Groningen Stochastics Seminar

If you are interested in giving a talk, please send an email with the title and abstract to a member of the Probability and Statistics Group.

Future Talks

Past Talks

Jun 11, 2021 16:00-17:00 | Eric Cator
(Radboud University) | Contact process on low-rank graphs |
---|---|---|

May 28, 2021 16:00-17:00 | Piet Lammers
(IHES Paris) | Height function delocalisation on cubic planar graphs |

May 21, 2021 16:00-17:00 | Serte Donderwinkel
(University of Oxford) | What do the strongly connected components of a large directed graph with i.i.d. degree tuples look like? |

May 14, 2021 16:00-17:00 | Balázs Ráth (Budapest University of Technology) | Percolation of worms |

Apr 30, 2021 16:00-17:00 | Jun Yu
(Umeå University) | Statistical learning and inference for spatiotemporal data |

Apr 16, 2021 16:00-17:00 | Xavier Pérez Giménez
(University of Nebraska-Lincoln) | Perfect matchings in the random bipartite geometric graph |

Apr 9, 2021 16:00-17:00 | Aernout van Enter
(University of Groningen) | Dyson models with random boundary conditions |

Mar 19, 2021 13:15-14:15 | Guillem Perarnau (Universitat Politècnica de Catalunya) | Extremal stationary values for random digraphs |

Feb 25, 2021 17:00-18:00 | Tobias Müller
(University of Groningen) | Percolation on hyperbolic Poisson-Voronoi tessellations |

Feb 23, 2021 13:15-14:15 | Christian Hirsch
(University of Groningen) | Maximum likelihood estimation in stochastic channel models |

Feb 12, 2021 16:00-17:00 | Victor Bernal
(University of Groningen) | Improved network reconstruction with shrinkage-based Gaussian graphical models |

Jan 22, 2021 16:00-17:00 | Luca del Core
(University of Groningen) | Stochastic modelling of COVID-19 spread in Italy |

Jan 8, 2021 16:00-17:00 | Abdul Salam
(University of Groningen) | Non Homogeneous Dynamic Bayesian Network Models |

Dec 11, 2020 16:00-17:00 | Rangel Baldasso
(University of Leiden) | Sharp thresholds for Glauber dynamics percolation |

Nov 27, 2020 16:00-17:00 | Deepan Basu
(University of Groningen) | A class of stick breaking models |

Oct 23, 2020 16:00-17:00 | Jo Ellis-Monaghan
(University of Amsterdam) | Graph theoretical design strategies for tile-based DNA self-assembly |

Oct 09, 2020 16:00-17:00 | Daniel Valesin
(University of Groningen) | The contact process on random hyperbolic graphs |

Sep 25, 2020 16:00-17:00 | Markus Heydenreich
(LMU Munich) | The weight-dependent random connection model |

Sep 18, 2020 16:00-17:00 | Daniel Willhalm
(University of Groningen) | Large deviation estimates for subsets of size conditioned Galton-Watson trees |

Sep 04, 2020 16:00-17:00 | Christian Hirsch
(University of Groningen) | Functional CLT for persistent Betti numbers on point processes and networks |

Apr 20, 2020 14:00-15:00 | Leandro Chiarini
(IMPA/Utrecht) | TBA |

Apr 02, 2020 16:00-17:00 | Johannes Krebs
(TU Braunschweig) | On the law of the iterated logarithm and strong invariance principles in stochastic geometry |

January 23, 2020 Room 5161.0293 (Nijenborgh 4) 11:00-12:00 | Benjamin Hansen
(University of Groningen) | Voronoï percolation in the hyperbolic plane |

September 25, 2019 16:00-17:00 | Bernardo N. B. de Lima
(Federal University of Minas Gerais) | The Constrained-degree percolation model |

June 18-20, 2019 | Miloš Stojaković
(Novi Sad) | Mini Course: Introduction to Positional Games |

June 5, 2019 Room 5161.0267 11:00 - 12:00 | Fiona Skerman
(Masaryk) | Community structure in Networks - limits of learning |

March 21, 2019 Room 5161.0041B 14:00 - 15:00 | Guilherme Reis
(Bahia-UFBA) | Interacting diffusions on sparse random graphs |

