The Internet is a complex network of interconnected routers and the existence of collective behavior such as congestion suggests that the correlations between different connections play a crucial role. It is thus critical to measure and quantify these correlations. We use methods of random matrix theory (RMT) to analyze the cross-correlation matrix C of information flow changes of 650 connections between 26 routers of the French scientific network 'Renater'. We find that C has the universal properties of the Gaussian orthogonal ensemble of random matrices: The distribution of eigenvalues-up to a rescaling which exhibits a typical correlation time of the order 10 minutes-and the spacing distribution follows the predictions of RMT. There are some deviations for large eigenvalues which contain network-specific information and which identify genuine correlations between connections. The study of the most correlated connections reveal the existence of 'active centers' which are exchanging information with a large number of routers thereby inducing correlations between the corresponding connections. These strong correlations could be a reason for the observed self-similarity in the WWW traffic.

The Internet infrastructure is not virtual: its distribution is dictated by social, geographical, economical, or political constraints. However, the infrastructure's design does not determine entirely the information traffic and different sources of complexity such as the intrinsic heterogeneity of the network or human practices have to be taken into account. In order to manage the Internet expansion, plan new connections or optimize the existing ones, it is thus critical to understand correlations between emergent global statistical patterns of Internet activity and human factors. We analyze data from the French national 'Renater' network which has about 2 millions users and which consists in about 30 interconnected routers located in different regions of France and we report the following results. The Internet flow is strongly localized: most of the traffic takes place on a 'spanning' network connecting a small number of routers which can be classified either as 'active centers' looking for information or 'databases' providing information. We also show that the Internet activity of a region increases with the number of published papers by laboratories of that region, demonstrating the positive impact of the Web on scientific activity and illustrating quantitatively the adage 'the more you read, the more you write'.

Among the different indicators that quantify the spread of an epidemic such as the ongoing COVID-19, stands first the reproduction number which measures how many people can be contaminated by an infected person. In order to permit the monitoring of the evolution of this number, a new estimation procedure is proposed here, assuming a well-accepted model for current incidence data, based on past observations. The novelty of the proposed approach is twofold: 1) the estimation of the reproduction number is achieved by convex optimization within a proximal-based inverse problem formulation, with constraints aimed at promoting piecewise smoothness; 2) the approach is developed in a multivariate setting, allowing for the simultaneous handling of multiple time series attached to different geographical regions, together with a spatial (graph-based) regularization of their evolutions in time. The effectiveness of the approach is first supported by simulations, and two main applications to real COVID-19 data are then discussed. The first one refers to the comparative evolution of the reproduction number for a number of countries, while the second one focuses on French departments and their joint analysis, leading to dynamic maps revealing the temporal coevolution of their reproduction numbers.