# PhD positions in Probability Theory

The successful candidate will be working on fundamental questions in complex networks. This is a very active field within probability theory, with …

## Arbeidsvoorwaarden

- Standplaats:
- Niels Bohrweg, Leiden, Zuid-Holland
- Dienstverband:
- Tijdelijk contract / Tijdelijke opdracht
- Uren per week:
- 38 uur

- Salarisindicatie:
- € 2191 - € 2801 per maand
- Opleidingsniveau:
- WO

## Functieomschrijving

The successful candidate will be working on fundamental questions in complex networks. This is a very active field within probability theory, with applications ranging from physics and chemistry to computer science and population genetics.

The PhD positions are financed by NETWORKS, a collaborative research program funded by the Dutch science foundation, with partners from Amsterdam, Eindhoven and Leiden. See www.thenetworkcenter.nl. A brief description of each of the 4 PhD-projects is given below.

## Functie-eisen

- A MSc degree;
- A special focus on probability theory;
- An interest in statistical physics is desirable;
- Excellent grades and good reference letters.

## Conditions

We offer a one-year term position with the possibility of renewal based on need, funding and performance. Salary range from €2,191 to €2,801.-gross per month (pay scale P,in accordance with the Collective Labour Agreement for Dutch Universities). There is no fixed starting date for the position.

Leiden University offers an attractive benefits package with additional holiday (8%) and end-of-year bonuses(8.3 %), training and career development and sabbatical leave. Our individual choices model gives you some freedom to assemble your own set of terms and conditions. Candidates from outside the Netherlands may be eligible for a substantial tax break. Additional budget allows for research visits abroad and attendance of international conferences. More at https://www.universiteitleiden.nl/en/working-at/job-application-procedure-and-employment-conditions.

All our PhD students are embedded in the Leiden University Graduate School of Science www.graduateschools.leidenuniv.nl. Our graduate school offers several PhD training courses at three levels: professional courses, skills training and personal effectiveness. In addition, advanced courses to deepen scientific knowledge are offered by the research school.

**Diversity**

Leiden University is strongly committed to diversity within its community and especially welcomes applications from members of underrepresented groups.

## Additional information

Enquiries about the position can be made to Professor Frank den Hollander, Professor in Probability Theory, email denholla@math.leidenuniv.nl.

**Applications **

Applications using the vacancy number and including a full CV, a list of publications, brief past and future research statements, as well as the names and addresses of at least three persons who can be contacted for reference (and have agreed to be contacted), should be uploaded **no later than 15 April 2019** to the website jobs.math.leidenuniv.nl/2019/mathphd/.

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**Project descriptions**

**Project 1: Random walks on dynamic random graphs.***Supervisors:* Luca Avena (Leiden), Frank den Hollander (Leiden), Remco van der Hofstad (Eindhoven).

Search algorithms on networks are important tools for the organisation of large data sets. A key example is Google PageRank, which assigns a weight to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of measuring its relative importance within the set. The weights are assigned via exploration: a page that is linked to by many pages with a high rank receives a high rank itself. Complex networks are modelled as random graphs. Search algorithms are modelled as random walks, moving along the network by randomly picking an edge incident to the vertex currently visited and jumping to the vertex at the other end.

The goal of the project is to analyse the long-time behaviour of different classes of random walks (simple, non-backtracking, with resets) on different classes of sparse random graphs, evolving randomly over time according to different types of edge-rewiring dynamics. Key questions concern the characterisation of mixing times and cover times. Depending on the relative speeds at which the random graph and the random walk evolve, different speeds of mixing are expected.

The project is anchored in probability theory, but is of interest also in computer science for the design of exploration algorithms on complex networks. Mathematical techniques include the theory of Markov processes, coupling methods, combinatorial path-counting arguments, and branching process approximations.

