Searcharxiv⌕ Search

arXiv subjects

Mindaugas Bloznelis

Publications and source records attributed to Mindaugas Bloznelis.

At least 19 recordsLinked to original sources
arXiv2026

On the strength of connectedness of unions of random graphs

Let $G_1,\dots, G_m$ be independent identically distributed random subgraphs of the complete graph ${\cal K}_n$. We analyse the threshold behaviour of the strength of connectedness of the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Let $a=\min\{t\ge 1:\, {\bf P}\{δ(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $λ(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} {\bf E} X$, where $X$ stands for the number of non isolated vertices of $G_1$. Letting $n,m\to+\infty$ we show that ${\bf P}\{\cup_{i=1}^mG_i$ is $a(k+1)$-connected$\} \to 1 $ for $λ(k)\to -\infty$, and ${\bf P}\{\cup_{i=1}^mG_i$ is $ak+1$-connected$\} \to 0 $ for $λ(k)\to +\infty$. In particular, the connectivity strength of the union graph $\cup_{i=1}^mG_i$ increases in steps of size $a$. Our results are obtained in a more general setting where the contributing random subgraphs do not need to be identically distributed.

math.CO↗
arXiv2025

k-connectivity threshold for superpositions of Bernoulli random graphs

Let $G_1,\dots, G_m$ be independent identically distributed Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having vertex sets of random sizes $X_1,\dots, X_m\in \{0,1,2,\dots\}$ and random edge densities $Q_1,\dots, Q_m\in [0,1]$. Assuming that each $G_i$ has a vertex of degree $1$ with positive probability, we establish the $k$-connectivity threshold as $n,m\to+\infty$ for the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$.

math.PR↗
arXiv2024

Two models of sparse and clustered dynamic networks

We present two models of sparse dynamic networks that display transitivity - the tendency for vertices sharing a common neighbour to be neighbours of one another. Our first network is a continuous time Markov chain $G=\{G_t=(V,E_t), t\ge 0\}$ whose states are graphs with the common vertex set $V=\{1,\dots, n\}$. The transitions are defined as follows. Given $t$, the vertex pairs $\{i,j\}\subset V$ are assigned independent exponential waiting times $A_{ij}$. At time $t+\min_{ij} A_{ij}$ the pair $\{i_0,j_0\}$ with $A_{i_0j_0}=\min_{ij} A_{ij}$ toggles its adjacency status. To mimic clustering patterns of sparse real networks we set intensities $a_{ij}$ of exponential times $A_{ij}$ to be negatively correlated with the degrees of the common neighbours of vertices $i$ and $j$ in $G_t$. Another dynamic network is based on a latent Markov chain $H=\{H_t=(V\cup W, E_t), t\ge 0\}$ whose states are bipartite graphs with the bipartition $V\cup W$, where $W=\{1,\dots,m\}$ is an auxiliary set of attributes/affiliations. Our second network $G'=\{G'_t =(E'_t,V), t\ge 0\}$ is the affiliation network defined by $H$: vertices $i_1,i_2\in V$ are adjacent in $G'_t$ whenever $i_1$ and $i_2$ have a common neighbour in $H_t$. We analyze geometric properties of both dynamic networks at stationarity and show that networks possess high clustering. They admit tunable degree distribution and clustering coefficients.

math.PR↗
arXiv2023

Connectivity threshold for superpositions of Bernoulli random graphs. II

Let $G_1,\dots, G_m$ be independent Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having variable sizes $X_1,\dots, X_m\in \{0,1,2,\dots\}$ and densities $Q_1,\dots, Q_m\in [0,1]$. Letting $n,m\to+\infty$ we establish the connectivity threshold for the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Assuming that $(X_1,Q_1), (X_2,Q_2),\dots, (X_m,Q_m)$ are independent identically distributed bivariate random variables and $\ln n -\frac{m}{n}E\bigl(X_1(1-(1-Q_1)^{|X_1-1|}\bigr)\to c$ we show that $P\{\cup_{i=1}^mG_i$ is connected$\}\to e^{-e^c}$.The result extends to the case of non-identically distributed random variables $(X_1,Q_1),\dots, (X_m,Q_m)$ as well.

