The structural organization of higher-order networks: Measures, algorithms and software for hypergraphs

Traditional network models have revealed important properties of real-world systems, including small-world and clustering organization. Yet, they often face limitations in capturing the full complexity of real-world systems. Over the years, researchers have devoted a lot of attention to expanding the set of frameworks available to model complex systems, incrementally encoding more features about the interactions, such as direction, time and multiplexity. Mounting empirical evidence, from metabolic interactions to scientific publishing, now suggests that real-world interactions tend to involve more than two units simultaneously. Explicitly encoding such higher-order interactions in our network modeling with hypergraphs or other tools seems a natural progression in network science research, will this be the key to new insights hidden by traditional binary models? In this Thesis, we introduce computational and methodological tools to analyze the structural organization of systems with higher-order interactions at multiple scales. At the microscale, we propose efficient algorithms for higher-order motif analysis in hypergraphs, identifying overrepresented patterns of group interactions. At the mesoscale, we introduce hyperlink communities, which naturally capture both the hierarchical organization of hyperedges and community overlap in higher-order networks. To study group interactions with richer features, we introduce measures to analyze directed hypergraphs, including reciprocity definitions and motif analysis, providing insights into systems like metabolic networks and financial transactions. We also develop tools to study systems where nodes participate in different types of group interactions simultaneously, such as scientific collaborations across fields. To facilitate the adoption of higher-order network analysis, we develop Hypergraphx, an open-source Python library providing tools for hypergraph construction, visualization, and analysis. We complement this with Hypergraph-data, a curated repository of real-world datasets with rich metadata spanning different domains. These contributions aim to lower the entry barrier for higher-order network analysis and foster interdisciplinary collaboration. With this Thesis, we aim to advance our ability to characterize and understand complex systems beyond traditional network approaches, opening new perspectives for modeling and analyzing group interactions in real-world systems.