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
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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.) Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d, and connecting two points if their Euclidean 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 @math 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 give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability. In their seminal work which initiated random graph theory Erdos and Renyi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) = 0, 1 n denote the Hamming space endowed with the probability measure μp defined by μp( 1, 2, . . . , n) = pk · (1 − p)n−k, where k = 1 + 2 + · · · + n. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on 1, 2, . . . , n such that A is invariant under Γ. Theorem. For every symmetric monotone A, if μp(A) > then μq(A) > 1− for q = p+ c1 log(1 2 ) logn. (c1 is an absolute constant.)