February 28, 2019 Room 5161.0289 14:00 - 15:00 | Stein Andreas Bethuelsen
(Stavanger) | On projections of the supercritical contact process: uniform mixing and cutoff phenomenon |

December 12, 2018 Room 5172.0804 15:00 - 16:00 | Luca Avena
(Leiden) | Loop-erased partitioning: two-point correlations on mean-field models |

November 21, 2018 Room 5173.0176 (Linnaeusborg) 15:00 - 16:00 | Guus Regts
(Amsterdam) | On the location of zeros of the independence polynomial for bounded degree graphs |

November 7, 2018 Room 5161.0289 14:00 - 15:00 | Alessandra Cipriani
(Delft) | The discrete Gaussian free field on a compact manifold |

October 24, 2018 Room 5111.0006 15:00-16:00 | Clara Stegehuis
(Eindhoven) | Optimizing constrained and unconstrained network structures |

September 19, 2018 Room 5173.0045 (Linnaeusborg) 15:30 - 16:30 | Dieter Mitsche
(Nice) | k-regular subgraphs near the k-core threshold of a random graph |

June 20, 2018 Room 5161.0222 11:00 - 12:00 | Bernardo N. B. de Lima
(Federal University of Minas Gerais) | Truncated long-range percolation on oriented graphs |

June 18, 2018 Room 5161.0165 15:00 - 16:00 | Moritz Schauer
(Leiden) | A generator approach to stochastic monotonicity and propagation of order |

April 23, 2018 Room 5161.0293 16:00 - 17:00 | Evgeny Verbitskiy
(Leiden) | Solvable models from the ergodic point of view |

March 15, 2018 Room 5115.0020 11:00-12:00 | Rodrigo Ribeiro
(IMPA) | Building your path to escape from home |

February 16, 2018 Room 5161.0222 15:00-16:00 | Renato Santos
(Weierstrass Institute) | Random walk on random walks |

February 1, 2018 Room 5161.0293 16:00 - 17:00 | Conrado Costa
(Leiden) | Law of large numbers and LDP for the Random Walk in the Cooling random Environment |

January 24, 2018 Room 5161.0222 16:00 - 17:00 | Daniel Valesin
(Groningen) | Monotonicity for multi-range percolation on oriented trees |

Abstracts

**Title:** Contact process on low-rank graphs

**Speaker:** Eric Cator

**Abstract:**
In this talk I will introduce the contact process for general infection matrices (which is slightly more general than graphs) as a model for the spread of disease or information. Since in general the contact process is hard to analyse, many ways of approximating the process have been studied. An important example of such an approximation is a type of mean field approach. I will show that this corresponds to approximating the infection matrix with a rank 1 matrix. We will extend this to higher finite rank approximations, and we will give an exact description of the behaviour of the contact process on such a finite rank matrix, as the number of nodes goes to infinity. In particular, we will be able to describe the meta-stable state of the contact process as a Gaussian process with known covariance structure.

**Title:** Height function delocalisation on cubic planar graphs

**Speaker:** Piet Lammers

**Abstract:**
We consider models of integer-valued height functions on shift-invariant planar graphs whose maximum degree is three. We prove delocalisation for a large class of such models. This provides a simplified proof of the Kosterlitz-Thouless phase transition, and is, to the knowledge of the author, the first height function delocalisation proof which is not tied to a specific model. Included are: the discrete Gaussian and solid-on-solid models, as well as the uniformly random K-Lipschitz function, on the vertices of the hexagonal lattice. This talk is based on arXiv:2012.09687.

**Title:** What do the strongly connected components of a large directed graph with i.i.d. degree tuples look like?

**Speaker:** Serte Donderwinkel

**Abstract:**
Universal scaling limits of random graph models can be considered the Central Limit Theorems of graph theory and have attracted a lot of interest in recent years. We consider the strongly connected components (SCC) of a uniform directed graph on $n$ vertices with i.i.d. degree tuples, under suitable moment conditions on the degree distribution. The moment conditions ensure that the graph is critical: the largest SCC contains $O(n^{1/3})$ vertices, and there are many other SCC of that order. We show that the sequence of SCC ordered by decreasing length converges under rescaling of the edge lengths to a random sequence of directed multigraphs with edge lengths that are all 3-regular or loops. The talk is based on joint work with Zheneng Xie.

