Many real-world systems, from the human brain to engineered infrastructure, can be modeled as networks whose connectivity evolves over time and across multiple layers. In this dissertation, we explore two complementary problems in dynamical network analysis: predicting synchronization in multilayer systems and localizing seizure-generating regions in epilepsy using interpretable network biomarkers.

First, we propose an approximation method that extends the master stability function framework to multilayer networks of saddle-focus oscillators, including Rossler and piecewise linear systems. Our method reduces stability analysis to solving linear algebraic equations and reveals counterintuitive behaviors such as role reversal between coupling layers and the destabilizing effect of increasing the size of a synchronizing layer.

Second, we introduce node degree volatility, which we define as the rate of change in a node's functional connectivity over time. This measure serves as a novel biomarker for identifying dynamically critical nodes. We apply it to time-resolved functional networks reconstructed from intracranial EEG (iEEG) recordings, where nodes represent iEEG channels and edges denote directed causal influences estimated using Granger causality.

Third, we evaluate this method in a diverse cohort of 80 patients with drug-resistant epilepsy, analyzing 82 seizures. Of these, 56 seizures were from cases with successful surgical outcomes. We find that nodes with the highest degree volatility at seizure onset predict the clinically identified seizure onset zone (SOZ) in 73.2% of successful surgical cases, using a stringent 5% threshold to designate SOZ nodes. On the analyzed dataset, this biomarker outperforms established iEEG-based methods, including neural fragility and conventional network metrics such as node degree and betweenness centrality, highlighting the value of capturing local temporal dynamics in brain networks. Node degree volatility complements existing approaches and could also serve as a powerful feature within machine learning frameworks for SOZ localization. Importantly, unlike many machine learning methods that can be computationally intensive and often lack transparency, our approach is direct and interpretable, offering a clear and clinically actionable path to SOZ localization. Beyond epilepsy, node degree volatility may serve as a general framework for identifying dynamically critical nodes in diverse evolving networks, from biological systems to social and engineering domains.