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The book discusses the potential of higher-order interactions to model real-world relational systems. Over the last decade, networks have emerged as the paradigmatic framework to model complex systems. Yet, as simple collections of nodes and links, they are intrinsically limited to pairwise interactions, limiting our ability to describe, understand, and predict complex phenomena which arise from higher-order interactions. Here we introduce the new modeling framework of higher-order systems, where hypergraphs and simplicial complexes are used to describe complex patterns of interactions among any number of agents. This book is intended both as a first introduction and an overview of the state of the art of this rapidly emerging field, serving as a reference for network scientists interested in better modeling the interconnected world we live in.
List of contents
Higher-order systems as a modelling framework.- Graphs, simplicial complexes and hypergraphs: Spectral theory and topology.- Random Simplicial Complexes: Models and Phenomena.- Topological Data Analysis.- Flow-based Community Detection in Hypergraphs.- Pattern formation on hypergraphs.- Non-pairwise interaction in oscillatory ensembles: From theory to data analysis.- From symmetric networks to heteroclinic dynamics and chaos in coupled phase oscillators with higher-order interactions.- Explosive synchronization and multistability in large systems of Kuramoto oscillators with higher-order interactions.- Multiorder Laplacian for Kuramoto dynamics with higher-order interactions.- The Master Stability Function for Synchronization in Simplicial Complexes.- Geometry, Topology and Simplicial Synchronization.- Signal processing on simplicial complexes.- Social contagion on higher order structures.- Consensus Dynamics and Opinion Formation on Hypergraphs.- Collective games on hypergraphs.- Topological Data Analysis of Spatial Systems.- Higher-order description of brain function.- Higher-Order Interactions in Biology: The Curious Case of Epistasis.
