Robustness in sparse random-like graphs

Project Information

RobSparseRand

Grant agreement ID: 101038085

DOI

10.3030/101038085

Project closed

EC signature date 8 April 2021

Start date 20 September 2021

End date 19 September 2023

Funded under

• SPREADING EXCELLENCE AND WIDENING PARTICIPATION

Total cost

€ 147 463,68

EU contribution

€ 147 463,68

Coordinated by

FACULTY OF SCIENCE UNIVERSITY OF ZAGREB
Croatia

Periodic Reporting for period 1 - RobSparseRand (Robustness in sparse random-like graphs)

Reporting period: 2021-09-20 to 2023-09-19

Summary of the context and overall objectives of the project

Sparse, highly connected, random-like graphs are currently a focal point in discrete mathematics, theoretical computer science and network science, motivated by the insight that networks of this type are ubiquitous in computing, biology, economics, physics, social science, etc., and by the theoretical challenges of this setting. Random graph models used in statistical modelling of real-world networks include the Molloy--Reed model for scale-free graphs and the Watts--Strogatz model (designed to simultaneously exhibit the small-world phenomenon and formation of hubs, cited 43000 times). Barab\'asi devotes a chapter of his classic network science book to robustness of random graphs, specifically addressing robustness against adversarial attack.

The aim of the project RobSparseRand was to study robustness in sparse graphs, and specifically in random graphs. In our context, robustness can be understood as Ramsey-type properties, or the presence of specific structures, such as paths and cycles. More specifically, the aim is to solve important open problems in Ramsey theory, by advancing the essential tools in random graph theory (sparse regularity framework, concentration bounds, embedding techniques) and illuminating potentially useful classes of expander graphs.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The Researcher has submitted or published six research papers (with collaborators) on topics relevant to the project: sparse Ramsey graphs, Ramsey multiplicity, community structure in random graph models, paths in random temporal graphs, and extremal graph theory. Let us highlight two of these results. In the article Canonical Colourings in random graphs, the Researcher and Mathias Schacht have established the threshold for a rather strong (canonical) notion of robustness in random graphs at the correct threshold. They have also established the existence of `canonically Ramsey’ graphs with minimal possible clique number. The article was selected for the LAGOS 2023 Best Paper Award. Secondly, the article The Turán density of tight cycles in three-uniform hypergraphs (with Shoham Letzter and Alexey Pokrovskiy) contains a solution to a Turán-type problem for long cycles in hypergraphs, whose relevance and impact is testified by the publication in International Mathematics Research Notices.

At the University of Zagreb, the Researcher has supervised three masters’ theses and given a course attended by 8 PhD students and researchers. The training activities included advanced courses on Analytic number theory, Ergodic theory (University of Zagreb), Ramsey theory (University of Hamburg), as well as 6 workshops and numerous conferences.
The project results have been presented by the Researcher in 9 conferences (3 invited) and 12 research seminars, many of which are recorded and available online. The Researcher has also co-organised two workshops in Croatia (in Zagreb and in Brač) hosting over 40 internally acclaimed researchers and PhD students, making the project and the research area accessible to the local research community.
The activities conducted by the Researcher (including workshops at schools, public events, videos, and conferences for aspiring researchers) have reached audiences of various ages and backgrounds. In particular, the Researcher has held workshops and talks for over 200 high school students in collaboration with the association MNM.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

As mentioned, several important open problems in Ramsey Theory and Extremal Combinatorics were addressed during the project. Their relevance can be attested, for instance, by further work of other authors. As an example, let us mention the work of Balogh and Luo on Turan-type problems for hypergraph cycles; Versteegen; Kral and Lamaison on Ramsey multiplicity of linear patterns; Girao and Munha Correia on unavoidable patterns in graph colourings; Alvarado, Kohayakawa, Morris, Mota on canonical Ramsey properties of random graphs.

The Researcher and her collaborators have also used and developed powerful probabilistic and analytic tools, including the transference principle due to Green-Tao and Conlon-Gowers, branching processes, coupling methods, construction of spread measures. Using these techniques, extremal results on Erdős-Renyi random graphs, randomly augmented graphs, random temporal graphs and preferential attachment graphs have been obtained.

The research on community structure of sparse graphs and random temporal graphs crosses boundaries between mathematical disciplines, connecting probability, combinatorics and network science.

A hypergraph colouring which characterises the presence of tight cycles

Modularity of random graphs vs lower bound attained by Agdur, Kamčev and Skerman

A sketch of the community partitioning algorithm of Agdur, Kamčev and Skerman