Periodic Reporting for period 3 - DYNASNET (Dynamics and Structure of Networks)
Reporting period: 2022-09-01 to 2024-02-29
Contemporary world, diverse human activities and their effect on the environment, the manifold impact of technology on society, the global health problems, all these very diverse areas have one latent feature in common: they are all related to particular networks. Such networks can be seemingly unmanageably large and complex, but their underlying structure may be modeled as a graph or network. This allows us to utilize the large toolbox of graph theory, including the emerging field of graph limits, and yield a new understanding of the above phenomena.
DYNASNET is an important addition to network science by bringing together a group of leading experts in network science, mathematics, physics and computer science. This interdisciplinary group attacks some of the most challenging problems in this very diverse area, contributes to foundations of the emerging theory and brings a new paradigm on the boundary between these individual disciplines.
More and more branches of science recognize that networks are a substantial element of the appropriate description of the structures they study. These networks have an underlying graph structure, affected by their physical realizations, as well as inherent dynamics (propagation of material, information, or infection, for example). It is an important task to describe the interaction between the underlying structure and this dynamics. Better understanding of this interaction can lead to more reliable predictions on disease propagation, or better understanding of the working of the brain. The long-term objective of this project is to obtain nontrivial and important information about these interactions along the lines of real-life observations, simulations, mathematics modeling, and eventually rigorous mathematical results.
DYNASNET is an important addition to network science by bringing together a group of leading experts in network science, mathematics, physics and computer science. This interdisciplinary group attacks some of the most challenging problems in this very diverse area, contributes to foundations of the emerging theory and brings a new paradigm on the boundary between these individual disciplines.
More and more branches of science recognize that networks are a substantial element of the appropriate description of the structures they study. These networks have an underlying graph structure, affected by their physical realizations, as well as inherent dynamics (propagation of material, information, or infection, for example). It is an important task to describe the interaction between the underlying structure and this dynamics. Better understanding of this interaction can lead to more reliable predictions on disease propagation, or better understanding of the working of the brain. The long-term objective of this project is to obtain nontrivial and important information about these interactions along the lines of real-life observations, simulations, mathematics modeling, and eventually rigorous mathematical results.
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).
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).
The study of physical networks is an emerging field driven by the newly available data describing the detailed three-dimensional structure of complex networks. The work within DYNASNET contributes to establishing the basic tenets of the field, developing the mathematical language required to describe physical networks and identifying the most important consequences of physicality on the behavior of networks.
In epidemic modeling, besides a better understanding of the "switchover phenomenon", a move from specific randomized network models to general graphs satisfying reasonable conditions should bring in a new perspective and more advanced mathematical tools.
New probabilistic models for graph limits have been introduced, providing templates for very large networks. These models handle networks from the very sparse to the dense case. The connection with measure theory should bring substantial progress not only in network theory, but in measure theory as well. The model theoretic approach allows us to handle big Ramsey degrees, presenting challenging problems for higher dimensional structures (such as hypergraphs).
Both Markov chains (as generalizations of graphs) and physical networks are novel objects of study, and in their emerging theory one can "expect the unexpected". In dense graph limit theory, every natural limit object can be approximated by finite graphs, while for bounded-degree graphs, the analogous question is a central open problem. The intermediate case that is being developed could serve as a useful interpolation between these two extremes. The study of physical networks can lead to better understanding of the human brain, for example, by mathematically analyzing the bundling phenomenon of edges.
In epidemic modeling, besides a better understanding of the "switchover phenomenon", a move from specific randomized network models to general graphs satisfying reasonable conditions should bring in a new perspective and more advanced mathematical tools.
New probabilistic models for graph limits have been introduced, providing templates for very large networks. These models handle networks from the very sparse to the dense case. The connection with measure theory should bring substantial progress not only in network theory, but in measure theory as well. The model theoretic approach allows us to handle big Ramsey degrees, presenting challenging problems for higher dimensional structures (such as hypergraphs).
Both Markov chains (as generalizations of graphs) and physical networks are novel objects of study, and in their emerging theory one can "expect the unexpected". In dense graph limit theory, every natural limit object can be approximated by finite graphs, while for bounded-degree graphs, the analogous question is a central open problem. The intermediate case that is being developed could serve as a useful interpolation between these two extremes. The study of physical networks can lead to better understanding of the human brain, for example, by mathematically analyzing the bundling phenomenon of edges.