arXiv subjects

Dmitri Krioukov

Publications and source records attributed to Dmitri Krioukov.

At least 19 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↗

arXiv2025

Deterministic construction of typical networks in network models

It is often desirable to assess how well a given dataset is described by a given model. In network science, for instance, one often wants to say that a given real-world network appears to come from a particular network model. In statistical physics, the corresponding problem is about how typical a given state, representing real-world data, is in a particular statistical ensemble. One way to address this problem is to measure the distance between the data and the most typical state in the ensemble. Here, we identify the conditions that allow us to define this most typical state. These conditions hold in a wide class of grand canonical ensembles and their random mixtures. Our main contribution is a deterministic construction of a state that converges to this most typical state in the thermodynamic limit. This construction involves rounds of derandomization procedures, some of which deal with derandomizing point processes, an uncharted territory. We illustrate the construction on one particular network model, deterministic hyperbolic graphs, and its application to real-world networks, many of which we find are close to the most typical network in the model. While our main focus is on network models, our results are very general and apply to any grand canonical ensembles and their random mixtures satisfying certain niceness requirements.

physics.soc-ph↗

arXiv2025

Growing unlabeled networks

Models of growing networks are a central topic in network science. In these models, vertices are usually labeled by their arrival time, distinguishing even those node pairs whose structural roles are identical. In contrast, unlabeled networks encode only structure, so unlabeled growth rules must be defined in terms of structurally distinguishable outcomes; network symmetries therefore play a key role in unlabeled growth dynamics. Here, we introduce and study models of growing unlabeled trees, defined in analogy to widely-studied labeled growth models such as uniform and preferential attachment. We develop a theoretical formalism to analyze these trees via tracking their leaf-based statistics. We find that while many characteristics of labeled network growth are retained, numerous critical differences arise, caused primarily by symmetries among leaves in common neighborhoods. In particular, degree heterogeneity is enhanced, with the strength of this enhancement depending on details of growth dynamics: mild enhancement for uniform attachment, and extreme enhancement for preferential attachment. These results and the developed analytical formalism may be of interest beyond the setting of growing unlabeled trees.

physics.soc-ph↗

arXiv2025

Local d'Alembertian for causal sets

Causal set theory is an intrinsically nonlocal approach to quantum gravity, inheriting its nonlocality from Lorentzian nonlocality. This nonlocality causes problems in defining differential operators -- such as the d'Alembert operator, a cornerstone of any relativistic field theory -- within the causal set framework. It has been proposed that d'Alembertians in causal sets must themselves be nonlocal to respect causal set nonlocality. However, we show that such nonlocal d'Alembertians do not converge to the standard continuum d'Alembertian for some basic fields. To address this problem, we introduce a local d'Alembert operator for causal sets and demonstrate its convergence to its continuum counterpart for arbitrary fields in Minkowski spacetimes. Our construction leverages recent developments in measuring both timelike and spacelike distances in causal sets using only their intrinsic structure. This approach thus reconciles locality with Lorentz invariance, and paves a way toward defining converging discrete approximations to locality-based differential operators even in theories that are inherently nonlocal.

gr-qc↗

arXiv2025

Multiplexity amplifies geometry in networks

Many real-world network are multilayer, with nontrivial correlations across layers. Here we show that these correlations amplify geometry in networks. We focus on mutual clustering--a measure of the amount of triangles that are present in all layers among the same triplets of nodes--and find that this clustering is abnormally high in many real-world networks, even when clustering in each individual layer is weak. We explain this unexpected phenomenon using a simple multiplex network model with latent geometry: links that are most congruent with this geometry are the ones that persist across layers, amplifying the cross-layer triangle overlap. This result reveals a different dimension in which multilayer networks are radically distinct from their constituent layers.

physics.soc-ph↗

arXiv2024

Ensemble inequivalence and phase transitions in unlabeled networks

We discover a first-order phase transition in the canonical ensemble of random unlabeled networks with a prescribed average number of links. The transition is caused by the nonconcavity of microcanonical entropy. Above the critical point coinciding with the graph symmetry phase transition, the canonical and microcanonical ensembles are equivalent and have a well-behaved thermodynamic limit. Below the critical point, the ensemble equivalence is broken, and the canonical ensemble is a mixture of phases: empty networks and networks with average degrees diverging logarithmically with the network size. As a consequence, networks with bounded average degrees do not survive in the thermodynamic limit, decaying into the empty phase. The celebrated percolation transition in labeled networks is thus absent in unlabeled networks. In view of these differences between labeled and unlabeled ensembles, the question of which one should be used as a null model of different real-world networks cannot be ignored.

