arXiv subjects

Huck Stepanyants

Publications and source records attributed to Huck Stepanyants.

5 recordsLinked to original sources

arXiv2026

The rank and layer distributions in random recursive trees

The distribution of node depths in a network is crucial for analyzing network structure. Two measures, rank and layer, quantify how deep inside a network a node is. The rank is the node's distance to the closest leaf, a node of degree 1. The layer is the number of times all leaves must be stripped off for the node to become a leaf. We derive exact recursive expressions for the rank and layer distributions in random recursive trees. We show that the rank and layer distributions decay factorially and geometrically, respectively, and prove the self-averaging of the numbers of nodes with fixed rank or layer. Unlike previous studies, our approach does not depend on labels or a root node, providing a more versatile framework for analyzing rank and layer distributions in complex networks.

math.PR↗

arXiv2025

Projective limits of probabilistic symmetries and their applications to random graph limits

We couple projective limits of probability measures to direct limits of their symmetry groups. We show that the direct limit group is the group of symmetries of the projective limit probability measure. If projective systems of probability measures represent point processes in increasingly larger finite regions of the same infinite space, then we show that under some additional niceness and consistency assumptions, an extension of the direct limit group is the symmetry group of the projective limit point process in the whole infinite space. The application of these results to random graph limits provides ``shortest paths'' to graphons and graphexes as it recovers these random graph limits as trivial corollaries. Another application example encompasses a broad class of limits of random graphs with bounded average degrees. This class includes a representative collection of paradigmatic random graph models that have attracted significant research attention in diverse areas of science. Our approach thus provides a general unified framework to study limits of very different types of random graphs.

math.PR↗

arXiv2024

Computing distances on Riemann surfaces

Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1.

math.GT↗

arXiv2023

Diameter of Compact Riemann Surfaces

Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than $1$ are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold.

math.GT↗

arXiv2022

Entropy of labeled versus unlabeled networks

The structure of a network is an unlabeled graph, yet graphs in most models of complex networks are labeled by meaningless random integers. Is the associated labeling noise always negligible, or can it overpower the network-structural signal? To address this question, we introduce and consider the sparse unlabeled versions of popular network models, and compare their entropy against the original labeled versions. We show that labeled and unlabeled Erdos-Renyi graphs are entropically equivalent, even though their degree distributions are very different. The labeled and unlabeled versions of the configuration model may have different prefactors in their leading entropy terms, although this remains conjectural. Our main results are upper and lower bounds for the entropy of labeled and unlabeled one-dimensional random geometric graphs. We show that their unlabeled entropy is negligible in comparison with the labeled entropy. This means that in sparse networks the entropy of meaningless labeling may dominate the entropy of the network structure. The main implication of this result is that the common practice of using exchangeable models to reason about real-world networks with distinguishable nodes may introduce uncontrolled aberrations into conclusions made about these networks, suggesting a need for a thorough reexamination of the statistical foundations and key results of network science.

physics.soc-ph↗