The structure and dynamics of networks with higher order interactions

All beauty, richness and harmony in the emergent dynamics of a complex system largely depend on the specific way in which its elementary components interact. The last twenty-five years have seen the birth and development of the multidisciplinary field of Network Science, wherein a variety of distributed systems in physics, biology, social sciences and engineering have been modeled as networks of coupled units, in the attempt to unveil the mechanisms underneath their observed functionality. There is, however, a fundamental limit to such a representation: networks capture only pairwise interactions, whereas the functioning of many real-world systems not only involves dyadic connections, but rather is the outcome of collective actions at the level of groups of nodes. For instance, in ecological systems, three or more species may compete for food or territory, and similar multi-component interactions appear in functional and structural brain networks, protein interaction networks, semantic networks, multi-authors scientific collaborations, offline and online social networks, gene regulatory networks and spreading of consensus or contagious diseases due to multiple, simultaneous, contacts. Such multi-component interactions can only be grasped through either hypergraphs or simplicial complexes, which indeed have recently found a huge number of applications. In this report, we cover the extensive literature of the past years on this subject, and we focus on the structure and dynamics of hypergraphs and simplicial complexes. These are indeed becoming increasingly relevant, thanks to the enhanced resolution of data sets and the recent advances in data analysis techniques, which (concurrently and definitely) have shown that such structures play a pivotal role in the complex organization and functioning of real-world distributed systems.