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Guest Edited by Prof Ginestra Bianconi (Queen Mary University) in collaboration with our Editorial Board Member Prof Federico Battiston (Central European University).
Many real-world systems, from social relationships to the human brain, can be successfully described as graphs: a collection of elementary units (nodes), and their pairwise interactions (links). Over the years, network approaches have been successfully applied to a wide class of domains, from economics to ecology. Thanks to technological advances and an increasingly interconnected world, data availability has recently exploded, amplifying the potential and applicability of network science approaches. Despite being widespread, traditional network descriptions often do not provide a faithful representation of reality. In many systems, interactions among the units are not limited to pairs, but can occur in groups of greater size. This is the case of human face-to-face interactions, species interactions in an ecosystem, or neurological coupling among different brain regions. These ‘higher-order interactions’ are better described by simplicial complexes and hypergraphs, more complex mathematical structures with respect to traditional graphs.
Building on early mathematical work on topological data analysis and graph theory, and supported by new experimental evidence, the investigation of networks with higher-order interactions has become ubiquitous in the last decade. Taking into account the higher-order structure of real-world systems has revealed new patterns of interactions and functionality which arise from inherently high-order features and could not be understood by limiting the analysis of structural properties to pairwise links. From social contagion to synchronisation, the introduction of higher-order interactions in networked systems has already been shown to give rise to new emergent physical phenomena, which cannot be predicted by breaking higher-order interactions into simple low-order dyads.
The aim of this Collection is to provide, as a single resource, a venue for the latest and most important findings on higher-order interaction networks, which we believe will become an important reference for physicists working on the topic in the future years.
While first order phase transitions between incoherence and synchronization are critical for collective behavior in various oscillator system application, e.g., the brain and power grids, such transitions typically require finely tuned properties. In this work the authors show that first order phase transitions and bistability can emerge naturally as a consequence of the presence of higher-order interactions between oscillators.
A general theory for dynamical processes in higher-order systems is still missing. Here, the authors provide a general mathematical framework based on linear stability analysis that allows to assess the stability of classes of processes on arbitrary hypergraphs.
Higher-order networks including many-body interactions among nodes are ubiquitous in complex systems. The authors propose several growing mechanisms which are able to generate synthetic simplicial complexes displaying desired and customized statistical properties fitting those observed in the real world.
Higher-order contagion models capture opinion dynamics and adoption of behavior in social networks. In this paper, the authors propose a mathematical framework able to accurately characterize the phase diagram of these contagion processes in social higher-order networks.
Synchronization phenomena, where coupled oscillators coordinate their behavior, are ubiquitous in physics, biology, and neuroscience. In this work the authors investigate a framework of coupled topological signals where oscillators are defined both on the nodes and the links of a network, showing that this leads to new topologically induced explosive transitions.
Real-world networks are typically characterised by a non-trivial organization at the mesoscale, such that groups of nodes are preferentially connected within distinguishable network regions known as communities. In this work the authors define unipartite, bipartite, and multilayer network representations of hypergraph flows to extract the community structure of social and biological systems with higher-order interactions.
Higher-order interactions intervene in a large variety of networked phenomena, from shared interests known to influence the creation of social ties, to co-location shaping networks embedded in space, like power grids. This work introduces a Bayesian framework to infer higher-order interactions hidden in network data.
Recent studies have shown that complex systems are often best represented by generalized networks such as hypergraphs, multilayer networks, and temporal networks. Here, the authors propose a unified framework to investigate cluster synchronization patterns in generalized networks and demonstrate the existence of chimera states that emerge exclusively in the presence of higher-order interactions.
Evaluating the importance of nodes and hyperedges in hypergraphs is relevant to link detection, link prediction and matrix completion. Here, the authors define a family of nonlinear eigenvector centrality measures for both edges and nodes in hypergraphs, propose an algorithm to calculate them, and illustrate their application on real-world data sets.
The increasing availability of new data on biological and sociotechnical systems highlights the importance of well grounded filtering techniques to separate meaningful interactions from noise. In this work the authors propose the first method to detect informative connections of any order in statistically validated hypergraphs, showing on synthetic benchmarks and real-world systems that the highlighted hyperlinks are more informative than those extracted with traditional pairwise approaches.
A common problem in reconstructing weighted networks to represent real-world systems is that low-weight edges might appear due to noise, affecting the topology of the inferred network. Here, the authors propose a method based on persistent homology that allows one to investigate the higher-order network organization that can be created by low-weight, noisy edges.
The study of high-order networks as attracted significant attention recently. The authors introduce the concept of computability for searching maximum clique in large networks and an optimized algorithm for finding cavities with different orders.
Group interactions can dramatically alter social contagion dynamics and lead to the emergence of new phenomena like abrupt transitions and critical mass effects. The authors develop an approximate master equation framework to analytically describe contagion in heterogeneous hypergraphs and study the impact of large influential groups in seeding and sustaining epidemics.
While the Deffuant-Weisbuch model, one of the paradigmatic models of opinion dynamics, represents homophily through pairwise interactions, many real world interactions are not pairwise but happen in groups. Here, the authors take a numerical approach and extend the Deffuant model to hypergraphs of different topology, and show how introducing higher order interactions affects the phase transition to consensus.
Critical mass dynamics, where a minority of committed individuals reaching a critical size can overturn stable social conventions, is typically modelled by taking into account pairwise interactions only. Here, the authors generalise the Naming Game model to higher-order social interactions, and show both numerically and analytically how the interplay of local interactions and group size influences global social outcomes.
Recent research has shown that pair interactions in a given network are superseded by higher-order interactions and to incorporate these features into our understanding of a network additional mathematical tools, such as hypergraphs, are required. Here, the authors develop an algorithm to detect motifs in hypergraphs and show how they can be used to identify structural differences in a variety of real-world systems.
Synchronization is a widespread emergent feature of complex systems. Here, the authors investigate the optimization of synchronization in phase oscillators with higher-order interactions, and find that optimized networks are more homogeneous in the nodes’ degree for undirected interactions, and for directed interactions they are generally structurally asymmetric, but can be symmetric, which differs from the pairwise case.
Synchronization dynamics in the presence of higher order interactions is well represented through variations of the Kuramoto model and subject of current interest. Here, the authors study and characterize the behavior of the simplicial Kuramoto model with weights on any simplices and in the presence of linear and nonlinear frustration, defined as the simplicial Sakaguchi-Kuramoto model.
Topological signals are dynamical variables that can be associated to nodes, links, triangles, etc. Here, the authors formulate Dirac synchronization that uses the Dirac operator to couple locally topological signals defined on nodes and links and show, numerically and analytically, that this gives rise to an explosive synchronization transition and to a coherent rhythmic phase.