
Counting directed acyclic and elementary digraphs
Directed acyclic graphs (DAGs) can be characterised as directed graphs w...
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BDDBased Algorithm for SCC Decomposition of EdgeColoured Graphs
Edgecoloured directed graphs provide an essential structure for modelli...
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LinearTime Algorithms for Computing Twinless Strong Articulation Points and Related Problems
A directed graph G=(V,E) is twinless strongly connected if it contains a...
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The Phase Transition of Discrepancy in Random Hypergraphs
Motivated by the BeckFiala conjecture, we study the discrepancy problem...
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Probabilistic Properties of GIG Digraphs
We study the probabilistic properties of the Greatest Increase Grid (GIG...
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Phase Transitions in the Edge/Concurrent Vertex Model
Although it is wellknown that some exponential family random graph mode...
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Interlayer Transition in Neural Architecture Search
Differential Neural Architecture Search (NAS) methods represent the netw...
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The birth of the strong components
Random directed graphs D(n,p) undergo a phase transition around the point p = 1/n, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as n →∞ when p = (1 + μ n^1/3)/n, 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 μ goes from ∞ to ∞. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of μ 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 saddlepoint 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 p, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2cycles 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.
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