Related papers: The 27 geodesic networks in the directed landscape

Transport in crowded, complex environments occurs across many spatial scales. Geometric restrictions can hinder the motion of individuals and, combined with crowding between individuals, can have drastic effects on global transport…

Real complex systems are not rigidly structured; no clear rules or blueprints exist for their construction. Yet, amidst their apparent randomness, complex structural properties universally emerge. We propose that an important class of…

Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This…

We develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges, that is, edges pointing in both directions simultaneously. These two ingredients alter the previously…

Many complex networks in nature have directed links, a property that affects the network's navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in- and…

Let Z be a so-called well-behaved percolation, i.e. a certain random closed set in the hyperbolic plane, whose law is invariant under all isometries; for example the covered region in a Poisson Boolean model. The Hausdorff-dimension of the…

Spatially embedded networks have attracted increasing attention in the last decade. In this context, new types of network characteristics have been introduced which explicitly take spatial information into account. Among others, edge…

The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The…

We consider the geodesic of the directed last passage percolation with iid exponential weights. We find the explicit one-point distribution of the geodesic location joint with the last passage times, and its limit as the parameters go to…

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…

The projective shape of a configuration of k points or "landmarks" in RP(d) consists of the information that is invariant under projective transformations and hence is reconstructable from uncalibrated camera views. Mathematically, the…

The fundamental idea of embedding a network in a metric space is rooted in the principle of proximity preservation. Nodes are mapped into points of the space with pairwise distance that reflects their proximity in the network. Popular…

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of…

A geometrical pattern is a set of points with all pairwise distances (or, more generally, relative distances) specified. Finding matches to such patterns has applications to spatial data in seismic, astronomical, and transportation…

Recent genomic and bioinformatic advances have motivated the development of numerous random network models purporting to describe graphs of biological, technological, and sociological origin. The success of a model has been evaluated by how…

We define the log-gamma sheet and the log-gamma landscape in terms of the 2-parameter and 4-parameter free energy of the log-gamma polymer model and prove that they converge to the Airy sheet and the directed landscape, which are central…

Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…

We consider the probability that a dense wireless network confined within a given convex geometry is fully connected. We exploit a recently reported theory to develop a systematic methodology for analytically characterizing the connectivity…

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive…

All types of networks arise as intricate combinations of dyadic building blocks formed by pairs of vertices. In directed networks, the dyadic patterns are entirely determined by reciprocity, i.e. the tendency to form, or to avoid, mutual…