Observability of Hypergraphs

In this paper we develop a framework to study observability for uniform hypergraphs. Hypergraphs are generalizations of graphs in which edges may connect any number of nodes, thereby representing multi-way relationships which are ubiquitous in many real-world networks including neuroscience, social networks, and bioinformatics. We define a canonical multilinear dynamical system with linear outputs on uniform hypergraphs which captures such multi-way interactions and results in a homogeneous polynomial system. We derive a Kalman-rank-like condition for assessing the local weak observability of this resulting system and propose techniques for its efficient computation. We also propose a greedy heuristic to determine the minimum set of observable nodes, and demonstrate our approach numerically on different hypergraph topologies, and hypergraphs derived from an experimental biological dataset.Comment: 7 pages, 3 figures, 2 algorithms, lots of math

Multilinear Control Systems Theory and its Applications

In biological and engineering systems, structure, function, and dynamics are highly coupled. Such multiway interactions can be naturally and compactly captured via tensor-based representations. Exploiting recent advances in tensor algebraic methods, we develop novel theoretical and computational approaches for data-driven model learning, analysis, and control of such tensor-based representations. In one line of work, we extend classical linear time-invariant (LTI) system notions including stability, reachability, and observability to multilinear time-invariant (MLTI) systems, in which the state, inputs, and outputs are represented as tensors, and express these notions in terms of more standard concepts of tensor ranks/decompositions. We also introduce a tensor decomposition-based model reduction framework which can significantly reduce the number of MLTI system parameters. In another line of work, we develop the notion of tensor entropy for uniform hypergraphs, which can capture higher order interactions between entities than classical graphs. We show that this tensor entropy is an extension of von Neumann entropy for graphs and can be used as a measure of regularity for uniform hypergraphs. Moreover, we employ uniform hypergraphs for studying controllability of high-dimensional networked systems. We propose another tensor-based multilinear system representation for characterizing the multidimensional state dynamics of uniform hypergraphs, and derive a Kalman-rank-like condition to identify the minimum number of control nodes (MCN) needed to achieve full control of the whole hypergraph. We demonstrate these new tensor-based theoretical and computational developments in a variety of biological and engineering examples.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169968/1/canc_1.pd

Disentangling the 4D Nucleome

The dynamical relationship between 3D genome structure, genome function, and cellular phenotype is referred to as the 4D Nucleome (4DN). 4DN analysis remains difficult, since multiple data modalities must be integrated and comprehensively studied in order to obtain new insights. In my dissertation work, I present a computational toolbox which offers both novel and established methods to integrate and analyze time series genome structure and function data. I also provide an extension of the 4DN that captures the contributions of the maternal and paternal genomes. I uncover differences between the two genomes’ structural and functional features across the cell cycle, and reveal an allele-specific relationship between local genome structures and gene expression. In addition, I present a computational framework for analyzing multi-way genomic interactions which allow us to identify transcription clusters in the human genome. Finally, I introduce a computational method to characterize the differences between memory and plasma B cells in the adaptive immune system, which guide us to develop an immune system inspired learning system.PHDBioinformaticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169908/1/lindsly_1.pd