**Title:** Percolation of worms

**Speaker:** Balázs Ráth

**Abstract:**
We consider the following correlated percolation model on the d dimensional lattice $Z^d, d \geq 5$. From each vertex of $Z^d$, we start POI$(v)$ number of worms $(v>0)$, where the number of worms are i.i.d. for different vertices. Each worm $w$ has a random length $L_w$, moreover the lengths are i.i.d. for different worms. Each worm $w$ (independently of other worms) performs a simple symmetric random walk of length $L_w-1$ on $Z^d$. Let us consider the set $S^v$ of sites visited by these worms. How do the percolation properties of $S^v$ depend on the length distribution of worms?

We give an easy sufficient condition under which $S^v$ exhibits percolation phase transition and a more involved sufficient condition under which $S^v$ percolates for all $v>0$. We compare our model to other known models (e.g., finitary random interlacements, Poisson-Boolean percolation model) and argue that the percolative behaviour of our model is quite close to being "extremal" in a natural family of similar models. Joint work with Sándor Rokob.

**Title:** Statistical learning and inference for spatiotemporal data

**Speaker:** Jun Yu

**Abstract:**
In this presentation, I will talk about what we as a statistician can do for our society and how we can help to find solutions for real-life problems, by a number of examples from statistical learning and inference for spatiotemporal data, coming from real applications.

**Title:** Perfect matchings in the random bipartite geometric graph

**Speaker:** Xavier Pérez Giménez

**Abstract:**
We consider the standard random bipartite geometric graph process in which n red vertices and n blue vertices are placed at random on the unit d-dimensional cube and edges are added sequentially, between vertices of different colors, in increasing order of edge-length. A natural question is to ask whether the first edge in the process that results in the minimum degree being at least one coincides, with high probability, with the first edge that creates a perfect matching. While this was already known to be false when d=2, as the thresholds are not even of the same order, we are able to positively answer it for d at least 3. This is joint work with Abigail Raz.

**Title:** Dyson models with random boundary conditions

**Speaker:** Aernout van Enter

**Abstract:**
I discuss the low-temperature behaviour of Dyson models (long-range Ising models in one dimension) in the presence of random boundary conditions. As for typical random (i.i.d.) boundary conditions Chaotic Size Dependence occurs, that is, the pointwise thermodynamic limit of the finite-volume Gibbs states for increasing volumes does not exist, but the sequence of states moves between various possible limit points, as a consequence it makes sense to study distributional limits, the so-called metastates which are measures on the possible limiting Gibbs measures. The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2. Joint work with Eric Endo and Arnaud Le Ny.

**Title:** Extremal stationary values for random digraphs

**Speaker:** Guillem Perarnau

**Abstract:**
In this talk, we will discuss the extremal values of the stationary distribution of a random walk on a directed random graph with given degrees. While for undirected graphs the stationary distribution is simply determined by the degrees, the graph geometry plays a major role in the directed case. Understanding typical stationary values is key to determine the mixing time of the walk. However, typical results provide no information on extremal values, which are important for many applications. In this talk, we will show that the minimum value may be polynomially smaller than the average value, and that the correct exponents comes from the combination of two factors: (1) the contribution of atypically thin in-neighborhoods, controlled by subcritical branching processes; and (2) the contribution of atypically “light” trajectories, controlled by large deviation rate functions. This result has direct consequences for the understanding of random walks in random digraphs. In particular, we will show how one can obtain an estimate on the cover time of the random digraph. This is joint work with Xing Shi Cai.

**Title:** Percolation on hyperbolic Poisson-Voronoi tessellations

**Speaker:** Tobias Müller

**Abstract:**
I will discuss percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. That is, we colour each cell of the hyperbolic Poisson-Voronoitessellation black with probability and white with probability 1−p, independently of the colours of all other cells. We say that percolation occurs if there is an infinite connected cluster of black cells. I will sketch joint work with the doctoral candidate Ben Hansen that resolves a conjecture and an open question, posed by Benjamini and Schramm about twenty years ago, on the behaviour of the“critical probability for percolation” as a function of the intensity parameter of the underlying Poisson process. (Unlike in Euclidean Poisson-Voronoi percolation, this critical value depends on the intensity of the Poisson process.) Based on joint work with Benjamin Hansen.

