The Study of Structures and Dynamics on Hypergraphs Using the Spectra of General Operators

Parui, Samiron (2023) The Study of Structures and Dynamics on Hypergraphs Using the Spectra of General Operators. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Samiron Parui (17RS062)) 17RS062.pdf - Submitted Version Restricted to Repository staff only Download (3MB)

Abstract

This thesis is based on four papers (reference numbers 7-10), that study the interplay of the operator associated with hypergraph with the structure and dynamics on hypergraphs. Since almost all the information associated with the adjacency and incidence relation of hypergraphs can be encoded in terms of the connectivity tensor or hypermatrix, studying hypergraphs with these hypermatrices is a convenient approach in spectral theory for hypergraphs. High computational complexity and non-linearity of the hypermatrices are the major drawbacks of this approach. Despite a certain scope of information loss, the spectral theory of hypergraphs via matrices or linear operators is becoming more popular because of the linearity and abundance of tools and techniques in matrix theory. Unlike graphs, The notion of hypergraph-adjacency is not unique. We refer the reader to the references numbers 5 and 17 for two notions of hypergraph-adjacency matrices. Variations in the notion of adjacency lead us to variations in Laplacian and signless Laplacian. We use two positive-valued functions on the vertices and hyperedges to define the general diffusion operator. By changing the positive valued functions, we can cover several variations of hypergraph connectivity operators. We prove here the existence of some specific structures in hypergraphs is reflected in the spectra of the general operators. This study also reveals that, conversely, the spectra of the general operators encode information about some specific structures of a hypergraph. We observe that the set of vertices of a hypergraph can be expressed as a disjoint union of these structures. Each hyperedge is disjoint union of some of these structures. Thus, we named these structures as building blocks. In addition to the spectra of general operators, we find that these building blocks affect hypergraph colouring, hypergraph automorphisms, random walks, and some other dynamical processes on hypergraphs. We also include some applications of our study in dynamical networks. Our study of the diffusion process in dynamical networks with multi-body interactions shows that the operator, which incorporates the influence of hyperedge couplings, is a variation of our introduced diffusion operator associated with the underlying hypergraph. We use the spectral theory of hypergraphs to incorporate and study the multi-body interactions in dynamical networks. We also study synchronization in dynamical networks with multi-body interaction. Here we introduce the notion of synchronization-preserving clusters. A synchronization-preserving cluster is a subset of the vertex set of the underlying hypergraph of a dynamical network such that if the dynamical systems in the cluster are synchronized, then the synchronization is retained in the subsequent time steps. We prove that the units and unions of twin units are synchronization-preserving clusters. This thesis ends with some possible extensions and applications.

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