A random geometric graph, @math , is formed by choosing @math points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most @math . For a given constant @math , we show that @math is a distance threshold function for @math to have a connected subgraph on @math points. Based on this, we show that @math is a distance threshold for @math to be plane, and @math is a distance threshold to be planar. We also investigate distance thresholds for @math to have a non-crossing edge, a clique of a given size, and an independent set of a given size.
An instance of Erd " o s-R ' e nyi graph @cite_1 is obtained by taking @math vertices and connecting any two with probability @math , independently of all other pairs; the graph derived by this scheme is denoted by @math . In @math the threshold is expressed by the edge existence probability @math , while in @math the threshold is expressed in terms of @math . In both random graphs and random geometric graphs, property thresholds are of great interest @cite_19 @cite_10 @cite_11 @cite_7 @cite_13 . Note that edge crossing configurations in @math have a geometric nature, and as such, have no analogues in the context of the Erd " o s-R ' e nyi model for random graphs. However, planarity, and having a clique or an independent set of specific size are of interest in both @math and @math .
@cite_13 @cite_7 @cite_1 @cite_19 @cite_10 @cite_11
2949304946 2037549896 2128738155 2964292545
Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process. ean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0, 1] d . We present upper bounds on the threshold width, and show that our bound is sharp for d = 1 and at most a sublogarithmic factor away for d ≥ 2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs. We look at a model of random graphs suggested by Gilbert: given an integer @math and @math , scatter @math vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than @math apart.We consider the asymptotics when the metric space is the interval [0, 1], and @math is a function of @math , for @math . We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.) We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math .