Related papers: Hyperharmonic analysis for the study of high-order…
Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies…
Comparing networks is essential for a number of downstream tasks, from clustering to anomaly detection. Despite higher-order interactions being critical for understanding the dynamics of complex systems, traditional approaches for network…
In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on…
Battiston et al. (arXiv:2110.06023) provide a comprehensive overview of how investigations of complex systems should take into account interactions between more than two elements, which can be modelled by hypergraphs and studied via…
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of…
Multivariate information decompositions hold promise to yield insight into complex systems, and stand out for their ability to identify synergistic phenomena. However, the adoption of these approaches has been hindered by there being…
Complex systems are difficult to study not only because they are nonlinear, multiscale, and often nonstationary, but because their scientifically relevant organization is often invisible at the level of individual components, pairwise…
Capturing complex high-order interactions among data is an important task in many scenarios. A common way to model high-order interactions is to use hypergraphs whose topology can be mathematically represented by tensors. Existing methods…
The study of irreducible higher-order interactions has become a core topic of study in complex systems. Two of the most well-developed frameworks, topological data analysis and multivariate information theory, aim to provide formal tools…
Higher-order interactions provide a nuanced understanding of the relational structure of complex systems beyond traditional pairwise interactions. However, higher-order network analyses also incur more cumbersome interpretations and greater…
Higher-order networks are gaining significant scientific attention due to their ability to encode the many-body interactions present in complex systems. However, higher-order networks have the limitation that they only capture many-body…
Higher-order information is crucial for relational learning in many domains where relationships extend beyond pairwise interactions. Hypergraphs provide a natural framework for modeling such relationships, which has motivated recent…
Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These…
Hypergraphs are useful mathematical models for describing complex relationships among members of a structured graph, while hyperdigraphs serve as a generalization that can encode asymmetric relationships in the data. However, obtaining…
The aim of this paper is to introduce a novel dictionary learning algorithm for sparse representation of signals defined over combinatorial topological spaces, specifically, regular cell complexes. Leveraging Hodge theory, we embed topology…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…
Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the…
Despite growing interest in synchronization dynamics over "higher-order" network models, optimization theory for such systems is limited. Here, we study a family of Kuramoto models inspired by algebraic topology in which oscillators are…
The Laplace transform is an algebraic method that is widely used for analyzing physical systems by either solving the differential equations modeling their dynamics or by evaluating their transfer function. The dynamics of the given system…
One of the key challenges in many research fields is uncovering how different interconnected systems interact within complex networks, typically represented as multi-layer networks. Capturing the intra- and cross-layer interactions among…