The graph structure of a deterministic automaton chosen at random

Abstract

An n‐state deterministic finite automaton over a k‐letter alphabet can be seen as a digraph with n vertices which all have k labeled out‐arcs. Grusho (Publ Math Inst Hungarian Acad Sci 5 (1960), 17–61). proved that whp in a random k‐out digraph there is a strongly connected component of linear size, i.e., a giant, and derived a central limit theorem. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of tree‐like structures, and present a new proof of Grusho's theorem. Among other things, we pinpoint the phase transition for strong connectivity. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 428–458, 2017

The diameter of the directed configuration model

“…Penrose studied the emergence of a linear order strongly connected component [22] and its diameter was determined by Addario-Berry, Balle and the second author [1]. In [10], the first author and Devroye studied the diameter outside the giant strongly connected component and other properties of the d-out model.…”

Analysis of Regular Operations Application and Finite Automa

“…In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largest SCC with high probability [17,19]. This is also true for the largest induced sub-digraph of minimum in-degree one, which can be seen as a directed version of 1-core [18].…”

“…It is known that when k ≥ 2 is a fixed integer, with high probability, the largest strongly connected component (SCC) in such a k-out digraph chosen uniformly at random is of linear size and it can be reached by all vertices. [8] It has also been proven that if k is allowed to increase as n increases, then the whole digraph has a phase transition for strong connectivity similar to Erdős-Rényi model for connectivity [18].…”

The giant component of the directed configuration model revisited

“…Remark 6.2. In Cai and Devroye (2017), it was showed that the number of cycles outside the giant of a uniform random k-out digraph with k ≥ 2 converges to a Poisson distribution. We believe that similar methods can be applied to derive that the law of C n,≥1 converges to a Poisson distribution with mean log 1 1−ν .…”