Related papers: Geodesic Distance in Planar Graphs
We develop parametric classes of covariance functions on linear networks and their extension to graphs with Euclidean edges, i.e., graphs with edges viewed as line segments or more general sets with a coordinate system allowing us to…
Geodesic paths and distances are among the most popular intrinsic properties of 3D surfaces. Traditionally, geodesic paths on discrete polygon surfaces were computed using shortest path algorithms, such as Dijkstra. However, such algorithms…
Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may tame the infinities that arise from quantizing gravity, and dispense with the machinery of…
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
We present a self-contained analysis of theories of discrete 2D gravity coupled to matter, using geometric methods to derive equations for generating functions in terms of free (noncommuting) variables. For the class of discrete gravity…
We present a new derivation of the distance-dependent two-point function for planar Eulerian triangulations and give expressions for more refined generating functions where we also control hull perimeters. These results are obtained in the…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into $S^2$. On the…
We consider Teichm\"uller geodesics in strata of translation surfaces. We prove lower and upper bounds for the Hausdorff dimension of the set of parameters generating a geodesic bounded in some compact part of the stratum. Then we compute…
The search for a mathematical foundation for the path integral of Euclidean quantum gravity calls for the construction of random geometry on the spacetime manifold. Following developments in physics on the two-dimensional theory, random…
For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, it is known that given any non-axial direction, all the semi-infinite geodesics starting from points in $\mathbb{Z}^2$ in that direction almost surely…
We study the random planar maps obtained from supercritical Galton--Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and…
Active contour models based on partial differential equations have proved successful in image segmentation, yet the study of their geometric formulation on arbitrary geometric graphs is still at an early stage. In this paper, we introduce…
We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq…
The method of Hamilton-Jacobi is used to obtain geodesics around non- Riemannian planar torsional defects.It is shown that by perturbation expansion in the Cartan torsion the geodesics obtained are parabolic curves along the plane x-z when…
Understanding network structure and having access to realistic graphs plays a central role in computer and social networks research. In this paper, we propose a complete, and practical methodology for generating graphs that resemble a real…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work…
A new class of distances for graph vertices is proposed. This class contains parametric families of distances which reduce to the shortest-path, weighted shortest-path, and the resistance distances at the limiting values of the family…
We consider the geodesic motion on the symmetric moduli spaces that arise after timelike and spacelike reductions of supergravity theories. The geodesics correspond to timelike respectively spacelike $p$-brane solutions when they are lifted…