**Title:** Maximum likelihood estimation in stochastic channel models

**Speaker:** Christian Hirsch

**Abstract:**
We propose Monte Carlo maximum likelihood estimation as a novel approach in the context of calibration and selection of stochastic channel models. First, considering a Turin channel model with inhomogeneous arrival rate as a prototypical example, we explain how the general statistical methodology is adapted and refined for the specific requirements and challenges of stochastic multipath channel models. Then, we illustrate the advantages and pitfalls of the method on the basis of simulated data. Finally, we apply our calibration method to wideband signal data from indoor channels. Based on joint work with Ayush Bharti, Troels Pedersen, Rasmus Waagepetersen.

**Title:** Improved network reconstruction with shrinkage-based Gaussian graphical models

**Speaker:** Victor Bernal

**Abstract:**
Gaussian graphical models (GGMs) are undirected network models where the nodes represent the random variables and the edges their partial correlations. GGMs are straightforward to implement, easy to interpret, and have an acceptable computational cost. These advantages have made GGMs popular to reconstruct gene regulatory networks from high throughput data. In this talk, I will discuss the reconstruction of GGMs in the high dimensional case (n « p). In this scenario, the inference becomes challenging and requires regularization/shrinkage. I will present a novel approach to infer GGMs structures based on the Ledoit-Wolf (LW) shrinkage, which accounts for the sample size, the number of variables, and the shrinkage value.

**Title:** Stochastic modelling of COVID-19 spread in Italy

**Speaker:** Luca del Core

**Abstract:**
Italy was particularly hard hit during COVID-19 pandemic and the aim of this work is to describe the dynamics of infections within each region and the geographical spread of the virus over time. To this end we extend the standard SIRD framework by introducing pairwise interaction terms between the region-specific infected compartments. That is, we check how the increments of infected people of one region depends on that of another region. That information could be used as a proxy to measure how the epidemics spread geographically. Furthermore, we make all the transition parameters dependent on time so as to capture potential changes in the epidemic spread due to the effect of external factors such as containment measures. We also include the information of the number of tests performed as predictor for the infection parameters. We then use the master equation as probabilistic model to describe the evolution of the temporal process of the states and we focus on its first two-order moments by considering a local linear approximation. Finally we infer the corresponding parameters using a step-wise optimisation procedure which leads to the maximum likelihood estimate.

**Title:** Non Homogeneous Dynamic Bayesian Network Models

**Speaker:** Abdul Salam

**Abstract:**
One of the main objectives of systems biology is to learn network topologies from gene expression time series. The ultimate goal is to infer the dependency between random variables in the form of a network. To deal with this task, the class of dynamic Bayesian network (DBNs) models have been widely applied. The essential assumption for DBNs is that the regulatory processes are homogenous, so that the interaction parameters are constant over time. However, this assumption is too restrictive for many real-world applications. To overcome this issue, non-homogenous dynamic Bayesian networks (NH-DBNs) are considered as the best alternatives. In this work, we applied the class of NH-DBNs models to infer the network structure from gene expression time series. Particularly, we will discuss the results for two types of gene networks, firstly, the yeast network (true structure of the network is known) and secondly, mTOR network (true structure of this network is unknown). In the presentation we will put the main focus on the mTOR network, as we currently struggle with the data analysis. All our network models are based on the assumption that measurements have been taken at equidistant time points. But this assumption is not fulfilled for the mTOR data, where after a stimulation of the system the measurements were taken after 0, 1 ,3, 5, 10, 15, 30, 45, 60 and 120 minutes.

**Title:** Sharp thresholds for Glauber dynamics percolation

**Speaker:** Rangel Baldasso

**Abstract:**
Sharp threshold is a phenomena that characterizes abrupt change of behavior in a phase transition. In this talk, we prove that the two-dimensional model obtained by running Glauber dynamics up to some finite time yields a percolation model that presents a sharp threshold. The proof is based on special properties of a graphical construction of the model that allows us to pass from annealed to quenched measures. Based on a joint work with C. Alves, G. Amir, and A. Teixeira.

