Higher-Order Networks

Abstract

Higher-order networks describe the many-body interactions of a large variety of complex systems, ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. Having been used extensively in quantum gravity to describe discrete or discretized space-time, simplicial complexes have only recently started becoming the representation of choice for capturing the underlying network topology and geometry of complex systems. This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics. This Elements aims to demonstrate that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.

Dynamics on higher-order networks: a review

“…It is unquestionably true that all these developments have helped us to perceive many scenarios better, but we have specifically assumed dyadic or pairwise interactions as the backbone for connections among the units of the system. However, for a complete explanation of many complex systems, one needs to further improve the structural modelling of networked systems [ 8 , 9 ]. For instance, group interactions take place predominantly in systems arising in neurobiology [ 10 , 11 ], social systems [ 9 , 12 ] and ecology [ 13 , 14 ].…”

“…However, for a complete explanation of many complex systems, one needs to further improve the structural modelling of networked systems [8,9]. For instance, group interactions take place predominantly in systems arising in neurobiology [10,11], social systems [9,12] and ecology [13,14]. The network framework has been intrinsically limited to explanation through pairwise interactions, which are sufficient only to model the dyadic relationships, and so larger group interactions need a better formulation for networked systems [15][16][17].…”

Dynamics on higher-order networks: A review

“…However, a complete explanation of many complex systems requires more to furnish. Indeed, real-world systems often exhibit higherorder relationships beyond dyadic interactions [8,9]. For instance, group interactions take place predominantly in systems arising in neurobiology [10,11] and social sys-tems [9,12] to ecology [13,14].…”

“…Indeed, real-world systems often exhibit higherorder relationships beyond dyadic interactions [8,9]. For instance, group interactions take place predominantly in systems arising in neurobiology [10,11] and social sys-tems [9,12] to ecology [13,14]. Network framework has been intrinsically limited to explanation through the pairwise interactions which are sufficient only to model the dyadic relationships and so larger group interactions do need a better formulation for networked systems [15][16][17].…”

Graph Ricci Curvatures Reveal Atypical Functional Connectivity in Autism Spectrum Disorder

“…In recent years there has been an increasing interest in the development of geometric tools for analyzing complex networks (Boguñá et al, 2021), which enables the study of higher-order correlations in networks beyond pairwise interactions (Bianconi, 2021;Iacopini et al, 2019;Kartun-Giles and Bianconi, 2019). A fundamental concept in geometry is Ricci curvature (Jost, 2017), which quantifies the extent to which a space differs from being flat.…”