# Annual DYNASNET Meeting

### Description

This is an all DYNASNET meeting where each site and each group present some of their achievements.

### Annual DYNASNET Meeting 2022

October 17 – 21, 2022

**Monday, October 17, 2022
Arrival**

**Tuesday, October 18, 2022
CEU**

**Márton Pósfai:**

Meta-graphs of model and real physical networks

Meta-graphs of model and real physical networks

Physical networks -- networks with nodes and links that are physical objects embedded in space -- are shaped by constraints such as volume exclusion, the cost of building and maintaining nodes and links, and local assembly. First, I discuss linear physical networks (LPNs), a minimal model that allows us to explore the co-evolution of physical and network structure. I introduce the meta-graph, an auxiliary graph capturing physical conflict between physical links in the network, which allows us to analytically estimate various properties of the random LPNs. In the second half of the talk, we turn our attention to real physical networks obtained from experiments. After discussing the representation of the data, I introduce a quantity that captures how physically confined a link is in the network via link trajectory randomizations. Using a collection of real networks, we then investigate correlations between the physical and network structure of real systems.

**Ivan Bonamassa:
Bundle formation in physical networks by random deposition**

Bundles of long hard-core filaments are ubiquitous structures of real-world physical networks.

From the streamlines connecting the brain's white matter to the $\alpha$-actinin filaments shaping up the hierarchical organization of nerves and cytoskeleton, to name a few, bundles play also a fundamental role in the correct functioning of soft and living matter. We show that bundle formation can emerge only due to volume exclusion in a system of hard-core rods anchored to the opposite faces of a cubic box. We provide numerical evidence that bundles can always form in this system for every thickness, $\lambda^{-1}$, of the rods and that a dynamical phase transition occurs at threshold time $t_c\sim \lambda^{-1}$.

We characterize this transition and study its relation to the saturation limit of the deposition process analyzed.

**Cory Glover:
The Fabric of Physical Networks**

In this talk, we discuss how to model physical networks as spatial graphs and utilize the tools of knot theory to understand the tangledness of said networks. This is done in a simple format called the Gauss adjacency list (GAL). We discuss the utility of the GAL and present a set of moves which can be performed to isotope from one GAL to another.

**Wednesday, October 19, 2022
Budapest Rényi
Balázs Szegedy:
Subgraph densities in Markov kernels**

We define subgraph densities in large families of Markov kernels that are represented by singular measures. This extends several tools from dense graphlimit theory to sequences of sparse graphs.

**Endre Csóka:
3D-embedding and expansion**

We show that every connected graph with k edges has a blown-up 3D embedding into a sphere of radius sqrt(k). For expander graphs, this is the smallest possible volume up to a constant factor. Except if we allow embeddings where the vertices can be represented by an unbounded-size body.

**Balázs Ráth:
From metapopulation modelling of epidemics to the mathematics of the switchover phenomenon**

Based on the work of Gergely Ódor, Domonkos Czifra, Júlia Komjáthy, László Lovász, Márton Karsai, Tamás Móri, Dániel Keliger

Abstract: We present some of the features of our stochastic metapopulation model of the spread of covid in Hungary. The simulation results of this model gave rise to the so-called switchover phenomenon: if the parameters of the model are chosen appropriately then the epidemic is more dangerous if it starts from the periphery of the network as opposed to the center of the network. We present some conditions on the network under which switchover can be proved with mathematical rigour.

**Gabor Kun (CUNI guest):
Perfect matchings in graphings**

We give an overview of the history of equidecompositions from the Banach-Tarski paradox to circle squaring. We characterize hyperfinite bipartite graphings with measurable perfect matchings and give applications including two new, simpler proofs of the measurable circle squaring, and the extension of the Lyons-Nazarov theorem to amenable groups. Joint work with Matt Bowen and Marcin Sabok.

**Thursday, October 20, 2022
Prague, Charles University**

**Robert Šámal:
Random embeddings of (complete) graphs**

We study (orientable 2-cell) random embeddings of graphs. These are defined by generating a local rotation around every vertex, which determines the facial structure of the embedding, assuming every face is a disk. We determine asymptotically the expected number of faces in such embedding of complete graphs and in various models of random graphs and discuss the relevance for the minimum genus problem.

Joint work with J. Campion Loth, K. Halasz, B. Mohar and T. Masařík.

**Tomáš Hons:
Gadget construction and structural convergence**

We examine convergence properites of structures created by the gadget construction (from study of graph homomorphism) in order to create interesting examples of first order convergent sequences. In particular, we show that such sequences are elementarily convergent, while the full first order convergence is more subtle.

**Matěj Konečný:
Extending partial automorphisms**

We will outline recent developments regarding extending partial automorphisms of various structures with focus on sparse graphs and large networks.

**Zdeněk Dvořák:
Representation of short distances in structurally sparse graphs**

A partial orientation of a graph G is a weak r-guidance system if for any two vertices at distance at most r in G, there exists a shortest path P between them containing an edge e such that all other edges in P are directed towards e. In case that the orientation has bounded maximum outdegree Delta, this gives an efficient representation of shortest paths of length at most r in G: For any pair of vertices, we can either determine the distance between them or decide the distance is more than r, and in the former case, find a shortest path between them, in time O(Delta^r). We show that graphs from many natural

graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion. We also apply the notion to obtain approximation algorithms for distance variants of the independence and domination number in graph classes that admit weak guidance systems of bounded maximum outdegree.

**Samuel Braunfeld:
The map of tame graph classes**

We will discuss the map of tame graph classes and some recent results that simplify the picture.

**Patrice Ossona de Mendez:
Flipper game**

In this talk, we review three characterizations of nowhere dense classes (quasi wideness, splitter game, and spreading) and their generalization to dense hereditary classes of graphs.

**Friday, October 21, 2022
Departure**

The Lectures were complemeted by active dissucussions. Several new problems were isolated and will eventually lead to publication.