**Title:** A class of stick breaking models

**Speaker:** Deepan Basu

**Abstract:**
(Joint work with Krishanu Maulik and Subhabrata Sen). We start with a white stick of unit length and for a uniformly chosen point, we color the left or right side black with certain probability functions of the distance of the chosen point from one end. We keep iterating this on the white part of the stick, and attempt at characterizing the distribution of the location of ’vanishing white point’ of the stick by the aforementioned function.

**Title:** Graph theoretical design strategies for tile-based DNA self-assembly

**Speaker:** Jo Ellis-Monaghan

**Abstract:**
Recent advances in DNA self-assembly have resulted in increasingly sophisticated nanoscale wire-frame structures that can be modeled by graphs. These constructs serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis. One construction method uses k-armed branched junction molecules, called tiles, whose arms are double strands of DNA with one strand extending beyond the other, forming a ‘sticky end’ at the end of the arm that can bond to any other sticky end with complementary Watson-Crick bases. A vertex of degree k in the target graph is formed from a k-armed tile, and joined sticky ends form the edges. We use graph theory to determine optimal design strategies for scientists producing these nanostructures. This includes computational complexity results, algorithms, and pragmatic approaches for special classes of graphs, as well as new mathematical theory and new graphs invariants.

**Title:** The contact process on random hyperbolic graphs

**Speaker:** Daniel Valesin

**Abstract:**
We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with finite mean and infinite second moment. We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on the power law exponent and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of the graph goes to infinity. Joint work with Amitai Linker, Dieter Mitsche and Bruno Schapira.

**Title:** The weight-dependent random connection model

**Speaker:** Markus Heydenreich

**Abstract:**
We investigate a large class of random graphs on the points of a Poisson process in d-dimensional space, which combine scalefree degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. This generalises many spatial random graph models. Our focus is on the question whether infinite components are recurrent or transient, and we demonstrate that the answer depends on the model parameters. In a plain version of the random connection model, where weights are ignored, we can even analyse the model at the phase transition point. Indeed, we obtain an infrared bound for the critical connectivity function if the dimension is sufficiently large or if the pair connection function has sufficiently slow decay. This is achieved through an adaptation of the percolation lace expansion for Poisson processes. Based on joint work with Peter Gracar, Remco van der Hofstad, Günter Last, Kilian Matzke, Christian Mönch, and Peter Mörters.

**Title:** Large deviation estimates for subsets of size conditioned Galton-Watson trees

**Speaker:** Daniel Willhalm

**Abstract:**
We prove a Large Deviation Principle for the logarithmic number of subtrees of size conditioned Galton-Watson trees, provided that two assumptions about Large Deviation Principles being transferable are fulfilled. The proof is based on general large deviation techniques that are linked to conditioned Galton-Watson trees via so-called additive functionals.

**Title:** Functional CLT for persistent Betti numbers on point processes and networks

**Speaker:** Christian Hirsch

**Abstract:**
Persistent Betti numbers form a key tool in topological data analysis as they track the appearance and disappearance of topological features in a sample. In this talk, we derive a goodness of fit test of point patterns and random networks based on the persistence diagram in large volumes. On the conceptual side, the tests rely on functional central limit theorems for the sub-level filtration in cylindrical networks and for bounded-size features of the Čech-complex of planar point patterns. The proof is based on methods from a recently developed framework for CLTs on point processes with fast decay of correlations. We analyze the power of tests derived from this statistic on simulated point patterns and apply the tests to a point pattern from an application context in neuroscience.