math.PR↗
arXiv2023

Connectivity threshold for superpositions of Bernoulli random graphs

Let $G_1,\dots, G_m$ be independent Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having variable sizes $x_1,\dots, x_m\in [n]$ and densities $q_1,\dots, q_m\in [0,1]$. Letting $n,m\to+\infty$, we study the connectivity threshold for the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Assuming that the empirical distribution $P_{n,m}$ of the pairs $(x_1,q_1),\dots, (x_m,q_m)$ converges to a probability distribution $P$ we show that the threshold is defined by the mixed moments $κ_n=\iint x(1-(1-q)^{|x-1|})P_{n,m}(dx,dq)$. For $\ln n-\frac{m}{n}κ_n\to-\infty$ we have $P\{\cup_{i=1}^mG_i$ is connected$\}\to 1$ and for $\ln n-\frac{m}{n}κ_n\to+\infty$ we have $P\{\cup_{i=1}^mG_i$ is connected$\}\to 0$. Interestingly, this dichotomy only holds if the mixed moment $\iint x(1-(1-q)^{|x-1|})\ln(1+x)P(dx,dq)<\infty$.

math.PR↗
arXiv2022

Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

The clustering property of complex networks indicates the abundance of small dense subgraphs in otherwise sparse networks. For a community-affiliation network defined by a superposition of Bernoulli random graphs, which has a nonvanishing global clustering coefficient and a power-law degree distribution, we establish normal and $α$--stable approximations to the number of small cliques, cycles and more general $2$-connected subgraphs.

math.PR↗
arXiv2021

Assortativity and bidegree distributions on Bernoulli random graph superpositions

A probabilistic generative network model with $n$ nodes and $m$ overlapping layers is obtained as a superposition of $m$ mutually independent Bernoulli random graphs of varying size and strength. When $n$ and $m$ are large and of the same order of magnitude, the model admits a sparse limiting regime with a tunable power-law degree distribution and nonvanishing clustering coefficient. In this article we prove an asymptotic formula for the joint degree distribution of adjacent nodes. This yields a simple analytical formula for the model assortativity, and opens up ways to analyze rank correlation coefficients suitable for random graphs with heavy-tailed degree distributions. We also study the effects of power laws on the asymptotic joint degree distributions.

math.PR↗
arXiv2021

Edgeworth approximations for distributions of symmetric statistics

We study the distribution of a general class of asymptoticallylinear statistics which are symmetric functions of $N$ independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order $o(N^{-1})$. The Edgeworth expansion is based on Hoeffding's decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cramér's condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.

math.ST↗
arXiv2020

Clustering and percolation on superpositions of Bernoulli random graphs

A simple but powerful network model with $n$ nodes and $m$ partly overlapping layers is generated as an overlay of independent random graphs $G_1,\dots,G_m$ with variable sizes and densities. The model is parameterised by a joint distribution $P_n$ of layer sizes and densities. When $m$ grows linearly and $P_n \to P$ as $n \to \infty$, the model generates sparse random graphs with a rich statistical structure, admitting a nonvanishing clustering coefficient together with a limiting degree distribution and clustering spectrum with tunable power-law exponents. Remarkably, the model admits parameter regimes in which bond percolation exhibits two phase transitions: the first related to the emergence of a giant connected component, and the second to the appearance of gigantic single-layer components.

math.PR↗
arXiv2020

The cover time of a sparse random intersection graph

Many known networks have structure of affiliation networks, where each of $n$ network's nodes (actors) selects an attribute set from a given collection of $m$ attributes and two nodes (actors) establish adjacency relation whenever they share a common attribute. We study behaviour of the random walk on such networks. For that purpose we use commonly used model of such networks -- random intersection graph. We establish the cover time of the simple random walk on the binomial random intersection graph ${\cal G}(n,m,p)$ at the connectivity threshold and above it. We consider the range of $n,m,p$ where the typical attribute is shared by (stochastically) bounded number of actors.

