Random Structures and Dynamics Arising from Questions in Social, Biological, and Physical Sciences

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Analysis of random structures and stochastic dynamics has been a topic of interest across many disciplines. Study of various spatial and non-spatial random graph models, stochastic spatial models, and stochastic processes taking place on random graphs, and network-based algorithms are key to these efforts. Although these problems have received significant attention in the literature, rigorous analysis and foundational work is needed in many areas to have reliable results and to cope with upcoming challenges and the complexity of real-world data. This research addresses several theoretical and some empirical challenges involving a wide class of questions arising in physics, social sciences, and biosciences. Specific research areas that would benefit from this research include determining functional connectivity within neural networks in brains, pandemic management, pest control strategies, and the evolution of social interactions. This research will provide new insights into the mechanistic underpinning of the underlying complex structures and associated covariates on different social and biological phenomena which are the central objective of research in many disciplines. This analysis will help scientists to understand experimental and observational data in biosciences and social sciences and develop suitable control strategies. Community detection for temporal networks would enhance the understanding of brain function across multiple spatial and temporal scales. The proposed work also has significant potential to increase our capacity to develop efficient and cost-effective intervention strategies to mitigate some of the most potentially damaging and fastest-spreading invasive epidemics of humans, livestock, and plants (including wheat, soybean, citrus, and hop). To guarantee that the results are relevant to scientists, parts of the project will be carried out in collaboration with scientists of different fields. Presentations on the relevant research will be given in stakeholder meetings and community outreach events. Undergraduate and graduate students will be involved in the research activities at the intersection of mathematics and other disciplines. Opportunities to learn theoretical and empirical analysis of various probabilistic models, and interactions with scientists will enhance their ability to work at the interface between mathematics and various related areas. One of the primary goals of this project is to understand and analyze different structural properties and limiting behavior of various spatial and non-spatial random graph models, and several stochastic dynamics taking place on graphs. The random graph models include open clusters of percolation and related models, multi-layer, and temporal network models. Percolation models originated in the physics literature as a model for a porous medium. There are many useful tools and a well-developed theory for studying the percolation models on two-dimensional lattices. However, for higher dimensional lattices, several key aspects, including the near-critical regime and the behavior of the model in subgraphs such as sectors, are poorly understood. This project will address some of these issues. Multi-layer networks are natural models for numerous datasets arising in various scientific fields, including genomics, biomedical sciences, neuroscience, economics, sociology, ecology, epidemiology, and technological networks. Depending on the context, various parametric probabilistic models have been used for the formation of multi-layer, multiplex, and temporal networks. For the estimation of those parameters from data, and for model selection purposes, it is very important to understand the behavior of some key functionals (e.g., subgraph counts or the concentration of aggregated adjacency matrices). These functionals will be analyzed for a wide variety of temporal network models, particularly where the snapshots have some correlation structure. The stochastic dynamics include some important variants of the standard models for infection spreading and opinion evolution in presence of additional restrictions (e.g., temporary isolation). The project also includes studying several theoretical and empirical aspects (e.g., bounds for detectability threshold, efficient detection algorithms) of various detection problems, which includes detection of anomalous structures, community detection from a correlated sequence of networks, detection of the source of the epidemic. The rigorous analyses of the models discussed above will require the development of novel mathematical techniques and research tools. These techniques and tools would be instrumental to obtain a deeper understanding of a broader class of random structures and dynamics arising in multiple scientific disciplines.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.