**Title:** On the law of the iterated logarithm and strong invariance principles in stochastic geometry

**Speaker:** Johannes Krebs

**Abstract:**
The law of the iterated logarithm (LIL) dates back to the findings of Khinchin (1933) and Kolmogorov (1929). The LIL operates in between the law of large numbers and the central limit theorem and describes the order of fluctuations of a random walk, resp. an empirical process. Today's formulation is due to Hartman–Wintner (1940): Let $X_1,X_2,\dots$ be iid random variables with mean zero and unit variance, then
\[
\limsup_{n\to\infty} \frac{\sum_{k=1}^n X_k}{\sqrt{2n \log\log n} } = 1 \quad a.s.
\]
In this talk we study the LIL and related strong invariance principles (SIP) in stochastic geometry for certain functionals $H$. On the domain $W_n = [-n^{1/d}/2, n^{1/d}/2]^d$ we observe a homogeneous Poisson or binomial process $X_n$. The LIL takes then the form
\[
\limsup_{n\to\infty} \frac{ H(X_n) - \mathbb{E}[H(X_n) ] }{\sqrt{2n \log\log n} } = \sigma \quad a.s.
\]
where $\sigma^2>0$ is the limiting variance of $n^{-1/2} H(X_n)$.
Strong invariance principles, describe the approximability of an empirical process by a Brownian motion. Using the Skorokhod embedding, we show that in the case of a one-dimensional stretched domain $H(X_n) - \mathbb{E}[H(X_n) ] $ can be approximated with a rescaled Brownian motion up to an accuracy of order $O_{a.s.}(n^{1/4} (\log n)^{1/2} (\log\log n)^{1/4} )$ in a Poisson setting. A similar result is valid in a binomial setting.
As potential applications, we think of well-known functionals defined on the k-nearest neighbors graph and important functionals in topological data analysis such as the Euler characteristic and persistent Betti numbers.

**Title:** Voronoï percolation in the hyperbolic plane

**Speaker:** Benjamin Hansen

**Abstract:**
We consider site percolation on Voronoï cells generated by a Poisson point process on the hyperbolic plane H^{2}. Each cell is coloured black independently with probability p, otherwise the cell is coloured white. Benjamini and Schramm proved the existence of three phases: For p ∈ [0, p_{c}] all black clusters are bounded and there is a unique infinite white cluster. For p ∈ (p_{c}, p_{u}), there are infinitely many unbounded black and white clusters. For p ∈ [p_{u}, 1] there is a unique infinite black cluster and all white clusters are bounded. They also showed p_{u} = 1 − p_{c}. The critical values p_{c} and p_{u} depend on the intensity of the Poisson point process. We prove that p_{c} tends to 1/2 as the intensity tends to infinity. This confirms a conjecture of Benjamini and Schramm.

Joint work with Tobias Müller.

**Title:** The Constrained-degree percolation model

**Speaker:** Bernardo N. B. de Lima

**Abstract:**
In the Constrained-degree percolation model on a graph $(V,E)$ there are a sequence, $(U_e)_{e\in\E}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open at time $U_e$, it succeeds if both its end-vertices would have degrees at most $k-1$. We prove a phase transition theorem for this model on the square lattice $\mathbb{L}^2$, as well on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase. Joint work with R. Sanchis, D. dos Santos, V. Sidoravicius and R. Teodoro.

**Title:** Mini Course: Introduction to Positional Games

**Speaker:** Miloš Stojaković

**Locations and Times:**

Tue 18, 10.00-13.00, Room 5161.0222 (Bernoulliborg)

Wed 19, 13.00-16.00, Room 5161.0293 (Bernoulliborg)

Thu 20, 14.00-17.00, Room 5161.0151 (Bernoulliborg)

**Abstract:**
Positional Game Theory provides a solid mathematical footing
for a variety of two-player games of perfect information, usually
played on discrete objects (like graphs), with a number of
applications in other branches of mathematics and computer science.
The field is just a few decades old, and it has experienced a
considerable growth in recent years. Our goal is to introduce some
basic concepts and notions, followed by recent results and numerous
open problems.

The prerequisites include undergraduate knowledge of discrete
mathematics and probability, and the lectures could be of interest to
people with a wide range of backgrounds and different levels of
seniority.

**Title:** Community structure in Networks - limits of learning

**Speaker:** Fiona Skerman

**Abstract:**
Modularity is at the heart of the most popular algorithms for community detection. For a given network G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in G. The (max) modularity q*(G) of the network G is defined to be the maximum over all vertex partitions of the modularity score, and satisfies 0 ≤ q*(G) ≤ 1.

1. "When is there Modularity from fluctuations in Random Graphs?" Guimera et al 2004 showed that the Erdös-Rényi random graph Gn,p with n vertices and edge-probability p can have surprisingly high modularity and that this should be taken into account when determining statistical significance of community structures in networks. Further, they predicted Gn,p to have modularity order (np)-1/3.

Our two key findings are that the modularity is 1 + o(1) with high probability (whp) for np up to 1 + o(1) and no further; and when np ≥ 1 and p is bounded below 1, it has order (np)−1/2 whp, in accord with a competing prediction by Reichardt and Bornholdt in 2006 based on spin glass methods.