math.CO↗
arXiv2019

Local probabilities of randomly stopped sums of power law lattice random variables

Let $X_1$ and $N\ge 0$ be integer valued power law random variables. For a randomly stopped sum $S_N=X_1+\cdots+X_N$ of independent and identically distributed copies of $X_1$ we establish a first order asymptotics of the local probabilities $P(S_N=t)$ as $t\to+\infty$. Using this result we show the $k^{-δ}$, $0\le δ\le 1$ scaling of the local clustering coefficient (of a randomly selected vertex of degree $k$) in a power law affiliation network.

math.PR↗
arXiv2018

Correlation between clustering and degree in affiliation networks

We are interested in the probability that two randomly selected neighbors of a random vertex of degree (at least) $k$ are adjacent. We evaluate this probability for a power law random intersection graph, where each vertex is prescribed a collection of attributes and two vertices are adjacent whenever they share a common attribute. We show that the probability obeys the scaling $k^{-δ}$ as $k\to+\infty$. Our results are mathematically rigorous. The parameter $0\le δ\le 1$ is determined by the tail indices of power law random weights defining the links between vertices and attributes.

cs.SI↗
arXiv2017

Degree-degree distribution in a power law random intersection graph with clustering

The bivariate distribution of degrees of adjacent vertices (degree-degree distribution) is an important network characteristic defining the statistical dependencies between degrees of adjacent vertices. We show the asymptotic degree-degree distribution of a sparse inhomogeneous random intersection graph and discuss its relation to the clustering and power law properties of the graph.

math.PR↗
arXiv2016

Diclique clustering in a directed random graph

We discuss a notion of clustering for directed graphs, which describes how likely two followers of a node are to follow a common target. The associated network motifs, called dicliques or bi-fans, have been found to be key structural components in various real-world networks. We introduce a two-mode statistical network model consisting of actors and auxiliary attributes, where an actor i decides to follow an actor j whenever i demands an attribute supplied by j. We show that the digraph admits nontrivial clustering properties of the aforementioned type, as well as power-law indegree and outdegree distributions.

cs.SI↗
arXiv2014

Preferred attachment model of affiliation network

In an affiliation network vertices are linked to attributes and two vertices are declared adjacent whenever they share a common attribute. For example, two customers of an internet shop are called adjacent if they have purchased the same or similar items. Assuming that each newly arrived customer is linked preferentially to already popular items we obtain a preferred attachment model of an evolving affiliation network. We show that the network has a scale-free property and establish the asymptotic degree distribution.

physics.soc-ph↗
arXiv2013

Large cliques in sparse random intersection graphs

Given positive integers n and m, and a probability measure P on {0, 1, ..., m} the random intersection graph G(n,m,P) on vertex set V = {1,2, ..., n} and with attribute set W = {w_1, w_2, ..., w_m} is defined as follows. Let S_1, S_2, ..., S_n be independent random subsets of W such that for any v \in V and any S \subseteq W we have \pr(S_v = S) = P(|S|) / \binom (m, |S|). The edge set of G(n,m,P) consists of those pairs {u,v} V for which S_u and S_v intersect. We study the asymptotic order of the clique number ω(G(n,m,P)) in random intersection graphs with bounded expected degrees. For instance, in the case m = Θ(n) we show that if the vertex degree distribution is power-law with exponent α\in (1;2), then the maximum clique is of a polynomial size, while if the variance of the degrees is bounded, then the maximum clique has (ln n)/(ln ln n) (1 + o_P(1)) vertices whp. In each case there is a polynomial algorithm which finds a clique of size ω(G(n,m,P)) (1-o_P(1)).

math.CO↗