**Project 2: Random forests and network data sets***Supervisors:* Luca Avena (Leiden), Alex Gaudilliere (Marseille)

Many real-world data sets are encoded into weighted graph structures (e.g. migration flows, airline connections, energy networks). There is an increasing interest in designing efficient algorithm to analyse such data. Big data representing non-regular modular structures pose serious computational and conceptual challenges even in simple visualisation procedures, node classifications or clustering methods. For such problems randomised algorithms can be crucial and can outperform deterministic algorithms.

The project aims at investigating random forests and related random-walk based sampling algorithms to probe the architecture of an arbitrary weighted graph. From the fundamental side, the focus is on the study of scaling limits of random spanning forests on weighted graphs representing hierarchical structures, which has connections to the theory of uniform spanning trees and determinantal processes. From the applied side, the focus is on designing a renormalisation algorithmic method to identify, in a multi-scale fashion and with the help of random partitionings induced by spanning forests, densely connected subgroups of nodes in data sets that are encoded into a network. The renormalisation scheme to be developed will be tested on data sets from different areas.

The project combines ideas from probability theory, combinatorics and algorithmics. Familiarity with the basic theory of random walks is required, and with programming languages such as C++ or Python is highly desirable.

**Project 3: Breaking of ensemble equivalence in constrained random graphs***Supervisors:* Diego Garlaschelli (Leiden/Lucca), Frank den Hollander (Leiden), Michel Mandjes (Amsterdam)

Complex networks are often modelled as random graphs subject to certain constraints, e.g. on the number of edges and triangles or on the degree sequence. Statistical physics prescribes what probability distribution on the set of possible graphs should be chosen given a particular type of constraint. Two important choices are the microcanonical ensemble (where the constraints are hard) and the canonical ensemble (where the constraints are soft, i.e., hold as ensemble averages only). For random graphs that are large but finite, the two ensembles are obviously different and, in fact, represent different empirical situations. As the size of the graph gets large, the two ensembles are traditionally assumed to become equivalent, i.e., the soft constraints are expected to behave asymptotically like hard constraints. This assumption of ensemble equivalence is one of the corner stones of statistical physics, but it does not hold in general.

The goal of the project is to classify when breaking of ensemble equivalence occurs and to quantify to what extent it affects the scaling properties of the network. The implication of breaking of ensemble equivalence is that the proper choice of model for describing a real-world network depends on the a priori knowledge that is available about the constraints on the network, so that a principled choice of the ensemble to be used in practical applications is required. Applications are pattern detection, community detection and network reconstruction. Both spare and dense networks are of interest. It is also interesting to investigate what happens when the networks is dynamic.

The project lies at the interface between probability theory, combinatorics and statistical physics. Mathematical tools come from information theory, large deviation theory and the theory of graphons.

**Project 4: Reaction-diffusion processes on random graphs ***Supervisors:* Frank den Hollander (Leiden), Wolfgang Konig (Berlin), Renato dos Santos (Shanghai)

The Parabolic Anderson Model describes is a heat equation with a random potential. It describes the diffusion of heat or particles in the presence of sources and sinks. So far the PAM has been studied mostly on lattices and on Euclidean space, and mostly for random potentials that have a very short correlation length. The main interest is in the long-time asymptotics of the total mass of the solution to the PAM equation and the question where the bulk of the total mass is mainly concentrated, i.e., where is it located and how large is it as a function of time?

A key quantity of interest is the so-called Lyapunov exponent, the logarithmic growth rate of the total mass. Here one may distinguish between the annealed setting (i.e., taking the expectation with respect to the random potential) and the quenched setting (i.e., almost surely with respect to the random potential). The main contribution to the total mass typically comes from a single `intermittent island’ in space that is far away from the location of the initial mass. On this island the potential typically has a certain characteristic shape.

The above picture has been verified in a variety of different settings. The goal of the project is to study the PAM on sparse random graphs, which have the property that they are `locally tree-like’, i.e., on a small scale look like a random tree. The target is to understand what the intermittent island looks like and how its size and shape can be characterised.

The project relies on probability theory, functional analysis and spectral theory. Mathematical techniques are rooted in large deviation theory and variational calculus.

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