockdown. Here we provide the first quantitative assessment of the impact of such measures on the mobility and the spatial proximity of Italians, through the analysis of a large-scale dataset on de-identified, geo-located smartphone users. With respect to pre-outbreak averages, we estimate a reduction of 50% of the total trips between Italian provinces, following the lockdown. In the same week, the average users’ radius of gyration has declined by about 50% and the average degree of the users’ proximity network has dropped by 47% at national level.

Assessing the impact of mobility on epidemic spreading is of crucial importance for understanding the effect of policies like mass quarantines and selective re-openings. While many factors affect disease incidence at a local level, making it more or less homogeneous with respect to other areas, the importance of multi-seeding has often been overlooked. Multi-seeding occurs when several independent (non-clustered) infected individuals arrive at a susceptible population. This can lead to independent outbreaks that spark from distinct areas of the local contact (social) network. Such mechanism has the potential to boost incidence, making control efforts and contact tracing less effective. Here, through a modeling approach we show that the effect produced by the number of initial infections is non-linear on the incidence peak and peak time. When case importations are carried by mobility from an already infected area, this effect is further enhanced by the local demography and underlying mixing patterns: the impact of every seed is larger in smaller populations. Finally, both in the model simulations and the analysis, we show that a multi-seeding effect combined with mobility restrictions can explain the observed spatial heterogeneities in the first wave of COVID-19 incidence and mortality in five European countries. Our results allow us for identifying what we have called epidemic epicenter: an area that shapes incidence and mortality peaks in the entire country. The present work further clarifies the nonlinear effects that mobility can have on the evolution of an epidemic and highlight their relevance for epidemic control.