Related papers: Average-weight percolation on the complete graph
For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…
We prove that if $(G_n)_{n\geq1}=((V_n,E_n))_{n\geq 1}$ is a sequence of finite, vertex-transitive graphs with bounded degrees and $|V_n|\to\infty$ that is at least $(1+\epsilon)$-dimensional for some $\epsilon>0$ in the sense that…
The mean weight of a cycle in an edge-weighted graph is the sum of the cycle's edge weights divided by the cycle's length. We study the minimum mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge weights drawn…
We consider first passage percolation on the Erd\H{o}s--R\'{e}nyi graph with $n$ vertices in which each pair of distinct vertices is connected independently by an edge with probability $\lambda/n$ for some $\lambda>1$. The edges of the…
If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with…
The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0, 1\}^n$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned…
We consider the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Given $p>1$ and an initial opinion profile $f_0:V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v=v_t$ is selected, and the opinion…
We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\mathbf{n},P)$ every vertex has one of two types, the…
In this paper we explore first passage percolation (FPP) on the Erd\H{o}s-R\'enyi random graph $G_n(p_n)$, where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when $np_n\to \lambda>1,$ we…
We study in this paper, the first passage percolation on a random graph model, the configuration model. We first introduce, the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two…
The generalised random graph contains $n$ vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments and…
We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(\log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise…
We study the mean time for a random walk to traverse between two arbitrary sites of the Erdos-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as…
The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of…
Consider a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$. The weight of the shortest (minimum-weight) path $P_1$ between two given vertices is known to be $\ln n / n$, asymptotically. Define…
In 1999, Zhang proved that, for first passage percolation on the square lattice $\mathbb{Z}^2$ with i.i.d. non-negative edge weights, if the probability that the passage time distribution of an edge $P(t_e = 0) =1/2 $, the critical value…
The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $\rho/n$ for some $\rho>0$ constant. In this paper we add i.i.d. exponential edge weights…
Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =\Omega(d^{\lfloor…
We prove non-universality results for first-passage percolation on the configuration model with i.i.d. degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of…
A Poisson point process of unit intensity is placed in the square $[0,n]^2$. An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process…