CEU Electronic Theses and Dissertations, 2025
| Author | Di Gaetano, Leonardo |
|---|---|
| Title | Higher-Order Networks: Structural Models, Rare Events and Brain Functional Organization |
| Summary | In many complex systems, interactions occur not just between pairs but among groups involving three or more entities. Higher-order networks are mathematical frameworks that extend traditional network models by capturing these multi-way interactions. Recognizing and modeling higher-order interactions is crucial because they provide a more accurate representation of real-world systems, leading to a deeper understanding of emergent behaviors that cannot be explained by pairwise interactions alone. In the first part of this thesis, we develop a mathematical framework to analyze networks incorporating higher-order interactions. We introduce an extension of the Hidden Variables formalism tailored for higher-order networks, which allows for the characterization of systems with multi-way interactions. Through this formalism, we explore key structural properties such as hyper-degree distributions, degree correlations, and the overall connectivity of the network, revealing that higher-order interactions significantly influence network topology, particularly in aggregated structures generated by higher-order interactions that accumulate over time. Building upon this methodology, future research can extend the Hidden Variables formalism to a broader class of generative network models based on intrinsic node properties, such as fitness models or embedding space models, while accounting for any order of interactions. In the second part of this thesis, we examine how higher-order interactions influence the dynamics of random walks on hypergraphs. Specifically, we focus on rare events in which the behavior of the random walk deviates significantly from what is typically expected. By exploring both quenched and annealed scenarios, corresponding respectively to cases where we compute fluctuations over static networks and ensembles of networks, we investigate how higher-order interactions impact dynamical fluctuations in different settings. Our analysis reveals that higher-order interactions can either suppress or amplify fluctuations from the typical behavior depending on the network configuration. The approach proposed in this thesis can be further used in the future to investigate rare events in a wider class of dynamical systems whenever they can be mapped onto Markovian processes opening possibilities for studying dynamical fluctuations beyond random walks on hypergraphs, such as investigating the controllability of epidemic models or other types of spreading processes not exclusively on higher-order structures. In the final part of this thesis, higher-order network analysis is applied to the study of brain networks in epilepsy patients. A neighborhood-based description of brain connectivity is introduced to identify pathological hubs, which are regions that play a crucial role in the spread of seizures but are not the primary epileptogenic focus (i.e., not the initial source of epileptic activity). By employing higher-order network metrics, the study offers new perspectives on brain network organization. The findings suggest that surgical strategies should account for the higher-order structure of the neighborhoods of these pathological hubs, potentially leading to more effective treatments that reduce seizure recurrence while preserving essential brain functions. This higher-order representation of brain data offers innovative perspectives for investigating neurological disorders beyond epilepsy, such as Alzheimer's disease and schizophrenia. |
| Supervisor | Battiston, Federico |
| Department | Network Science PhD |
| Full text | https://www.etd.ceu.edu/2025/di-gaetano_leonardo.pdf |
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