2. Sampling sufficiency for determining modularity. We analyse when community structure of the underlying graph can be determined from observations of the network which we model by random sampling of the graph.

Joint work with Colin McDiarmid and based on the paper 'Modularity of Erdös-Rényi random graphs' arxiv.org/abs/1808.02243.

**Title:** Interacting diffusions on sparse random graphs

**Speaker:** Guilherme Reis

**Abstract:**
We consider systems of diffusion processes whose interactions are described by a graph. For example, traditional mean-field interacting diffusions correspond to a complete interaction graph. In recent years some effort has been directed to understanding more general interactions. When the interaction graph is random, in the particular case of the Erdős–Rényi random graph, we show how the behavior of this particle system changes whether the mean degree of the Erdős–Rényi graph diverges to infinity or converges to a constant. In this talk we will focus on the case that the mean degree converges to a constant. In this case we prove a strong law of large numbers for the empirical measures. To achieve this we exploit the local weak convergence and a locality property of this system. Loosely speaking, the locality property states that information does not propagate too fast over the graph for this kind of particle system. This tool allow us to manage the dependencies of the system. Joint work with Roberto Imbuzeiro Oliveira.

**Title:** On projections of the supercritical contact process: uniform mixing and cutoff phenomenon

**Speaker:** Stein Andreas Bethuelsen

**Abstract:**
In this talk I will present the findings in https://arxiv.org/abs/1810.06840. This paper concerns the supercritical contact process on general graphs and the study of the asymptotic properties of its projections onto finite subsets. We will see that, under minimal assumptions on the graph structure, such projections have a surprisingly strong temporal loss-of-memory property as soon as the infection rate is large enough. Furthermore, I will discuss certain relations between these results and similar findings for the projection of the 2d Ising model onto a line, as studied in https://arxiv.org/abs/1802.02059 (joint work with Diana Conache (TUM)). When time permits, I will also discuss a particular motivation for these two papers stemming from random walks on (dynamic) random environments.

**Title:** Loop-erased partitioning: two-point correlations on mean-field models

**Speaker:** Luca Avena

**Abstract:**
We consider a certain random partitioning of the vertex set of an arbitrary graph that can be efficiently sampled using loop-erased random walks with killing. The related random blocks tend to cluster nodes visited by the random walk-with infinitesimal generator the graph Laplacian- on time scale 1/q, with q > 0 being a tuning parameter. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, we define an interaction potential between pair of vertices of a graph, as the probability that they do not belong to the same block of the random partition. This interaction potential can be seen as an affinity measure for “densely connected nodes” and capture well-separated communities in network models presenting non-homogenous landscapes. In this spirit, we compute the potential and its scaling limits on a complete graph and on a non-homogenous weighted version with community structures. For the latter we show a phase-transition for detectability as a function of the tuning parameter and the edge weights.

Joint work with ALEXANDRE GAUDILLIEĚRE, PAOLO MILANESI (Marselle) AND MATTEO QUATTROPANI (Rome)

**Title:** Local weak convergence for PageRank

**Speaker:** Nelly Litvak

**Abstract:**
PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called `power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees. Up to date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, in this talk we do not focus on one random graph model but investigate the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. The local weak convergence is one of recent notions of graph limits, originally developed for undirected graphs. We will extend this notion to directed graphs and use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. This may help to identify general network structures in which the power-law hypothesis holds. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as preferential attachment models.

This is a joint work with Alessandro Garavaglia (TU/e) and Remco van der Hofstad (TU/e)

**Title:** On the location of zeros of the independence polynomial for bounded degree graphs

**Speaker:** Guus Regts

**Abstract:**
In this talk I will introduce the independence polynomial (a.k.a. the partition function of the hard-core model in statistical physics) and motivate the study of the location of its zeros by applications to statistical physics and theoretical computer science.

Along the way I will indicate how the theory of complex dynamical systems has been used to settle a conjecture of Alan Sokal on the location of the zeros of the independence polynomial for bounded degree graphs. I will end with some open problems.

**Title:** The discrete Gaussian free field on a compact manifold

**Speaker:** Alessandra Cipriani

**Abstract:**
In this talk we aim at defining the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a suitable random graph that replaces the square lattice Z^d in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field. Joint work with Bart van Ginkel (TU Delft).

