Διαδικασία Αχλιόπτα

Η διαδικασία Αχλιόπτα δημιουργήθηκε το 2000, από πρόταση του Δημήτρη Αχλιόπτα σε εργαστήριο του Ινστιτούτου Φιλντς. Είναι η γενικοποίηση μιας κοινής διαδικασίας σχηματισμού ενός τυχαίου γραφήματος Έρντους-Ρένι. Ο Αχλιόπτας πρότεινε μια τάξη μεταβλητών στην διαδικασία τυχαίου γραφήματος G ~ n {\textstyle {\widetilde {G}}_{n}} , η οποία περιγράφεται ως εξής: Ξεκινώντας με το άδειο γράφημα με n {\textstyle n} κορυφές και χωρίς καμία άκρη. Σε κάθε βήμα, δύο πιθανές άκρες e 1 {\textstyle e_{1}} και e 2 {\textstyle e_{2}} , επιλέγονται ανεξάρτητα και ομοιόμορφα τυχαία από όλες τις πιθανές άκρες ( n 2 ) {\textstyle {\binom {n}{2}}} (ή από τις άκρες που δεν είναι πάντα παρόντες). Μία από αυτές τις άκρες επιλέγεται σύμφωνα με κάποιον κανόνα R {\textstyle {\mathcal {R}}} και προστίθεται στο γράφημα.. Το αποτέλεσμα είναι μια διαδικασία τυχαίου γραφήματος G ~ n R {\textstyle {\widetilde {G}}_{n}^{\mathcal {R}}} του οποίου η διανομή στη διαδρομή εξαρτάται από τον κανόνα R {\textstyle {\mathcal {R}}} . Αυτές οι διαδικασίες λέγονται διαδικασίες Αχλιόπτα. Βεβαίως, αν επιλεγεί ένα e 1 {\textstyle e_{1}} , δίνει (ακριβώς ή προσεγγιστικά, ανάλογα με τους ακριβείς ορισμούς), την κλασική διαδικασία τυχαίου γραφήματος G ~ n {\textstyle {\widetilde {G}}_{n}} . Η αρχική ερώτηση του Αχλιόπτα ήταν εάν κάποιος μπορεί να μεταβάλλει το κρίσιμο σημείο της διαδικασίας τυχαίου γραφήματος επιλέγοντας ένα κατάλληλο κανόνα.

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