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.
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 $n,m,p$ …
The largest component in an inhomogeneous random intersection graph with clustering
给定b>0，整数n，m=bn和{0，1，.，m}，考虑顶点集[n]={1，.上的随机交图，n}，其中，当S（i）和S（j）相交时，i和j被声明为相邻。这里S（1），.，[001 pdf 1st-31files][001 pdf 1st-31files][001 pdf 1st-31files]（|S（i）|=k）=Q（k）。对于稀疏随机交图，我们建立了最大连通分支N=n（1-Q（0））g+o（n）阶的概率一阶渐近性.这里g是一个相关的多类型泊松分支过程的非灭绝概率的平均值。
Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where ${\cal F}$ runs over all intersecting sets in $G$ and $G_v$ is a stabilizer of $v\in V$. In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be eit…
Nearly complete intersection ideals were introduced by A. Boocher and J. Seiner (2018) and defines a special class of monomial ideals in a polynomial ring. These ideals were used to give a lower bound of the total sum of betti numbers that appear a minimal free resolution of a monomial ideal. In this note we give a graph theoretic classification of nearly complete intersection ideals generated in degree two. In doing so, we define a novel graph operation (the inversion) that is motivated by the definition of this new class of ideals.
Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and ${\cal F}$ runs over all intersecting sets in $G$. Intersection densities of transitive groups of degree $pq$, where $p>q$ are odd primes, is considered. In particular, t…
On the intersection distribution of degree three polynomials and related topics
最近Li和Pott（arXiv：2003.06678v1）提出了多项式$f$在有限域$\mathbb{F}_q$上的交集分布，它涉及多项式集合$\{f(x)+cx \mid c \in \mathbb{F}_q\}$的集体行为.相交分布具有潜在的几何解释，其指示$f$的图形与仿射平面$AG(2,q)$中的线之间的相交图案。当$q$是偶数时，长期存在的分类o-多项式的公开问题可以用简单的方式重新表述，即分类所有具有与$x^2$相同的交集分布的多项式。受这种联系的启发，我们继续考虑下一个最简单的情况，并推导出$\mathbb{F}_q$上所有三次多项式的交集分布，其中$q$为奇数和偶数。此外，我们首次将所有具有相同交集分布的单项式分类为$x^3$，得到了这类单项式的一些特征，并提出了一个猜想。此外，还给出了三次多项式的交分布的两个应用。第一个是构造非同构的Steiner三元系，第二个是在仿射平面上产生具有先前未知大小的Kakeya集的无限族。
A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $ω\in Ω$ such that $ω^g = ω^h$. The \emph{intersection density} $ρ(G)$ of $G$ is the maximum of $\left\{ \frac{|\mathcal{F}|}{|G_ω|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}$, where $G_ω$ is the stabilizer of $ω$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $ρ(G) = 1$. Moreover, if $G$ is primitive of de…
Recently, we have witnessed tremendous applications of algebraic intersection theory to branches of mathematics, that previously seemed very distant. In this article we review some of them. Our aim is to provide a unified approach to the results e.g. in the theory of chromatic polynomials (work of Adiprasito, Huh, Katz), maximum likelihood degree in algebraic statistics (Drton, Manivel, Monin, Sturmfels, Uhler, Wiśniewski), Euler characteristics of determinental varieties (Dimca, Papadima), characteristic numbers (Aluffi, Schubert, Vakil) and the degree of semidefinite programming (Bothmer, Ni…
