Related papers: Hypernetworks: From Posets to Geometry
Statistical ensembles of networks, i.e., probability spaces of all networks that are consistent with given aggregate statistics, have become instrumental in the analysis of complex networks. Their numerical and analytical study provides the…
We review an approach that uses binary relations as the fundamental constituents of the universe, utilizing them as building blocks for both space and matter. The model is defined by an ultraviolet continuous fixed point of a statistical…
This work describes how the formalization of complex network concepts in terms of discrete mathematics, especially mathematical morphology, allows a series of generalizations and important results ranging from new measurements of the…
We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation…
We continue studying net bundles over partially ordered sets (posets), defined as the analogues of ordinary fibre bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, giving versions…
We present the application of topological data analysis (TDA) to study unweighted complex networks via their persistent homology. By endowing appropriate weights that capture the inherent topological characteristics of such a network, we…
This work explores hypernetworks: an approach of using a one network, also known as a hypernetwork, to generate the weights for another network. Hypernetworks provide an abstraction that is similar to what is found in nature: the…
A geometric entropy is defined as the Riemannian volume of the parameter space of a statistical manifold associated with a given network. As such it can be a good candidate for measuring networks complexity. Here we investigate its ability…
As complex networks find applications in a growing range of disciplines, the diversity of naturally occurring and model networks being studied is exploding. The adoption of a well-developed collection of network taxonomies is a natural…
We consider a Gaussian statistical model whose parameter space is given by the variances of random variables. Underlying this model we identify networks by interpreting random variables as sitting on vertices and their correlations as…
Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain…
Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as…
Motivated by the relevance of higher-order interactions in quantum physics and materials science at the nanoscale, recently a model has been introduced for new classes of networks that grow by the geometrically constrained aggregation of…
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can…
A common approach to modeling networks assigns each node to a position on a low-dimensional manifold where distance is inversely proportional to connection likelihood. More positive manifold curvature encourages more and tighter…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Understanding the origins of complexity is a fundamental challenge with implications for biological and technological systems. Network theory emerges as a powerful tool to model complex systems. Networks are an intuitive framework to…
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…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
Most complex systems can be captured by graphs or networks. Networks connect nodes (e.g.\ neurons) through edges (synapses), thus summarizing the system's structure. A popular way of interrogating graphs is community detection, which…