Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis

Abstract

In this work, we developed new mathematical methods for analyzing network topology and applied these methods to the analysis of brain networks. More specifically, we rigorously developed quantitative methods based on complexes constructed from digraphs (digraph-based complexes), such as path complexes and directed clique complexes (alternatively, we refer to these complexes as "higher-order structures," or "higher-order topologies," or "simplicial structures"), and, in the case of directed clique complexes, also methods based on the interrelations between the directed cliques, what we called "directed higher-order connectivities." This new quantitative theory for digraph-based complexes can be seen as a step towards the formalization of a "quantitative simplicial theory." Subsequently, we used these new methods, such as characterization measures and similarity measures for digraph-based complexes, to analyze the topology of digraphs derived from brain connectivity estimators, specifically the estimator known as information partial directed coherence (iPDC), which is a multivariate estimator that can be considered a representation of Granger causality in the frequency-domain, particularly estimated from electroencephalography (EEG) data from patients diagnosed with left temporal lobe epilepsy, in the delta, theta and alpha frequency bands, to try to find new biomarkers based on the higher-order structures and connectivities of these digraphs. In particular, we attempted to answer the following questions: How does the higher-order topology of the brain network change from the pre-ictal to the ictal phase, from the ictal to the post-ictal phase, at each frequency band and in each cerebral hemisphere? Does the analysis of higher-order structures provide new and better biomarkers for seizure dynamics and also for the laterality of the seizure focus than the usual graph theoretical analyses?

Version 4: I corrected the text and added new notations to clarify the distinction between maximal and lower q-digraphs. Clarifications regarding density effects are added in Chapter 7

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