Networks are a simple but enormously effective model to represent the interaction of a large number of physical, social or economic systems. Nowadays networks are present in all areas of science, technology and society, from biology to economy and social sciences. Relevant examples are the Internet, transportation and communication networks, gene regulatory networks, large scale brain networks or protein-protein-interaction networks, among others. The main goal of RandNET is to analyse in depth the various aspects of randomness in large networks, to provide solid theoretical foundations for modeling randomness, and to apply them to combinatorial statistics and learning in networks. In RandNET we advocate for building solid mathematical and computational foundations for network analysis, and for a fruitful interplay between theoretical models and applications.
The mathematical model of a network is a graph, which may be directed (the internet is a paramount example), weighted (as is common in combinatorial optimization), or have different categories for nodes and/or edges (as happens often in biological and social networks). Graph theory has been the most active area in discrete mathematics for the past decades, which has produced a rich and solid corpus covering many aspects, such as graph coloring, topological graph theory, extremal graph theory and, in particular, the theory of random graphs, an extremely active area of current research.
The Erdös-Rényi model of random graphs has been the theoretical framework inspiring the central topics of randomness in networks. Alternative more realistic models, like the preferential attachment networks, inhomogeneous random graphs, random graphs with given degree sequences or geometric random graphs, have been proposed that fit much better properties of real networks such as the power-law for the distribution of the vertex degrees. The extensive research on the Erdös-Rényi model has provided us with powerful tools and insightful questions that are now being applied to more realistic models. A cornerstone example is the emergence of the giant component and the associated phase transition discovered by Erdös and Rényi in their 1960 seminal paper; now the paradigm of the giant component is a central topic in the analysis random networks, with applications to community detection and related topics.
The scientific work packages of the network constitute a highly innovative approach to network analysis, combining solid theoretical models with cutting-edge applications. The RandNET team consists of renowned specialists in several areas, including discrete mathematics, probability theory, computer science, statistics and data science. One of the common goals of the participants in RandNET is to exchange perspectives and knowledge from their different backgrounds in order to build a larger interdisciplinary toolkit for the analysis of large networks, combining techniques from combinatorics, probability, statistics and computer science. The involvement of an industrial partner shapes the development of research with a view towards applications, particularly in the area of learning in networks. All participants are committed to strengthen and enlarge previous collaborations by launching new research projects, and through the exchange of senior and junior researchers, joint participation in workshops and summer schools, and dissemination, training and outreach activities. The dissemination and outreach activities are carefully planned and rely on already existing and successful communication platforms, existing outreach activities, and include a far-reaching goal of producing on-line lectures on network analysis addressed to the scientific community at large.
