Related papers: The phase transition in site percolation on pseudo…
Attach to each edge of the complete graph on $n$ vertices, i.i.d. exponential random variables with mean $n$. Aldous [1] proved that the longest path with average weight below $p$ undergoes a phase transition at $p=\frac{1}{e}$: it is…
We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for…
We introduce a bond percolation procedure on a $D$-dimensional lattice where two neighbouring sites are connected by $N$ channels, each operated by valves at both ends. Out of a total of $N$, randomly chosen $n$ valves are open at every…
We study site and bond percolation on directed simple random graphs with a given degree distribution and derive the expressions for the critical value of the percolation probability above which the giant strongly connected component emerges…
We consider a one-dimensional network in which the nodes at Euclidean distance $l$ can have long range connections with a probabilty $P(l) \sim l^{-\delta}$ in addition to nearest neighbour connections. This system has been shown to exhibit…
Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1…
We consider classes of pseudo-random graphs on $n$ vertices for which the degree of every vertex and the co-degree between every pair of vertices are in the intervals $(np - Cn^\delta,np+Cn^\delta)$ and $(np^2- C n^\delta, np^2 +C…
We consider the contact process on a dynamic graph defined as a random $d$-regular graph with a stationary edge-switching dynamics. In this graph dynamics, independently of the contact process state, each pair $\{e_1,e_2\}$ of edges of the…
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase…
It is well-known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant (i.e., order $\Omega(n)$)…
We characterize the size of the largest common induced subgraph of two independent random uniform $d$-hypergraphs of different sizes with $d\geq 3$. More precisely, its distribution is asymptotically concentrated on two points, and we…
For $d\ge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of…
In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…
We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue…
Consider the following model of strong-majority bootstrap percolation on a graph. Let r be some positive integer, and p in [0,1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every…
Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of…
Consider a 2-dimensional soft random geometric graph $G(\lambda,s,\phi)$, obtained by placing a Poisson($\lambda s^2$) number of vertices uniformly at random in a square of side $s$, with edges placed between each pair $x,y$ of vertices…
Consider the interchange process on a connected graph $G=(V,E)$ on $n$ vertices. I.e.\ shuffle a deck of cards by first placing one card at each vertex of $G$ in a fixed order and then at each tick of the clock, picking an edge uniformly at…
We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical…
We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for…