Overview 

My group develops methodology and theory for network-based models of complex systems and high-dimensional data including AI algorithm development. At the same time, we strive to integrate emerging advances in mathematics, computing and AI to engineering new methodologies that cater specifically to new application domains. Much of this work focuses on generalizing graph-based models to use generalizations of graphs including multilayer networks, temporal networks, simplicial complexes and hypergraphs. We also collaborate in interdisciplinary teams with domain practitioners to address domain challenges. Current foci include AI algorithm development for wildlife and tourism application.

My group is generously funded with current support from the NSF program on Algorithms for Threat Detection (ATD), the Wyoming Center For Wildlife, Technology and Computing (WyldTech), the UW Worth Initiative, and Wyoming's AI Match program for an award from Alumbra LLC. Prior support has come from the NSF program on Mathematical Biology, Simons Foundation, Nakatani Foundation, and the State University of New York.

Algorithms, methods, and theory for high-dimensional data 

My group develops methodology and theory for studying structural and organizational patterns in data describing social, physical, biological and technological networks. Our focus is on generalizations of graphs including multilayer, multiplex and temporal networks in which “network layers” encode different types of edges (e.g., categorical social ties, coupled infrastructures, or a network at different instances in time). Our mathematical techniques include spectral perturbation theory, random matrix theory, causal inference, nonlinear dimension reduction, and topological data analysis.

• Tunable eigenvector-based centralities for multiplex and temporal networks

• Noise-response analysis of deep neural networks quantifies robustness and fingerprints structural malware

• Continuous-in-depth neural networks

• Eigenvector-based centrality measures for temporal networks

• Super-resolution community detection for layer-aggregated multilayer networks

• Enhanced detectability of community structure in multilayer networks through layer aggregation

• Topological data analysis of contagion maps for examining spreading processes on networks

• Causal inference by optimal causation entropy

Multiscale models and theory for self-organizing systems

My group develops and analyzes models for network-coupled dynamical systems to study self-organization for applications ranging from neuronal networks and biological systems to social networks. For these and other complex systems, the mathematical mechanisms that give rise to emergent behavior often stem from a complicated interplay between nonlinear dynamics and network structure. We are identifying and analyzing new ways in which structural/dynamical mechanisms can support emergent behavior. At the same time, we strive to develop rigorous mathematical analysis to place this pursuit on a strong mathematical foundation. Our current focus is addressing the important issue that network-based modeling has traditionally focused on pairwise (i.e., dyadic) models of interactions that can be encoded by graphs. Graphs, unfortunately, are insufficient for modeling most real-world systems, and it is important to develop more accurate modules using generalizations of graphs including multilayer, multiplex and temporal networks as well as simplicical complexes and hypergraphs. These more-comprehensive modeling frameworks are revealing new structural/dynamical mechanisms that support emergent behavior and are deepening our mathematical understanding for how complex systems function.

• • Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes 

  • Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

  • Impact of community structure on consensus machine learning

  • Asymmetric coupling optimizes interconnected consensus systems

• • Multiplex Markov chains: Convection cycles and optimality

  • Optimal synchronization of complex networks

Data, ML and AI applications through interdisciplinary collaborations

We collaborate with domain experts in the biological, physical, social and computer sciences to maintain a strong connection between theory development and domain-driven research pursuits.

• Human mobility data and machine learning reveal geographic differences in alcohol sales and alcohol outlet visits across U.S. states during COVID-19

• Role of alcohol outlet visits derived from mobile phone location data in enhancing domestic violence prediction at the neighborhood level

• Transient crosslinking kinetics optimize gene cluster interactions

• Inferring causal molecular networks: Empirical assessment through a community-based effort