Connectivity of Metric Random Graphs

For any given r ≥ 0, a random set of n points X1, . . . , Xn in the unit disc in the Euclidean plane induces a metric random graph G(n, r) with vertex set {1, . . . , n} and edge set { {i, j} : |Xi − Xj| ≤ r } where |Xi − Xj| denotes the Euclidean distance between the points Xi and Xj. Connectivity results analogous to the classical results of Erdös and Renyi forGn,p are shown to hold in this setting. In particular, if c is any real constant and r = rn = q 1 n ` logn + c + o(1) ́ then the number of isolated vertices in the graph G(n, r) asymptotically satisfies a Poisson law with mean e and, a fortiori, the probability that G(n, r) is connected tends to e −c . These results strengthen and expand on earlier results of Gupta and Kumar in this setting for the Euclidean case. It is also shown that these results continue to hold in general if the Euclidean norm is replaced by an `-norm (1 ≤ p ≤ ∞). Analogous results are shown in one dimension and extensions to higher dimensions sketched. 1 Metric Random Graphs For x ∈ R write |x| for the usual Euclidean norm of x. Let S = { x ∈ R : |x| ≤ 1 } denote the unit radius disc in the Euclidean plane (centred at the origin for convenience). Suppose X1, . . . , Xn is a random sequence of points drawn by independent sampling from the uniform distribution on S. For a given r > 0we drop edges between any pair of points Xi and Xj within a distance r of each other. The points Xi then induce a “metric” random graph G(n, r)with vertex set {1, . . . , n} indexed by the Xi and edge set { {i, j} : |Xi−Xj| ≤ r } . If {i, j} is an edge of the graph we say that the vertices i and j are adjacent or communicating in G(n, r). A vertex i of the random graph G(n, r) is isolated if there are no vertices adjacent to it. WriteN0 for the number of isolated vertices. Our main result asserts a sharp limit theorem for N0 in a suitable range. THEOREM 1 (POISSON LAW FOR ISOLATED VERTICES) Let c be any fixed real number and suppose r = rn varies with n such that rn = √ 1 n ( logn+ c+ o(1) ) as n→∞. Then the number of vertices of G(n, rn) that are isolated asymptotically has a Poisson distribution with mean e. More precisely, for every fixed non-negative integerm, P{N0 = m}→ e−e−c(e−c)m/m! as n→∞. The graph G(n, r) turns out to be almost surely comprised of isolated nodes and a single giant component. A sharp threshold function for connectivity analogous to the classical result of Erdös and Renyi [3] follows.