**Title:** Optimizing constrained and unconstrained network structures

**Speaker:** Clara Stegehuis

**Abstract:**
Subgraphs contain important information about network structures and their functions. We investigate the presence of subgraphs in inhomogeneous random graphs with infinite-variance degrees. We introduce an optimization problem which identifies the dominant structure of any given subgraph. The unique optimizer describes the degrees of the vertices that together span the most likely subgraph and allows us count and characterize all subgraphs.

We then show that this optimization problem easily extends to other network structures, such as clustering, which expresses the probability that two neighbors of a vertex are connected. The optimization problem is able to find the behavior of clustering in a wide class of random graph models.

**Title:** k-regular subgraphs near the k-core threshold of a random graph

**Speaker:** Dieter Mitsche

**Abstract:**
We prove that $G_{n,p}$ whp has a k-regular subgraph if $c$ is at least $e^{-\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least k; i.e. an non-empty k-core. In particular, this pins down the threshold for the appearance of a k-regular subgraph to a window of size $e^{-\Theta(k)}$.

Joint work with Mike Molloy and Pawel Pralat.

**Title:** Truncated long-range percolation on oriented graphs

**Speaker:** Bernardo N. B. de Lima

**Abstract:**
We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process.

Joint work with A. Alves, A. van Enter, M. Hilário and D. Valesin

**Title:** A generator approach to stochastic monotonicity and propagation of order

**Speaker:** Moritz Schauer

**Abstract:**
We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying Φ f ≥ 0 for some linear operator Φ.

We introduce a new functional analytic technique based on the generator A of a Markov process and its resolvent. We show that the existence of an operator B with positive resolvent such that Φ A - B Φ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order that can be applied in a wide range of settings including various orders for diffusion processes with or without boundary conditions and orders for discrete interacting particle systems.

Joint work with Richard Kraaij

**Title:** Solvable models from the ergodic point of view

**Speaker:** Evgeny Verbitskiy

**Abstract:** In this review talk, I will discuss a link between solvable models of statistical mechanics (dimers, spanning trees, sandpiles) and algebraic dynamical systems. Even though the question about the existence of such a link was raised almost two decades ago, this problem remained largely inaccessible. The development of the theory of symbolic covers of algebraic dynamical systems has only recently provided a suitable framework.

**Title:** Building your path to escape from home

**Speaker:** Rodrigo Ribeiro

**Abstract:** We consider a random walker in a dynamic random environment
given by a system of independent simple symmetric random walks.
We will describe some perturbative results that can be obtained via multi-scale analysis,
including regimes of high density, low density and large drift on particles.
Based on joint works with Oriane Blondel, Marcelo Hilário, Frank den Hollander,
Vladas Sidoravicius and Augusto Teixeira.

**Title:** Random walk on random walks

**Speaker:** Renato Santos

**Abstract:** In this talk we introduce one of the simplest conceivable model of a random walk that can modify its domain. The model works as follows: before every walker step, with probability p a new leaf is added to the vertex currently occupied by the walker. We discuss questions related to the walker such as the speed it moves away from its initial position and how its neighborhood may looks like asymptotically. This is a joint work with D. Figueiredo, G. Iacobellu, R. Olivera and B. Reed.

**Title:** Law of large numbers and LDP for the Random Walk in the Cooling random Environment

**Speaker:** Conrado Costa

**Abstract:** In this talk we discuss 2 main new results: the Strong Law of Large Numbers and a quenched Large deviation principle for the Random Walk in the Cooling random Environment introduced by Avena, den Hollander in 2016. For the proof of these two results, we present two auxiliary results. The first, an ergodic theorem for cooling random variables and the second, a concentration result for the cumulant generating function of the static Random Walk in the random environment.

**Title:** Monotonicity for multi-range percolation on oriented trees

**Speaker:** Daniel Valesin

**Abstract:** Percolation theory contains a variety of monotonicity-related conjectures which are easy and natural to state but challenging to prove. In this talk, we will survey a few of these problems. We will then specialize to the setting of Bernoulli percolation on multi-range oriented trees. The trees we consider are oriented regular trees where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there is percolation. We also show that this curve decreases with respect to the length of the long bonds. Joint work with Bernardo N. B. de Lima and Leonardo T. Rolla.