Johan Lord: From relations to simplicial complexes - A toolkit for the topological analysis of networks
MSc Thesis Presentation
Tid: To 2021-06-17 kl 13.00 - 14.00
Respondent: Johan Lord, KTH
We present a rigorous yet accessible introduction to structures on finite sets foundational for a formal study of complex networks. This includes a thorough treatment of binary relations, distance spaces, their properties and similarities. A correspondence between relations and graphs is given followed by a study of cohesiveness and centrality. We show how graph degeneracy is equivalent to the concept of k-cores, which give a measure of the cohesiveness or interconnectedness of a subgraph. We then further extend this to d-cores of directed graphs. After a brief introduction to topology focusing on topological spaces from distances, we give an introduction to algebraic topology, with an emphasis on homology. In the context of algebraic topology, the d-cores of a directed graph give rise to a partially ordered set of subgraphs, leading to a set of filtrations that is two-dimensional in nature. Finally, we apply these methods to the neuronal network of C.elegans. Persistent homology with respect to directed core filtrations as well as robustness of homology to targeted edge percolations in different directed cores is analyzed. Much importance is placed on intuition and on unifying methodologies of such dispersed disciplines as sociology and network neuroscience, by rooting them in pure mathematics.