# Wolfram Summer SchoolFounded in 2003

17th Annual Wolfram Summer School, held at Bentley University June 23–July 12, 2019

Alumni

# József Konczer

Science

Class of 2016

## Bio

I'm doing my PhD in theoretical physics at the Eötvös Loránd University, Budapest, and also working in the Wigner Research Centre. My topic is integrable methods in AdS/CFT correspondence, and I do practical computer-aided numerical and analytical calculations to determine quantities of the N=4 SYM theory and/or IIB superstring theory on the AdS5xS5 background. Sometimes we also investigate simple toy models that show similar behaviors in which we are interested.

I'm mainly interested in fundamental physics, foundations of quantum mechanics, error estimation of approximate calculations in various models and in making simple models for phenomena appearing in nature. Besides science, I really like to try out new things and meet different people to collect authentic experiences.

My quasi-permanent hobbies are waveboarding, drawing and reading.

## Project: Graph Evolution under Deterministic Subgraph Replacement Rule

The main goal of the project is to show a deterministic subgraph replacement rule, which can lead to a nontrivial behavior from the simplest possible graph or from a grid-like initial graph.

The first step of the project is to clarify the notation, determine the data structure and precisely define what the replacement rule means. During the project, only special undirected sparse graphs will be investigated, where every vertex has degree 3. After that, the possible subgraph replacement rules can be listed, if we set the number of vertices of initial and final subgraphs to given numbers. One can then investigate what rules fulfill the requirements of deterministic time evolution. The understanding of rules can be supported by appropriate visualizations of the graphs. If one has the rules for the subgraph replacements, we have to be able to apply them effectively to even a relatively big graph. Here I plan to first find all possible places where a given rule could be used and store these subgraphs in an association; then, during the evolution, pick randomly from the subgraphs in the list; after the replacement, re-investigate the finite neighborhood of the chosen subgraph for possible new subgraph patterns; and finally, update the subgraph list by these. This method relies on the sparseness of the graph and the locality of substitution rules.

After implementing these algorithms, we can start to explore the evolution of different initial graphs under different deterministic replacement rules.

My first natural choice for the initial graph would be the simplest possible, where the rules can be applied. In the case of this initial graph, the rules should be able to increase the number of vertices, to see some nontrivial behavior. In this case, for example, the emergent dimension would be an interesting numerical quantity that could be determined during the evolution.

The second natural initial graph choice is a grid-like configuration, which represents a space with given dimension (for example, 1, 2, 3). In this case, I would choose rules that would not change this configuration (the grid would be a fixed point of the evolution), then randomly (or if one wishes, in a localized way) do some perturbation to this fixed point and see how the disturbances propagate, observe if some graph properties remain the same and find out if some kind of temperature can be defined, etc. For the investigation of perturbed grids, an appropriate visualization would also be useful; however, this task can turn out to be challenging.

In fortunate cases, even the emergent casual networks can be constructed and investigated.

This project can lead to a more quantitative understanding of deterministic graph evolution models, and can possibly help to construct the definitions of standard quantities in field theories—like partition functions—in the graph theory framework.