The theory of random graphs was founded by Erdős and Rényi in 1959 after Erdős had discovered that the probabilistic method is useful in attacking problems of extremal graph theory. The breakthrough result of Erdős and Rényi was that giant components in random graphs appear suddenly. Their model is by now well understood, and two excellent monographs by Bollobás and by Janson, Łuczak, Ruciński describe the model in detail. Shortly afterwards, Gilbert introduced the random model of the Gilbert disc, nowadays known as random geometric graphs.

Both models are simple, but they do not address all characteristics of complex networks. For example, the famous chain experiment by Milgram showed that between any two people in the world there are at most six degrees of separation. This phenomenon of small diameter is nowadays reflected in the small diameter of online social networks such as Facebook, and none of the classical models reflects this appropriately. Hence, more recently, new random graph models have been designed, such as the Preferential Attachment Model or the Spatial Preferential Attachment Model. These models have also been used to model the spread of viral diseases, thereby giving theoretical insight into the propagation of epidemics, insight into neuron activity in neural networks, and also in sensor networks, in protein-protein interaction networks, to mention a few. The models also allow for a theoretical analysis of algorithms in such networks, such as an analysis of different strategies for forwarding information packets in computer networks. Moreover, in statistical physics, random graphs serve as mathematical models (“mean-field models”) of disordered systems such as glasses that do not have a rigid "ordered" structure. Also, thanks to the availability of more and more data in bigger networks and because of their relevance to a multitude of applications, these models have been the subject of sustained research effort over the past five decades, and interest in the topic from mathematics, computer science, statistical physics, neuroscience and engineering communities has heightened in the last two decades. As a result, there is a growing body of impressive results that give a detailed description of several aspects of these models. There are two brand new monographs on random graphs and complex networks, by Frieze, Karoński, and by Remco van der Hofstad. Also, the theoretic community has been expanding: the bi-annual conference on "Random Structures and Algorithms" that started in 1983 with 30 participants, in 2013 already had more than 200 participants. The school is aimed at PhD and MSc students as well as postdoctoral fellows in mathematics, computer science and physics, working in the broad area of graphs and networks.

The following four instructors will present interdisciplinary flavor of the subject: