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Andrei Raigorodskii

Publications and source records attributed to Andrei Raigorodskii.

11 recordsLinked to original sources
arXiv2026

On the Stability of the Independence Number in Random Distance Graphs

We consider a random subgraph $G_p(n,r,<s)$ of the complete distance graph $G(n,r,<s)$ whose vertices are the $r$-element subsets of the set $\{1,\dots,n\}$ and whose edges join pairs of subsets that intersect in fewer than $s$ elements; each edge survives independently of the others with probability $p$. The independence number of the graph $G(n,r,<s)$ equals $C_{n-s}^{r-s}$ -- this is the classical Erdos-Ko-Rado theorem. We prove that, for $r=r(n)\to\infty$, $s=s(n)\to\infty$, $s=o(r)$, $r^2=o(n)$ and $p\ge 16\,sr^2\ln(n/r)/n$, with probability tending to 1 the independence number of the random graph $G_p(n,r,<s)$ also equals $C_{n-s}^{r-s}$, i.e., the Erdos-Ko-Rado result is stable under random sparsification of the graph. Thereby, in the range of parameters $s\to\infty$, $s=o(r)$, a recent result of Raigorodskii and Karas is strengthened: the lower bound on the probability $p$ that guarantees stability is lowered by a factor of about $r/s$.

math.CO↗
arXiv2025

Stochastic Origin Frank-Wolfe for traffic assignment

In this paper, we present the Stochastic Origin Frank-Wolfe (SOFW) method, which is a special case of the block-coordinate Frank-Wolfe algorithm, applied to the problem of finding equilibrium flow distributions. By significantly reducing the computational complexity of the minimization oracle, the method improves overall efficiency at the cost of increased memory consumption. Its key advantage lies in minimizing the number of shortest path computations. We refer to existing theoretical convergence guarantees for generalized coordinate Frank-Wolfe methods and, in addition, extend the analysis by providing a convergence proof for a batched version of the Block-Coordinate Frank-Wolfe algorithm, which was not covered in the original work. We also demonstrate the practical effectiveness of our approach through experimental results. In particular, our findings show that the proposed method significantly outperforms the classical Frank-Wolfe algorithm and its variants on large-scale datasets. On smaller datasets, SOFW also remains effective, though the performance gap relative to classical methods becomes less pronounced. In such cases, there is a trade-off between solution quality, iteration time complexity, and memory usage.

math.OC↗
arXiv2017

Modularity of complex networks models

Modularity is designed to measure the strength of division of a network into clusters (known also as communities). Networks with high modularity have dense connections between the vertices within clusters but sparse connections between vertices of different clusters. As a result, modularity is often used in optimization methods for detecting community structure in networks, and so it is an important graph parameter from a practical point of view. Unfortunately, many existing non-spatial models of complex networks do not generate graphs with high modularity; on the other hand, spatial models naturally create clusters. We investigate this phenomenon by considering a few examples from both sub-classes. We prove precise theoretical results for the classical model of random d-regular graphs as well as the preferential attachment model, and contrast these results with the ones for the spatial preferential attachment (SPA) model that is a model for complex networks in which vertices are embedded in a metric space, and each vertex has a sphere of influence whose size increases if the vertex gains an in-link, and otherwise decreases with time. The results obtained in this paper can be used for developing statistical tests for models selection and to measure statistical significance of clusters observed in complex networks.

math.PR↗
arXiv2017

PageRank in Undirected Random Graphs

PageRank has numerous applications in information retrieval, reputation systems, machine learning, and graph partitioning. In this paper, we study PageRank in undirected random graphs with an expansion property. The Chung-Lu random graph is an example of such a graph. We show that in the limit, as the size of the graph goes to infinity, PageR- ank can be approximated by a mixture of the restart distribution and the vertex degree distribution. We also extend the result to Stochastic Block Model (SBM) graphs, where we show that there is a correction term that depends on the community partitioning.

math.PR↗
arXiv2016

PageRank in undirected random graphs

PageRank has numerous applications in information retrieval, reputation systems, machine learning, and graph partitioning.In this paper, we study PageRank in undirected random graphs with expansion property. The Chung-Lu random graph representsan example of such graphs. We show that in the limit, as the size of the graph goes to infinity, PageRank can be represented by a mixture of the restart distribution and the vertex degree distribution.

cs.SI↗
arXiv2016

On the stability of the Erdős-Ko-Rado theorem

Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a random analogue of the Erdős-Ko-Rado theorem.

math.CO↗
arXiv2013

New bounds for the distance Ramsey number

In this paper we study the distance Ramsey number $R_{\it D}(s,t,d)$. The \textit{distance Ramsey number} $R_{\it D}(s,t,d) $ is the minimum number $n$ such that for any graph $ G $ on $ n $ vertices, either $G$ contains an induced $ s $-vertex subgraph isomorphic to a distance graph in $ \Real^d $ or $ \bar {G} $ contains an induced $ t $-vertex subgraph isomorphic to the distance graph in $ \Real^d $. We obtain the upper and lower bounds on $R_{\it D}(s,s,d),$ which are similar to the bounds for the classical Ramsey number $R(\lceil \frac{s}{[d/2]} \rceil, \lceil \frac{s}{[d/2]} \rceil)$.

math.CO↗
arXiv2012

Empirical Validation of the Buckley--Osthus Model for the Web Host Graph: Degree and Edge Distributions

There has been a lot of research on random graph models for large real-world networks such as those formed by hyperlinks between web pages in the world wide web. Though largely successful qualitatively in capturing their key properties, such models may lack important quantitative characteristics of Internet graphs. While preferential attachment random graph models were shown to be capable of reflecting the degree distribution of the webgraph, their ability to reflect certain aspects of the edge distribution was not yet well studied. In this paper, we consider the Buckley--Osthus implementation of preferential attachment and its ability to model the web host graph in two aspects. One is the degree distribution that we observe to follow the power law, as often being the case for real-world graphs. Another one is the two-dimensional edge distribution, the number of edges between vertices of given degrees. We fit a single "initial attractiveness" parameter $a$ of the model, first with respect to the degree distribution of the web host graph, and then, absolutely independently, with respect to the edge distribution. Surprisingly, the values of $a$ we obtain turn out to be nearly the same. Therefore the same model with the same value of the parameter $a$ fits very well the two independent and basic aspects of the web host graph. In addition, we demonstrate that other models completely lack the asymptotic behavior of the edge distribution of the web host graph, even when accurately capturing the degree distribution. To the best of our knowledge, this is the first attempt for a real graph of Internet to describe the distribution of edges between vertices with respect to their degrees.

cs.SI↗
arXiv2012

Distance graphs having large chromatic numbers and not containing cliques or cycles of given size

In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.

math.CO↗
arXiv2010

Counterexamples to Borsuk's conjecture on spheres of small radii

In this work, the classical Borsuk conjecture is discussed, which states that any set of diameter 1 in the Euclidean space $ {\mathbb R}^d $ can be divided into $ d+1 $ parts of smaller diameter. During the last two decades, many counterexamples to the conjecture have been proposed in high dimensions. However, all of them are sets of diameter 1 that lie on spheres whose radii are close to the value $ {1}{\sqrt{2}} $. The main result of this paper is as follows: {\it for any $ r > {1}{2} $, there exists a $ d_0 $ such that for all $ d \ge d_0 $, a counterexample to Borsuk's conjecture can be found on a sphere $ S_r^{d-1} \subset {\mathbb R}^d $.

math.CO↗