cond-mat.stat-mech↗

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↗

arXiv2024

Random hyperbolic graphs in $d+1$ dimensions

We consider random hyperbolic graphs in hyperbolic spaces of any dimension $d+1\geq 2$. We present a rescaling of model parameters that casts the random hyperbolic graph model of any dimension to a unified mathematical framework, leaving the degree distribution invariant with respect to the dimension. Unlike the degree distribution, clustering does depend on the dimension, decreasing to 0 at $d \rightarrow \infty$. We analyze all of the other limiting regimes of the model, and we release a software package that generates random hyperbolic graphs and their limits in hyperbolic spaces of any dimension.

physics.soc-ph↗

arXiv2024

Measuring spatial distances in causal sets via causal overlaps

Causal set theory is perhaps the most minimalistic approach to quantum gravity, in the sense that it makes next to zero assumptions about the structure of spacetime below the Planck scale. Yet even with this minimalism, the continuum limit is still a major challenge in causal sets. One aspect of this challenge is the measurement of distances in causal sets. While the definition and estimation of time-like distances is relatively straightforward, dealing with space-like distances is much more problematic. Here we introduce an approach to measure distances between space-like separated events based on their causal overlap. We show that the distance estimation errors in this approach vanish in the continuum limit even for smallest distances of the order of the Planck length. These results are expected to inform the causal set geometrogenesis in general, and in particular the development of evolving causal set models in which space emerges from causal dynamics.

gr-qc↗

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↗

arXiv2022

Random Simplicial Complexes: Models and Phenomena

We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.

math.PR↗

arXiv2021

Ollivier-Ricci curvature convergence in random geometric graphs

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.

math.PR↗

arXiv2021

Dynamic Hidden-Variable Network Models

Models of complex networks often incorporate node-intrinsic properties abstracted as hidden variables. The probability of connections in the network is then a function of these variables. Real-world networks evolve over time, and many exhibit dynamics of node characteristics as well as of linking structure. Here we introduce and study natural temporal extensions of static hidden-variable network models with stochastic dynamics of hidden variables and links. The rates of the hidden variable dynamics and link dynamics are controlled by two parameters, and snapshots of networks in the dynamic models may or may not be equivalent to a static model, depending on the location in the parameter phase diagram. We quantify deviations from static-like behavior, and examine the level of structural persistence in the considered models. We explore temporal versions of popular static models with community structure, latent geometry, and degree-heterogeneity. We do not attempt to directly model real networks, but comment on interesting qualitative resemblances, discussing possible extensions, generalizations, and applications.

physics.soc-ph↗

arXiv2020

Link prediction with hyperbolic geometry

Link prediction is a paradigmatic problem in network science with a variety of applications. In latent space network models this problem boils down to ranking pairs of nodes in the order of increasing latent distances between them. The network model with hyperbolic latent spaces has a number of attractive properties suggesting it must be a powerful tool to predict links, but the past work in this direction reported mixed results. Here we perform systematic investigation of the utility of latent hyperbolic geometry for link prediction in networks. We first show that some measures of link prediction accuracy are extremely sensitive with respect to inaccuracies in the inference of latent hyperbolic coordinates of nodes, so that we develop a new coordinate inference method that maximizes the accuracy of such inference. Applying this method to synthetic and real networks, we then find that while there exists a multitude of competitive methods to predict obvious easy-to-predict links, among which hyperbolic link prediction is rarely the best but often competitive, it is the best, often by far, when the task is to predict less obvious missing links that are really hard to predict. These links include missing links in incomplete networks with large fractions of missing links, missing links between nodes that do not have any common neighbors, and missing links between dissimilar nodes at large latent distances. Overall these results suggest that the harder a specific link prediction task is, the more seriously one should consider using hyperbolic geometry.

physics.soc-ph↗

arXiv2020

Weighted hypersoft configuration model

Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration models. If the degree sequence is constrained exactly, the corresponding microcanonical ensemble of random graphs with a given degree sequence is the configuration model per se. If the degree sequence is constrained only on average, the corresponding grand-canonical ensemble of random graphs with a given expected degree sequence is the soft configuration model. If the degree sequence is not fixed at all but randomly drawn from a fixed distribution, the corresponding hypercanonical ensemble of random graphs with a given degree distribution is the hypersoft configuration model, a more adequate description of dynamic real-world networks in which degree sequences are never fixed but degree distributions often stay stable. Here, we introduce the hypersoft configuration model of weighted networks. The main contribution is a particular version of the model with power-law degree and strength distributions, and superlinear scaling of strengths with degrees, mimicking the properties of some real-world networks. As a byproduct, we generalize the notions of sparse graphons and their entropy to weighted networks.

physics.soc-ph↗

arXiv2020

Network Geometry

Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are all intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale-invariance, self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments dealing with these approaches to network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.

physics.soc-ph↗