Susceptibility in subcritical random graphs

Abstract

Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal.

State space collapse for critical multistage epidemics

“…Gathering these two observations, we end up thanks to (6.1) with the relation E(N n,1 ) ≈ n 4/3−1 = n 1/3 , when K = 1. This answer coincides with Antal and Krapivsky's conjecture and the above reasoning through random partitioning can be made rigorous, see, e.g., Janson and Luczak [12].…”

A moment-generating formula for Erdős-Rényi component sizes

“…where W is the Lambert-W function. Now it is known that if p = 1 − e −t/n and n → ∞ then |C| converges in distribution to the total number of offspring in a subcritical Galton-Watson branching process with POI(t) offspring distribution (see [4, Theorem 11.6.1]), i.e., |C| has Borel distribution with parameter t (see [2,Section 2.2] or [13,Section 7]). The generating function G t of the Borel distribution with parameter t is known to be characterized by the identity G t (z) ≡ ze (Gt(z)−1)t (see [3,Section 10.4]), which is in turn equivalent to G t (z) = −W (−e −t tz)/t, therefore a more rigorous version of (1.6) can be used to show that the distribution |C| weakly converges to the Borel distribution with parameter t as n → ∞.…”

Random subgraph counts and U-statistics: multivariate normal approximation via exchangeable pairs and embedding

“…In applications such as U -statistics, this ordering is typically introduced 'artificially' (see [7, p. 118ff.] and also [5] for a graph related example), and, hence, bounds of the form of our Theorem 1 appear more natural in this context, as the order of the index set does not influence the bounds. In particular, to obtain optimal bounds in the Kolmogorov metric, Stein's method is typically better suited than a martingale approach in cases where martingales are only an auxiliary construct and are not intrinsic to the particular problem.…”