The graph structure of a deterministic automaton chosen at random: full version

TL;DR

This paper investigates the structure of random deterministic automata by analyzing their underlying digraphs, revealing the typical composition of the non-giant part and providing a new proof of a key theorem on strong connectivity.

Contribution

It offers a new, concise proof of Grusho's theorem on the giant component in random k-out digraphs and characterizes the structure outside the giant.

Findings

The outside part of the giant contains few short cycles.

The outside part mostly consists of overlapping tree-like structures.

The directed acyclic graph of a random k-out digraph closely resembles the digraph with the giant contracted.

Abstract

A deterministic finite automaton (DFA) of $n$ states over a $k$-letter alphabet can be seen as a digraph with $n$ vertices which all have exactly $k$ labeled out-arcs ($k$-out digraph). In 1973 Grusho first proved that with high probability (whp) in a random $k$-out digraph there is a strongly connected component (SCC) of linear size that is reachable from all vertices, i.e., a giant. He also proved that the size of the giant follows a central limit law. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of overlapping tree-like structures. Thus the directed acyclic graph (DAG) of a random $k$-out digraph is almost the same as the digraph with the giant contracted into one vertex. These findings lead to a new, concise and self-contained proof of Grusho's theorem. This work also contains some other results including the structure outside the giant, the phase transition phenomenon in strong connectivity, the typical distance, and an extension to simple digraphs.