Content deleted Content added
m convert special characters found by Wikipedia:Typo Team/moss (via WP:JWB) |
m link phase transition Tag: Reverted |
||
Line 70:
===Randomness===
When the start state and accept states are ignored, a DFA of {{mvar|n}} states and an alphabet of size {{mvar|k}} can be seen as a [[Directed graph|digraph]] of {{mvar|n}} vertices in which all vertices have {{mvar|k}} out-arcs labeled {{math|1, ..., ''k''}} (a {{mvar|k}}-out digraph). It is known that when {{math|''k'' ≥ 2}} is a fixed integer, with high probability, the largest [[strongly connected component]] (SCC) in such a {{mvar|k}}-out digraph chosen uniformly at random is of linear size and it can be reached by all vertices.<ref name=Grusho>{{cite journal|last1=Grusho|first1=A. A.|title=Limit distributions of certain characteristics of random automaton graphs|journal=Mathematical Notes of the Academy of Sciences of the USSR|date=1973|volume=4|pages=633–637|doi=10.1007/BF01095785|s2cid=121723743|ref=Grusho1973}}</ref> It has also been proven that if {{mvar|k}} is allowed to increase as {{mvar|n}} increases, then the whole digraph has a [[phase transition]] for strong connectivity similar to [[Erdős–Rényi model]] for connectivity.<ref name=Cai>{{cite journal |last1=Cai |first1=Xing Shi |last2=Devroye |first2=Luc |title=The graph structure of a deterministic automaton chosen at random |journal=Random Structures & Algorithms |date=October 2017 |volume=51 |issue=3 |pages=428–458 |doi=10.1002/rsa.20707|arxiv=1504.06238 |s2cid=13013344 }}</ref>
In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largest [[strongly connected component|SCC]] with high probability.<ref name=Grusho /><ref>{{cite conference |last1=Carayol |first1=Arnaud |last2=Nicaud |first2=Cyril |date=February 2012 |title=Distribution of the number of accessible states in a random deterministic automaton |volume=14 |pages=194–205 |conference=STACS'12 (29th Symposium on Theoretical Aspects of Computer Science) |___location=Paris, France |url=https://hal.archives-ouvertes.fr/hal-00678213}}</ref> This is also true for the largest [[Glossary of graph theory#Subgraphs|induced sub-digraph]] of minimum in-degree one, which can be seen as a directed version of [[Degeneracy (graph theory)#k-Cores|{{math|1}}-core]].<ref name=Cai />
| |||