Going beyond networks, to include higher-order interactions of arbitrary sizes, is a major step to better describe complex systems. In the resulting hypergraph representation, tools to identify structures and central nodes are scarce. We consider the decomposition of a hypergraph in hyper-cores, subsets of nodes connected by at least a certain number of hyperedges of at least a certain size. We show that this provides a fingerprint for data described by hypergraphs and suggests a novel notion of centrality, the hypercoreness. We assess the role of hyper-cores and nodes with large hypercoreness in higher-order dynamical processes: such nodes have large spreading power and spreading processes are localized in central hyper-cores. Additionally, in the emergence of social conventions very few committed individuals with high hypercoreness can rapidly overturn a majority convention. Our work opens multiple research avenues, from comparing empirical data to model validation and study of temporally varying hypergraphs.

The structure and behaviour of many social systems are shaped by the interactions among their individuals. Representing them as complex networks has shed light on the mechanisms that govern their formation and evolution. Such representations, however, focus on interactions between pairs of individuals (represented by the edges of the network), although many social interactions involve instead groups: these can be represented as hyperedges, that can comprehend any number of individuals, leading to the use of higher-order network representations. While recent research has investigated the structure of static higher-order networks, little is known about the mechanisms that govern their evolution. How do groups form and develop? How do people move between different groups? Here, we investigate the temporal dynamics of group change, using empirical social interactions collected in different settings. We leverage proximity records from two data-collection efforts that have tracked the temporal evolution of social interactions among students of a university and children in a preschool. We study the mechanisms governing the temporal dynamics both at the node and group level, characterising how individuals navigate groups and how groups form and disaggregate, finding robust patterns across contexts. We then propose a dynamical hypergraph model that closely reproduces the empirical observations. These results represent a further step in understanding the social dynamics of higher-order interactions, stressing the importance of considering their temporal aspect. The proposed model moreover opens up research directions to study the impact of higher-order temporal network patterns on dynamical processes that evolve on top of them.

Epidemic control often requires optimal distribution of available vaccines and prophylactic tools, to protect from infection those susceptible. Well-established theory recommends prioritizing those at the highest risk of exposure. But the risk is hard to estimate, especially for diseases involving stigma and marginalization. We address this conundrum by proving that one should target those at high risk only if the infection-averting efficacy of prevention is above a critical value, which we derive analytically. We apply this to the distribution of pre-exposure prophylaxis (PrEP) of the Human Immunodeficiency Virus (HIV) among men-having-sex-with-men (MSM), a population particularly vulnerable to HIV. PrEP is effective in averting infections, but its global scale-up has been slow, showing the need to revisit distribution strategies, currently risk-based. Using data from MSM communities in 58 countries, we find that non-selective PrEP distribution often outperforms risk-based, showing that a logistically simpler strategy is also more effective. Our theory may help design more feasible and successful prevention.

How can minorities of individuals overturn social conventions? The theory of critical mass states that when a committed minority reaches a critical size, a cascade of behavioural changes can occur, overturning apparently stable social norms. Evidence comes from theoretical and empirical studies in which minorities of very different sizes, including extremely small ones, manage to bring a system to its tipping point. Here, we explore this diversity of scenarios by introducing group interactions as a crucial element of realism into a model for social convention. We find that the critical mass necessary to trigger behaviour change can be very small if individuals have a limited propensity to change their views. Moreover, the ability of the committed minority to overturn existing norms depends in a complex way on the group size. Our findings reconcile the different sizes of critical mass found in previous investigations and unveil the critical role of groups in such processes. This further highlights the importance of the emerging field of higher-order networks, beyond pairwise interactions.

Complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units. Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems. Here, we highlight recent evidence of collective behaviours induced by higher-order interactions, and we outline three key challenges for the physics of higher-order systems..

Innovation is the driving force of human progress. Recent urn models reproduce well the dynamics through which the discovery of a novelty may trigger further ones, in an expanding space of opportunities, but neglect the effects of social interactions. Here we focus on the mechanisms of collective exploration and we propose a model in which many urns, representing different explorers, are coupled through the links of a social network and exploit opportunities coming from their contacts. We study different network structures showing, both analytically and numerically, that the pace of discovery of an explorer depends on its centrality in the social network. Our model sheds light on the role that social structures play in discovery processes.

The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review both the measures designed to characterize the structure of these systems, and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We then introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher- order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers.

Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.

We introduce a model for the emergence of innovations, in which cognitive processes are described as random walks on the network of links among ideas or concepts, and an innovation corresponds to the first visit of a node. The transition matrix of the random walk depends on the network weights, while in turn the weight of an edge is reinforced by the passage of a walker. The presence of the network naturally accounts for the mechanism of the adjacent possible, and the model reproduces both the rate at which novelties emerge and the correlations among them observed empirically. We show this by using synthetic networks and by studying real data sets on the growth of knowledge in different scientific disciplines. Edge-reinforced random walks on complex topologies offer a new modeling framework for the dynamics of correlated novelties and another example of co-evolution of processes and networks.