Predecessors in a random mapping

Abstract

A random mapping (T; q) of a finite set V, V={1, 2,…,n} into itself assigns independently to each i∈V its unique image j∈V with probability q if i=j and with probability P=(1−q)/(n−1) if i≠j. The number of predecessors of elements from a given subset of V is studied. Exact results and limit theorems for the distribution of this random variable, the quasi‐binomial distribution, are given. The results are applied to an inverse epidemic process on a random digraph GT representing (T; q). © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 501–519, 1998

A Loading-Dependent Model of Probabilistic Cascading Failure

“…Islam, O'Shaughnessy, and Smith (1996) interpret d and p as primary and secondary infection probabilities and apply the quasibinomial distribution to data on the final size of influenza epidemics. Jaworski (1998) generalizes the derivation to a random mapping with a general fixed point probability. The cascading failure model gives a new application and interpretation of the quasibinomial distribution.…”

Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

“…To obtain the distribution of sD n [a], we exploit the following 'duality' (see [16]) between the successors and predecessors of TD n . 16) where E t (X) denotes the tth factorial moment of the random variable X. By a similar argument, we also have…”

A cutting process for random mappings

“…for 0 ≤ t ≤ N. So the conditional distribution of t n (m) − m given L n = is a quasi-binomial distribution (QBD I) [29,31]. In Section 4 we need the following stronger version of a local limit theorem for QBD I given in [29].…”

“…To establish this limit, we start by fixing δ > 0 and k ≥ 1, and we choose ε > 0 arbitrarily small. Since m = o( √ n), it follows from asymptotics established for QBD I [10,29,30] that we can choose some bound f (n, m) = o(n) such that for n and m large enough…”

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