Intersections of Randomly Embedded Sparse Graphs are Poisson

Intersections of Randomly Embedded Sparse Graphs are Poisson

Suppose that t ≥ 2 is an integer, and randomly label t graphs with the integers 1 . . . n. We give sufficient conditions for the number of edges common to all t of the labelings to be asymptotically Poisson as n → ∞. We show by example that our theorem is, in a sense, best possible. For Gn a sequence of graphs of bounded degree, each having at most n vertices, Tomescu [7] has shown that the num...

On randomly colouring locally sparse graphs

We consider the problem of generating a random q-colouring of a graph G = (V, E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph Hv induced by the neighbours of v ∈ V is ∆ where ∆ is the maximum degree and ∆ > c1 ln n then for sufficiently large c1, this chain mixes rapidly provided q/∆ > α, where α ≈ 1.763 is the root of α = e. F...

On randomly olouring lo ally sparse graphs

Traveling in Randomly Embedded Random Graphs

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the BeardwoodHalton-Hammersl...

Are randomly grown graphs really random?

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phas...