## 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**

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

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.

- 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

### 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**

- letter of motivation
- curriculum vitae
- budget plan for travel costs

Each text preferably no longer than one page.

Please direct scientific questions about the school to the organisers:

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