Mindaugas Bloznelis
We establish the asymptotic degree distribution of the typical vertex of inhomogeneous and passive random intersection graphs under the minimal moment conditions.
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Vertex Degree of Random Intersection Graph
通过独立地将给定对象集$W,$的子集分配给单图$G.$的顶点集$V$的每个顶点来构造随机交图。在$V,$的两个顶点之间存在边当且仅当它们各自的子集（在$W$中）具有至少一个公共元素。导出了顶点集$V,$中任意两个顶点连通的强阈值。并给出了图$G.$的典型顶点的度的几乎必然概率界
A note on the intersections of two random walks in two dimensions
本文证明了二维中两个独立随机游走值域交集的大偏差结果，补充了Phetpradap关于三维及以上维度的研究。
Uniform probability distributions on $\ell_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a simplex and a sphere. Certain new and interesting features, such as phase transitions and localization phenomena emerge.
A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in which each vertex is given a random weight, and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be so as well. Furthermore, an as…
We study a variant of the standard random intersection graph model ($G(n,m,F,H)$) in which random weights are assigned to both vertex types in the bipartite structure. Under certain assumptions on the distributions of these weights, the degree of a vertex is shown to depend on the weight of that particular vertex and on the distribution of the weights of the other vertex type.
We show that the multi-species higher spin stochastic vertex model, also called the U_q(A_n^{(1)}) vertex model, satisfies a duality where the indicator function has the form {η^x_{[1,n]} \geq ξ^x_{[1,n]} }. In other words, for every particle in the ξconfiguration of species i at vertex x, there must be a particle of species j > i at vertex x in the ηconfiguration. The proof follows by applying charge reversal to previously discovered duality functions, which also results in open boundary conditions. As a corollary, we recover the duality for the stochastic six vertex model recently found by Y…
A note on the component structure in random intersection graphs with tunable clustering
我们研究了可调聚类的随机交图中的组件结构，并证明了平均度可以作为最大组件大小的相变阈值。也就是说，如果期望度小于1，则最大分量的大小为a.a.s.。对数级，但如果平均度数大于1，a.a.s.出现线性级的单个大分量，并且第二大分量的大小至多为对数级。
Random $s$-intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random $s$-intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least $s$ items (the graph is called a random intersection graph when $s=1$). A special case of particular interest is a uniform random $s$-intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another im…
Random bipartite graphs with i.i.d. weights and applications to inhomogeneous random intersection graphs
本文构建带独立同分布权重的随机二部图并导出非齐次随机交图，刻画了中等聚类临界区中大连通分量的三类标度极限。
We consider the typical distance between vertices of the giant component of a random intersection graph having a power law (asymptotic) vertex degree distribution with infinite second moment. Given two vertices from the giant component we construct O(log log n) upper bound (in probability) for the length of the shortest path connecting them.
Number of Edges in Random Intersection Graph on Surface of a Sphere
在这篇文章中，我们认为“$N$”球冠面积$4πp$均匀分布在一个单位球的表面上。我们研究了由这些帽构造的随机交图$G_N$。证明了对于$p = \frac{c}{N^{\al}},\:c >0$和$\al >2,$，图$G_N$的边数服从Poisson分布.我们还得到了关于$G_N$中孤立点个数的强律结果：对于$p = \frac{c}{N^{\al}},\:c >0$对于$\al < 1,$，几乎必然在$G_N$中没有孤立点，即，在$G_N$中至少存在$N/2$边并且对于$\al >3,$，$G_N$中的每个顶点是孤立的即，边集合$\cE_N.$中没有边
We establish asymptotic vertex degree distribution and examine its relation to the clustering coefficient in two popular random intersection graph models of Godehardt and Jaworski [Electron. Notes Discrete Math. 10 (2001) 129-132]. For sparse graphs with a positive clustering coefficient, we examine statistical dependence between the (local) clustering coefficient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of Foudalis et al. [In Algorithms and Models for Web Graph (2011) Springer] that, "clustering correlates negatively with degr…
We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph.
A note on critical intersections of classical and Schatten $p$-balls
本文研究了 $\mathbb{R}^n$ 中两个范数单位球相交的渐近体积，提供了通用框架并特化至经典 $\ell_p^n$-球（$p=\infty$）与 Schatten $p$-球（$p=2,\infty$）情形，揭示了非常规的极限行为。
Random $s$-intersection graphs have recently received considerable attention in a wide range of application areas. In such a graph, each vertex is equipped with a set of items in some random manner, and any two vertices establish an undirected edge in between if and only if they have at least $s$ common items. In particular, in a uniform random $s$-intersection graph, each vertex independently selects a fixed number of items uniformly at random from a common item pool, while in a binomial random $s$-intersection graph, each item in some item pool is independently attached to each vertex with t…
We give a simple proof for recurrence of vertex reinforced jump process on \(\mathbb{Z}^d\), under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on \ \(\mathbb{Z}^d\) is {recurrent} for strong reinforcement. Finally, we prove that the \(H^{(2|2)}\) model on \(\mathbb{Z}^d\) localizes at strong disorder. Even though these results are well-known, we propose a unified approach, {which also has the advantage to provide shorter proofs}, and relies on estimating fractional moments, introduced by Aizenman and Molchanov.
A note on random walks in a hypercube
研究了n维超立方体上的简单随机游动。对于任何起始位置，只要a和b共享同一条边，我们就可以找到在到达顶点b之前到达顶点a的概率。这推广了Doyle，P.，和Snell，J.，“随机游动和电网络”，美国数学协会，1984年（见练习1.3.7）。
Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m}, consider the random intersection graph on the vertex set [n]={1, ..., n}, where i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ..., S(n) denote iid random subsets of [m] such that P(|S(i)|=k)=Q(k). For sparse random intersection graphs we establish a first order asymptotic for the order of the largest connected component N=n(1-Q(0))g+o(n) in probability. Here g is an average of nonextinction probabilities of a related multi-type Poisson branching process.
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 instan…
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