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We investigate hierarchical structure in various complex systems according to Minimum Spanning Tree methods. Firstly, we investigate stock markets where the graphis obtained from the matrix of correlations coefficient computed between all…
Local explainability methods -- those which seek to generate an explanation for each prediction -- are becoming increasingly prevalent due to the need for practitioners to rationalize their model outputs. However, comparing local…
Patterns are fundamental to human cognition, enabling the recognition of structure and regularity across diverse domains. In this work, we focus on structural repeats, patterns that arise from the repetition of hierarchical relations within…
Topological techniques have become a popular tool for studying information flows in neural networks. In particular, simplicial homology theory is used to analyze how cognitive representations of space emerge from large conglomerates of…
Structural causal models (SCMs) are a widespread formalism to deal with causal systems. A recent direction of research has considered the problem of relating formally SCMs at different levels of abstraction, by defining maps between SCMs…
Spatial navigation in mammals is based on building a mental representation of their environment---a cognitive map. However, both the nature of this cognitive map and its underpinning in neural structures and activity remains vague. A key…
A simple but efficient spectral approach for analyzing the community structure of complex networks is introduced. It works the same way for all types of networks, by spectrally splitting the adjacency matrix into a "unipartite" and a…
Correspondence identifies relationships among objects via similarities among their components; it is ubiquitous in the analysis of spatial datasets, including images, weather maps, and computational simulations. This paper develops a novel…
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…
Graph-based representations such as Scene Graphs enable localization in structured indoor environments by matching a locally observed graph, constructed from sensor data, to a prior map. This process is particularly challenging in…
Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics.…
Network theory and inverse modeling are two standard tools of applied physics, whose combination is needed when studying the dynamical organization of spatially distributed systems from indirect measurements. However, the associated…
Recently, neural network architectures have been developed to accommodate when the data has the structure of a graph or, more generally, a hypergraph. While useful, graph structures can be potentially limiting. Hypergraph structures in…
We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen…
Economic transformation -- change in what an economy produces -- is foundational to development and rising standards of living. Our understanding of this process has been propelled recently by two branches of work in the field of economic…
This work seeks to tackle the inherent complexity of dataspaces by introducing a novel data structure that can represent datasets across multiple levels of abstraction, ranging from local to global. We propose the concept of a multilevel…
A simple construction is presented, which generalises piecewise linear one-dimensional Markov maps to an arbitrary number of dimensions. The corresponding coupled map lattice, known as a simplicial mapping in the mathematical literature,…
We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular…
Hypergraphs provide a natural way to represent polyadic relationships in network data. For large hypergraphs, it is often difficult to visually detect structures within the data. Recently, a scalable polygon-based visualization approach was…
In this article, we introduce a reference model for interaction semantics among communicating discrete systems to guide the discourse on interoperability. The necessary set of unifying concepts is small and comprises essentially the notion…