6 research outputs found
The component counts of random injections
A model of random injections is defined which has domain
A
∪
B
and codomain
A
∪
C
, where
A
,
B
and
C
are mutually disjoint finite sets such that
|
B
|
⩽
|
C
|
. The model encompasses both random permutations, which is the case
B
=
C
=
∅
, and random maximum matchings of a complete bipartite graph, which is the case
A
=
∅
. The possible components of random injections are cycles and paths. Results on the counts of cycles and paths of different sizes are obtained for this model
The birth of the strong components
Random directed graphs undergo a phase transition around the point
, and the width of the transition window has been known since the
works of Luczak and Seierstad. They have established that as
when , the asymptotic probability that the strongly
connected components of a random directed graph are only cycles and single
vertices decreases from 1 to 0 as goes from to .
By using techniques from analytic combinatorics, we establish the exact
limiting value of this probability as a function of and provide more
properties of the structure of a random digraph around, below and above its
transition point. We obtain the limiting probability that a random digraph is
acyclic and the probability that it has one strongly connected complex
component with a given difference between the number of edges and vertices
(called excess). Our result can be extended to the case of several complex
components with given excesses as well in the whole range of sparse digraphs.
Our study is based on a general symbolic method which can deal with a great
variety of possible digraph families, and a version of the saddle-point method
which can be systematically applied to the complex contour integrals appearing
from the symbolic method. While the technically easiest model is the model of
random multidigraphs, in which multiple edges are allowed, and where edge
multiplicities are sampled independently according to a Poisson distribution
with a fixed parameter , we also show how to systematically approach the
family of simple digraphs, where multiple edges are forbidden, and where
2-cycles are either allowed or not.
Our theoretical predictions are supported by numerical simulations, and we
provide tables of numerical values for the integrals of Airy functions that
appear in this study.Comment: 62 pages, 12 figures, 6 tables. Supplementary computer algebra
computations available at https://gitlab.com/vit.north/strong-components-au