Related papers: Average-weight percolation on the complete graph
We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We…
In this paper we consider first-passage percolation on certain 1-dimensional periodic graphs, such as the $\Z\times\{0,1,\ldots,K-1\}^{d-1}$ nearest neighbour graph for $d,K\geq1$. We find that both length and weight of minimal-weight paths…
We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $\lambda…
The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erd\"os-R\'enyi process. It is well known that this process undergoes a phase transition at n/2 edges when,…
We consider the infinite directed graph with vertices the set of integers ...,-2,-1,0,1,2,... . Let v be a random variable taking either finite values or value "minus infinity". Consider random weights v(j,k), indexed by pairs (j,k) of…
We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…
Analytical results are presented for the structure of networks that evolve via a preferential-attachment-random-deletion (PARD) model in the regime of overall network growth and in the regime of overall contraction. The phase transition…
Archdeacon and Grable (1995) proved that the genus of the random graph $G\in\mathcal{G}_{n,p}$ is almost surely close to $pn^2/12$ if $p=p(n)\geq3(\ln n)^2n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in…
Understanding what types of phenomena lead to discontinuous phase transitions in the connectivity of random networks is an outstanding challenge. Here we show that a simple stochastic model of graph evolution leads to a discontinuous…
Let $E$ be the set of edges of the $d$-dimensional cubic lattice $\mathbb{Z}^d$, with $d\geq2$, and let $t(e),e\in E$, be nonnegative values. The passage time from a vertex $v$ to a vertex $w$ is defined as $\inf_{\pi:v\rightarrow…
We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-\alpha}$ with $\alpha>0$.…
We investigate the statistics of extremal path(s) (both the shortest and the longest) from the root to the bottom of a Cayley tree. The lengths of the edges are assumed to be independent identically distributed random variables drawn from a…
The famous Erd\H{o}s-Gallai Theorem on the Tur\'an number of paths states that every graph with $n$ vertices and $m$ edges contains a path with at least $\frac{2m}{n}$ edges. In this note, we first establish a simple but novel extension of…
We study paths of time-length $t$ of a continuous-time random walk on $\mathbb Z^2$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The…
We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erd\H{o}s--R\'enyi random graphs on $n$ vertices with a fixed average degree $c>1$.…
In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in the sense that $p_c<1$). As a corollary, we obtain that the critical point of…
We consider the following oriented percolation model of $\mathbb {N} \times \mathbb{Z}^d$: we equip $\mathbb {N}\times \mathbb{Z}^d$ with the edge set $\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}$, and we say that each edge…
In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time $\tilde{O}(n^{\frac{3 + \omega}{2}}) = \tilde{O}(n^{2.686})$. Here $n$ is the number…
Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We prove that the component structure of a graph chosen uniformly at random from the class $\mathcal{S}_g(n,m)$ of all graphs on vertex set $[n]=\{1,\dotsc,n\}$ with $m$ edges…
Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random…