Related papers: Giant component of the soft random geometric graph
We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using…
We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random…
Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different…
We study the intersection of a random geometric graph with an Erd\H{o}s-R\'enyi graph. Specifically, we generate the random geometric graph $G(n, r)$ by choosing $n$ points uniformly at random from $D=[0, 1]^2$ and joining any two points…
The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while…
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a…
In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our…
Order the vertices of a directed random graph \math{v_1,...,v_n}; edge \math{(v_i,v_j)} for \math{i<j} exists independently with probability \math{p}. This random graph model is related to certain spreading processes on networks. We…
The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is…
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2…
In 1980, Ajtai, Komlos and Szemer{\'e}di defined "groupie": Let $G=(V,E)$ be a simple graph, $|V|=n$, $|E|=e$. For a vertex $v\in V$, let $r(v)$ denote the sum of the degrees of the vertices adjacent to $v$. We say $v\in V$ is a {\it…
We study the number of chords and the number of crossings in the largest component of a random chord diagram when the chords are sparsely crossing. This is equivalent to studying the number of vertices and the number of edges in the largest…
We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this…
Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability…
We introduce a new way to sample inhomogeneous random graphs designed to have a lot of flexibility in the assignment of the degree sequence and the individual edge probabilities while remaining tractable. To achieve this we run a Poisson…
We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a…
This paper provides an overview of results, concerning longest or heaviest paths, in the area of random directed graphs on the integers along with some extensions. We study first-order asymptotics of heaviest paths allowing weights both on…
In a geometric inhomogeneous random graph vertices are given by the points of a Poisson process and are equipped with independent weights following a heavy tailed distribution. Any pair of distinct vertices is independently forming an edge…
We study site percolation on a sequence of graphs $\{G_n\}_{n\geq1}$ on $n$ vertices where degree of each vertex is in the interval $(np -a_n, np+a_n)$ and the co-degree of every pair of vertices is at most ${n}p^2+ b_n$, where $p \in…
Resolving a conjecture of F\"uredi from 1988, we prove that with high probability, the random graph $G(n,1/2)$ admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most…