走向基于有向图的复杂体的定量理论及其在脑网络分析中的应用
Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis
AI总结：
本文提出基于有向图构建的复杂体的定量方法，用于分析脑网络拓扑结构，探索更高阶结构在癫痫发作不同阶段的变化及生物标志物的潜在价值。
摘要：
在本文中，我们开发了新的数学方法用于分析网络拓扑，并将其应用于脑网络分析。具体而言，我们严格建立了基于有向图构造的复杂体（如路径复杂体和有向团复杂体）的定量方法，这些复杂体也被称为“更高阶结构”或“更高阶拓扑”或“simplicial结构”。在有向团复杂体的情况下，我们还开发了基于有向团之间相互关系的方法，称为“有向更高阶连通性”。这种新的基于有向图的复杂体的定量理论可以被视为向“定量simplicial理论”形式化的一步。随后，我们使用这些新方法，如对有向图复杂体的特征刻画和相似性度量，来分析由脑连接估计器导出的有向图拓扑结构，特别是信息部分定向相干性（iPDC）估计器，这是一种多变量估计器，可被视为频域中Granger因果关系的表示，特别是从被诊断为左侧颞叶癫痫患者的大脑电图（EEG）数据中估计出的delta、theta和alpha频段数据，以寻找基于这些有向图更高阶结构和连通性的新生物标志物。特别是，我们试图回答以下问题：脑网络的更高阶拓扑结构在发作前、发作中和发作后阶段如何变化？在每个频段和每个大脑半球中如何变化？更高阶结构的分析是否能提供比传统图论分析更好的癫痫动态和发作焦点侧向性的生物标志物？
英文摘要：
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?
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