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- W4387641633 abstract "We consider some models of random graphs and directed graphs and investigate their behavior near thresholds for the appearance of certain types of connected components. Firstly, we look at the critical window for the appearance of a giant strongly connected component in binomial random digraphs. We provide bounds on the probability that the largest strongly connected component is very large or very small. Next, we study the configuration model for graphs and show new upper bounds on the size of the largest connected component in the subcritical and barely subcritical regimes. We also show that these bounds are tight in some instances. Finally we look at the configuration model for random digraphs. We investigate the barely sub-critical region and show that this model behaves similarly to the binomial random digraph whose barely sub- and super-critical behaviour was studied by Luczak and Seierstad. Moreover, we show the existence of a threshold for the existence of a giant weak component, as predicted by Kryven. En aquesta tesi considerem diversos models de grafs i graf dirigits aleatoris, i investiguem el seu comportament a prop dels llindars per l'aparició de certs tipus de components connexes. En primer lloc, estudiem la finestra crítica per a l'aparició d'una component fortament connexa en dígrafs aleatoris binomials (o d'Erdos-Rényi). En particular, provem diversos resultats sobre la probabilitat límit que la component fortament connexa sigui sigui molt gran o molt petita. A continuació, estudiem el model de configuració per a grafs no dirigits i mostrem noves cotes superiors per la mida de la component connexa més gran en els règims sub-crítics i quasi-subcrítics. També demostrem que, en general, aquestes cotes no poden ser millorades. Finalment, estudiem el model de configuració per a dígrafs aleatoris. Ens centrem en la regió quasi-subcrítica i demostrem que aquest model es comporta de manera similar al model binomial, el comportament del qual va ser estudiat per Luczak i Seierstad en les regions quasi-subcrítica i quasi-supercrítica. A més a més, demostrem l'existència d'una funció llindar per a l'existència d'una component feble gegant, tal com va predir Kryven." @default.
- W4387641633 created "2023-10-15" @default.
- W4387641633 creator A5079186550 @default.
- W4387641633 date "2023-10-14" @default.
- W4387641633 modified "2023-10-15" @default.
- W4387641633 title "Threshold phenomena involving the connected components of random graphs and digraphs" @default.
- W4387641633 doi "https://doi.org/10.5821/dissertation-2117-361624" @default.
- W4387641633 hasPublicationYear "2023" @default.
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