The existence of (shortest-path) interval routing schemes for random graphs that use at most one interval label per edge is an open problem posed in [8]. In this paper, we show that for any random graph G(n, p) with edge probability p > 0.765, there exists an interval routing scheme that uses at most one label per edge and has an additive stretch 1. In doing so, we provide an interesting constr...

Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a k-bounded colouring. We say that a subset of the edges of G is multicoloured if each edge is of a different colour. We say that the colouring is H-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ARk = {G : every k-bounded colouring of G is H-good}...

We study the concentration of the largest induced paths, trees and cycles (holes) in the Erdos-Renyi random graph model and prove a 2-point concentration for the size of the largest induced path and hole, for all p = Ω(n ln n). As a corollary, we obtain an improvement over a result of Erdos and Palka concerning the size of the largest induced tree in a random graph. Further, we study the path c...

Betweenness centrality is a quantity that is frequently used to measure how ‘central’ a vertex v is. It is defined as the sum, over pairs of vertices other than v, of the proportions of shortest paths that pass through v. In this paper, we study the distribution of the betweenness centrality in random trees and related, subcritical graph families. Specifically, we prove that the betweenness cen...

We study the Hamilton cycle Maker-Breaker game, played on the edges of the random graph G(n, p). We prove a conjecture from [13], asserting that the property that Maker is able to win this game, has a sharp threshold at log n n . Our theorem can be considered a game-theoretic strengthening of classical results from the theory of random graphs: not only does G(n, p) almost surely admit a Hamilto...

We examine how three different communication processes operating through social networks are affected by homophily – the tendency of individuals to associate with others similar to themselves. Homophily has no effect if messages are broadcast or sent via shortest paths; only network connection density matters. In contrast, homophily substantially slows learning based on repeated averaging of ne...

This paper proposes a new random walk strategy that minimizes the variance of the estimate using statistical estimations of local and global features of the scene. Based on the local and global properties, the algorithm decides at each point whether a Russian-roulette like random termination is worth performing, or on the contrary, we should split the path into several child paths. In this sens...

Claim: E(time to insert x) ≤ O(1). Proof: A cuckoo graph has m vertices (one per cell of A) and n edges (since for each x, we connect g(x) to h(x)). Consider the path we get from an insertion of x. We could get a simple path, a single cycle, or a double cycle. Let us define the following random variables: T , the runtime; Pk, the indicator random variable of a path being at least length k; Ck, ...

A popular Bayesian nonparametric approach to survival analysis consists in modeling hazard rates as kernel mixtures driven by a completely random measure. In this paper we derive asymptotic results for linear and quadratic functionals of such random hazard rates. In particular, we prove central limit theorems for the cumulative hazard function and for the path–second moment and path–variance of...

The family of random regular graphs is a classic topic in the realms of graph theory, combinatorics and computer science. In this paper we study the problem of learning random regular graphs from random paths. A random regular graph is generated uniformly at random and in a standard label-guided graph exploration setting, the edges incident from a node in the graph have distinct local labels. T...