To develop the mathematical foundations of network theory, it is important to describe appropriate ``templates’’ (simple descriptions which encode relevant properties) for very large networks. This can be achieved by defining convergence of graph sequences and their limit objects. This task has been completed before for the cases of dense and for very sparse graphs; in this project, graphs of medium density have been studied. Surprisingly, it turned out that Markov chains, basic notions in probability theory, can be considered as generalizations of graphs, and can serve as these templates (D. Kunszenti-Kovács, L. Lovász, B. Szegedy: Subgraph densities in Markov spaces, Advances in Math., to appear). A theory of Markov chains, representing limits of convergent graph sequences, has been developed, opening up a large set of problems of generalizing graph theory to Markov chains. Extending flow theory to these limit objects, defining subgraph measures in Markov spaces, and connecting the analytic and combinatorial theory of submodularity are three groups of results that validate this approach.
Many basic properties of graphs can be expressed through formal logic, making logic and model theory important tools. Among several applications, this approach leads to a new understanding of Ramsey theory through the theory of graph sparsity (J. Nešetřil, P- Ossona de Mendez (2020): A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tee Depth), Memoirs of the American Math. Soc. 263, Number 1272). Related to the last topic, a systematic framework for combinatorial constructions of graph classes with the extension property for partial automorphisms has been developed. Very concisely, “symmetry may be localized” (Hubicka, Konecný, Nešetril: All those EPPA classes (Strengthenings of the Herwig-Lascar theorem), Transaction of the Amer. Math. Soc. 375 (11), 2022, 7601-7667).
Physical networks have emerged as an important and active area of collaboration in the research team. Physical networks are composed of three-dimensional physical objects embedded in space, examples of such systems include biological networks from molecular networks to nervous systems and man-made systems from computer chips to critical infrastructure. Our work focuses on describing, modeling and predicting these systems and the joint participation of graph theorists and network scientists in the project allows us to tackle these questions by leveraging results of graph combinatorics, group theory and knot theory. Some of the key results of the project were published in two Nature Physics papers so far (Liu, Yanchen, Nima Dehmamy, and Albert-László Barabási, Nature Physics 17.2 (2021): 216-222. and by Posfai et al, accepted in Nature Physics, 2023).
For the study of disease propagation on society networks, databases about commutation in Hungary between towns, about the spread of Covid, and about the change of behavior of people under information about the disease (awareness) have been set up, and mathematical models were formulated and programmed. Various methods and results from graph theory (e.g. percolation and graph limit theory) have been applied in the study of disease propagation. Simulations on these models lead to the observation of a "switch-over" phenomenon, showing a non-trivial relationship between the location of the original infection and the infection rate (G. Ódor, D. Czifra, J. Komjáthy, L. Lovász and M. Karsai (PNAS (2021) 118 (41) e2112607118).
