Abstract
In this paper we characterise the structural transition in random mappings with in-degree restrictions. Specifically, for integers $0~\!\!\leq~\!\!r\leq~\!\!n$, we consider a random mapping model $\hat{T}_n^r$ from $[n]=\{1,2, \ldots , n\}$ into $[n]$ such that $\hat{G}_n^r$, the directed graph on $n$ labelled vertices which represents the mapping $\hat{T}_n^r$, has $r$ vertices that are constrained to have in-degree at most $1$ and the remaining vertices have in-degree at most 2. When $r=n$, $\hat{T}_n^r$ is a uniform random permutation and when $r<n$, we can view $\hat{T}_n^r$ as a 'corrupted' permutation. We investigate structural transition in $\hat{G}_n^r$ as we vary the integer parameter $r$ relative to the total number of vertices $n$. We obtain exact and asymptotic distributions for the number of cyclic vertices, the number of components, and the size of the typical component in $\hat{G}_n^r$, and we characterise the dependence of the limiting distributions of these variables on the relationship between the parameters $n$ and $r$ as $n\to\infty$. We show that the number of cyclic vertices in $\hat{G}_n^r$ is $\Theta({n\over\sqrt{a}})$ and the number of components is $\Theta(\log({n\over \sqrt{a}}))$ where $a=n-r$. In contrast, provided only that $a=n-r\to\infty$, we show that the asymptotic distribution of the order statistics of the normalised component sizes of $\hat{G}_n^r$ is always the Poisson-Dirichlet(1/2) distribution as in the case of uniform random mappings with no in-degree restrictions.
Highlights
The motivation for the results in this paper comes from earlier work on the component structure of random mapping models
Provided only that a = n − r → ∞, we show that the asymptotic distribution of the order statistics of the normalised component sizes of Grn is always the Poisson-Dirichlet(1/2) distribution as in the case of uniform random mappings with no in-degree restrictions
In applications such as the analysis of shift register data, it is natural to consider random mapping digraphs where the in-degree of each vertex is at most m, where m 2 is a fixed integer. Such models were considered by Arney and Bender [3] and, more recently, by the authors in [15]. This later work shows that even in the case m = 2, the ‘macroscopic’ structure of the constrained√random mapping digraph remains similar to the structure of Gn, e.g. there are still Θ( n) cyclic vertices in the constrained digraph and the joint distribution of the normalised order statistics of the component sizes still converges to the PD(1/2) distribution on ∇ as the number of vertices tends to infinity
Summary
The motivation for the results in this paper comes from earlier work on the component structure of random mapping models. Such models were considered by Arney and Bender [3] and, more recently, by the authors in [15] This later work shows that even in the case m = 2, the ‘macroscopic’ structure of the constrained√random mapping digraph remains similar to the structure of Gn, e.g. there are still Θ( n) cyclic vertices in the constrained digraph and the joint distribution of the normalised order statistics of the component sizes still converges to the PD(1/2) distribution on ∇ as the number of vertices tends to infinity. We are interested in characterising how these in-degree constraints influence the graphical structure of the random mapping For this model we determine (precisely) how the exact and asymptotic cycle and component structure of the digraph depends on the parameter r(n).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
