Dynamic geometric graph processes: adjacency operator approach

International audienceThe $d$-dimensional unit cube $[0; 1]^d$ is discretized to create a collection $V$ of vertices used to dene geometric graphs. Each subset of $V$ is uniquely associated with a geometric graph. Dening a dynamic random walk on the subsets of $V$ induces a walk on the collection of geometric graphs in the discretized cube. These walks naturally model addition-deletion networks and can be visualized as walks on hypercubes with loops. Adjacency operators are constructed using subalgebras of Cliord algebras and are used to recover information about the cycle structure and connected components of the $n$ graph of a sequence

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

Recognizing Partial Cubes in Quadratic Time

We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time solutions.Comment: 25 pages, five figures. This version significantly expands previous versions, including a new report on an implementation of the algorithm and experiments with i