Random Graphs and Complex Networks: Structure and Function

• Keywords: Random Graphs, Complex Networks, Network Statistics, Stochastic Processes in Random Media, Algorithms

The topic of random graphs is at the forefront of applied probability, and it is one of the central topics in multidisciplinary science where mathematical ideas are used to model and understand the real world. At the same time, random graphs pose challenging mathematical problems that have attracted the attention from probabilists and combinatorialists since the 1960, with the pioneering work of Erdös and Rényi. Around the change of the millennium, when data sets started to become available, several applied disciplines started to realize that many real-world networks, even though they are from various different origins, share many fascinating features.

The aim of this course is four-fold. First, we aim to describe the novel models invented since 2000 to describe real-world networks, as well as their topological properties. These models share that they are rather inhomogeneous. Various models were invented to model specific aspects of networks, such as their community structure, clustering, and degree structure. Their topological properties are by now relatively well understood, and we aim to discuss them. Key examples of such properties include their giant component, critical connectivity behavior, graph distances and degree structure. Secondly, we aim to study their local behavior, as described by their local weak limit, and how they can be constructed using random walks and related stochastic processes, particularly in their scaling limits. Thirdly, we aim to discuss network functionality, as described by stochastic processes on them. These processes include the metric of weighted random graphs, and distances in those, as well as the vulnerability of networks, as modeled by percolation on them. Finally, we discuss several statistical problems of networks and their functionality, including community detection problems, the estimation of preferential attachment functions, and estimation of the source of an epidemic.

The school will present basic material in the first week, while the second week will contain more advanced material, at the forefront of science. The aim is that students who have followed the course can read recent articles in the mathematics of networks and place their content in the broad field.

Part of this material comes from the basic random graphs book by van der Hofstad, as well as two follow-up works on more advanced random graphs topics and the Saint-Flour lecture notes in preparation on stochastic processes on random graphs. Aside from this, the course aims to cover some seminal papers on the subject.

Instructors

Lectures:
Remco van der Hofstad
(Department of Mathematics and Computer Science, Eindhoven University of Technology, NL)
Shankar Bhamidi 
(Department of Statistics and Operations Research, University of North Carolina, USA)

Tutorials:
Viktoria Vadon
(Eindhoven University of Technology, NL)
Clara Stegehuis
(Eindhoven University of Technology, NL)

Format

Morning: 3 hours/day lectures
Afternoon: 2 hours/day supervised tutorials as well as individual and team work.

Moreover, there will be a poster session, where participants, upon previous request, may present their research. A welcome cocktail will be offered during the poster session. More detailed info to follow.

Room and board

Accommodation is included in the registration fee.
The students will be hosted at the Guest House of Villa del Grumello and at the Ostello Bello .
The organizing committee will take care of the reservation.
Working days’ lunches are included in the registration fees.

Attendance and final certificate

Full attendance of the activities of the summer school is mandatory for the participants.
An attendance certificate will be awarded by Università Bocconi, subject to a positive participation to the program.