# Summer School: Mathematics of Large Networks

### Description

The Mathematics of Large Networks Summer School is part of the Large networks and their limits (2022 Spring) semester. This summer school aims to bring together mathematicians and network scientists to foster the exchange of ideas between these two fields. During the school four minicourses will be given by distinguished researchers in graph theory and network science for students from both fields, who are interested in multidisciplinary approaches to networks. The main topics of the summer school include geometry of networks, dynamics on/of networks, higher order structures, network inference and applications.

**Lecturers:**

**Ginestra Bianconi (Queen Mary University of London)****Remco van der Hofstad (TU Eindhoven)****Renaud Lambiotte (University of Oxford)****Gergely Palla (Eötvös Loránd University)****Kavita Ramanan (Brown University)**

**Mini-courses:**

**Ginestra Bianconi: Higher-order networks: An introduction to simplicial complexes**

Higher-order networks encode interactions between two or more nodes and allow us to describe complex systems going beyond the assumption of having exclusively pairwise interactions.

New results have shown that higher-order networks can sustain higher-order dynamical processes that take advantage of the topology and geometry of the higher-order networks. Most relevantly these processes cannot be accounted by considering only the pairwise interactions of higher-order networks.

For these reasons higher-order networks constitute a very hot topic in Network Science with applications spanning from brain networks to social collaboration networks.

In this course we will cover the main aspects of this exciting new research subject covering:

- the main structural combinatorial, topological and geometrical aspects of higher-order networks;
- important equilibrium (maximum entropy) and non-equilibrium (growing) models for higher-order networks;
- the interplay between higher-order structure and dynamics.

The module will be based on the recent book:

Bianconi, Ginestra. *Higher order network: An Introduction to Simplicial Complexes*. Cambridge University Press, 2021.

**Remco van der Hofstad: Local and global structure of random graphs and complex networks**

Abstract: Empirical findings have shown that many real-world networks share fascinating features. Indeed, many real-world networks are small worlds, in the sense that typical distances are much smaller than the size of the network, and are scale-free, in the sense that there is a high variability in the number of direct connections of the elements in the network.

Spurred by these empirical findings, many models have been proposed for such networks. In this lecture series, we discuss empirical findings of real-world networks, and describe some of the random graph models proposed for them, such as the classical Erdös-Rényi random graph, as well as the more relevant configuration model, generalized random graphs and preferential attachment models.

We discuss local convergence in random graphs, and its relation to branching process approximations in, or the locally tree-like nature of, random graphs. While local convergence is related to the local structure around typical vertices in random graphs, it also has indirect implications for many global quantities such as the giant component and small-world properties of random graphs. For example, we show how the statement that the `giant component is almost local' can be made precise, and how it can be related to the small-world nature of random graphs.

Outline of the lecture series:

Lecture 1: Real-world networks and random graphs

Lecture 2: Local convergence of random graphs

Lecture 3: The giant in random graphs is almost local

This lecture series is based on joint work with, amongst others:

Gerard Hooghiemstra, Shankar Bhamidi,

Júlia Komjáthy, Piet Van Mieghem,

Henri van den Esker, and Dmitri Znamenski.

**Renaud Lambiotte: Modularity and Random Walks on Networks**

In this short tutorial, we will explore the interplays between community structure and Dynamics on Networks. Starting from general principles in dimensionality reduction and, in particular, time-scale separation, we will then investigate how the graph structure affects the spectral properties of the transition matrix. As a next step, we will overview different approaches based on random walks for community detection.

**Gergely Palla: Hyperbolic networks: models, embedding and community structure**

Hyperbolic network models are centred around the idea of placing nodes at random in a metric space with hyperbolic geometry, and connecting node pairs according to a probability that is decreasing as a function of the hyperbolic distance. A fascinating feature of these models is that they inherently reproduce the most important universal properties of real networks: the graphs generated this way are small-world, highly clustered and scale-free. The success of hyperbolic models provides a strong motivation for the inverse problem given by hyperbolic embedding as well, where the task is to find an optimal arrangement of the nodes in the hyperbolic space based on a given input network structure. The resulting layout can be beneficial from various aspects, including greedy routing and link prediction problems. A recently discovered further interesting property of hyperbolic networks is that they can display a very pronounced community structure as well.

During the lectures, we shall discuss the above-mentioned aspects of hyperbolic network models and hyperbolic network geometry. The first part of the lectures will focus on the most well known hyperbolic models such as the popularity-similarity optimisation model and the random hyperbolic graph. This will be followed by an overview of different hyperbolic embedding techniques, together with the quality measures that can be applied on the resulting layouts for comparing different methods on a quantitative base. For closing, the community structure of hyperbolic networks will be discussed.

**Kavita Ramanan: Dynamics on Sparse and Heterogeneous Networks**

Abstract: We will outline some of the challenges that arise in the study of dynamics on sparse and heterogeneous networks and describe recently developed theory that allows one to provide approximations to empirical measure and marginal network dynamics that are provably accurate in a suitable asymptotic regime. We will also provide illustrative examples of insights that can be gained from these approximations, and discuss several open problems.

A limited number of participants will also get a chance to give a short talk about their work.

**Application is now open. Application deadline: 2022 February 1.**

The school covers full lodging for all accepted participants at the CEU Conference and Residence Center (H-1106 Budapest, Kerepesi út 87.). There is very limited funding for travel support. Please indicate in the application form if you wish to apply for this support.

MSc and PhD students applying for the summer school should also ask their supervisor to send a short recommendation (typically a few sentences) to the contact e-mail address *large.networks@renyi.hu*

Link to the registration web-form of the Summer School: