Probabilistic and statistical methods for networks

21 August - 1 September 2017
(arrival 20 August, departure 1 September)
Venue: Mathematics Building of TU Berlin

The summer school will focus on probabilistic and statistical methods for networks. This is an enormously rich topic that has many connections between branches of mathematics, and applications to many other scientific disciplines. The central theme is randomness that may arise in various forms: it can be used to construct models for networks, to analyse networks using statistical methods, or as part of stochastic processes on networks. One strand of the school will consider the theory of statistical physics models on networks; another strand will develop tools of statistical inference in network data; and a third strand investigates applications such as networks in neuroscience, traffic and telecommunication.

The School is primarily aimed at graduate students, but also to postdocs, working in applied probability or in one of the application fields with strong probabilistic flavour.

 

Speakers

Shankar Bhamidi (North Carolina) - Probabilistic and statistical problems pertaining to dynamic networks
The last few years have witnessed an explosion in the amount of empirical data on real networks motivating an array of mathematical models for the evolution of such networks. Examples range from biological networks (brain networks of interacting neurons), information transmission (Internet), transportation, social networks and swarm intelligence and the evolution of self-organized behavior through the interactions of simple agents. This has stimulated vigorous activity in a multitude of fields, including biology, statistical physics, statistics, mathematics and computer science to understand these models and quantify their predictions and relevance to real systems.

The aim of this course is to introduce junior researchers to one corner of this vast field, Dynamic networks: systems that evolve over time through probabilistic rules. The following two main themes will be pursued:
  • Emergence of macroscopic connectivity [2 lectures]: The first two lectures will delve into techniques for understanding how macroscopic connectivity in the network arises via microscopic interactions between agents in the network. We will consider various random graph models and study the nature of emergence of the giant component, in particular establishing sufficient conditions for these objects to belong to the same universality class as governed by Aldous's multiplicative coalescent. We will show how these techniques can be used to study not just sizes of maximal components in the critical regime but also show that the maximal components appropriately scaled converge to limiting random metric spaces.
  • Evolving networks and continuous time branching [1 lecture]: The last lecture will study the so-called preferential attachment family of network models emphasizing one particular technical tool: continuous time branching processes. We will show how this technique leads to rigorous asymptotic descriptions to a number of problems in the statistical modeling of real world systems ranging from Twitter event networks to change point detection in evolving networks.
Introductory reading:
"Branching process with Biological applications" by Peter Jagers.

"On the convergence of supercritical general (C-M-J) branching processes" by Nerman.
https://link.springer.com/article/10.1007/BF00534830

"Brownian excursions, critical random graphs and the multiplicative coalescent" by Aldous.
https://projecteuclid.org/euclid.aop/1024404421

Durrett: https://services.math.duke.edu/~rtd/RGD/RGD.pdf

van der Hofstad: http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf

Advanced reading:
"Scaling limits of critical random graphs" by Addario-Berry, Broutin and Goldschmidt.
https://projecteuclid.org/euclid.ejp/1464819810

"Universality for critical random graphs": https://arxiv.org/abs/1411.3417

"Twitter event networks and the Superstar model":
http://projecteuclid.org/euclid.aoap/1438261046

"Spectra of large random trees":
https://link.springer.com/article/10.1007/s10959-011-0360-9

Benedikt Jahnel (Berlin) - Stochastic geometry in telecommunications
 
Continuum percolation is the subfield of probability theory devoted to the analysis of geometric and topological properties of connected components in large random geometric graphs. In particular, it provides a powerful toolkit to address if and how it is possible to transmit information over large distances using device-to-device communications.
In theses lectures, we will introduce basic concepts in continuum stochastic geometry such as percolation, chemical distance or random tessellations and show how these concepts can be used to derive connectivity properties of ad-hoc telecommunication networks with multiple structural components.

Suggested reading materials: R. Meester and R. Roy. Continuum Percolation. Cambridge University Press, Cambridge, 1996.
Max Klimm (Berlin) - Selfish routing in networks
 
Many networks such as road and telecommunication networks are used by a multitude of users that follow their own interests. In these networks it is impossible to globally impose routing strategies that are optimal for the overall network performance. Instead, the state of the system is largely determined by the individual routing strategies of the network users. Such selfish behavior poses a number of important questions that are addressed in this course. First, we will discuss the issue of network stability. Specifically, we will be interested in the questions whether there exist equilibrium points in the network where all users are satisfied with their current routing decisions, and how the users can reach these equilibria. Second, we will examine the degradation of the network performance due to the users’ uncorrdinated behavior by comparing the network performance in a globally optimal state with the network performance in an equilibrium.
Peter Mörters (Bath) - Reinforced branching processes

In this minicourse we look at a class of stochastic processes with reinforcement, which can be described in terms of general branching processes, also known as Crump-Mode-Jagers processes. In the first part of the course I will explain the classical convergence theory of these processes under the assumption of existence of a Malthusian parameter. In the second part I will present recent research (joint with Steffen Dereich and C\'ecile Mailler) exploring the interesting phenomena that may occur if this assumption fails. The theory has applications to urn schemes, networks, and genetic house-of-cards models, some of which will be discussed in the course.
Tiago Peixoto (Bath) - Statistical inference of network structure and dynamics
Networks are shaped by evolutionary mechanisms, and determine the central aspects of how complex systems function. However, differently from systems that are naturally embedded in space, we cannot simply \lq look\rq\ at network in order to extract its most important structural patterns. Instead, we must rely on well-founded algorithmic methods to extract this information from data in an interpretable way. In this lecture, we review a principled approach to this problem based on the elaboration of probabilistic models of network structure, and their statistical inference from empirical data. We aim to cover the following topics:
  • The stochastic block model (SBM) and its variants (degree correction, overlapping groups, etc.)
  • Bayesian inference and model selection: Distinguishing structure from noise.
  • Generalizing from data: Prediction of missing and spurious links.
  • Model extensions: Layered, dynamic SBMs, and generalized models on continuous latent spaces.
  • Fundamental limits of inference: The undetectability transition
  • Efficient inference algorithms.


Suggested reading
T.P.Peixoto, "Bayesian stochastic blockmodeling": http://front.math.ucdavis.edu/1705.10225

Software implementations: https://graph-tool.skewed.de

Specific documentation: https://graph-tool.skewed.de/static/doc/demos/inference/inference.html

Jörg Polzehl / Karsten Tabelow (Berlin) - Connectivity networks in neuroscience - construction and analysis
 
In the lectures we will introduce the basic concepts of anatomical, effective and functional connectivity in neuroscience. We will describe the neuro-imaging experiments providing the imaging data and models used in the statistical analysis of these data, construction of connectivity networks and analysis of their properties.
Klaus Obermayer / Wilhelm Stannat (Berlin) - Stochastic mean-field theories for brain networks

Lenka Zdeborová (Saclay) - Inference on networks via cavity method and message passing

 

Registration and Application

In order to access the application form, you first have to register: Go to Registration.
Then please submit your application via the online submission form.



Application Deadline: 31 May 2017.

Applications for funding have to include

  1. letter of motivation
  2. curriculum vitae
  3. budget plan for travel costs

Each text preferably no longer than one page.

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