title: General and exact approach to percolation on random graphs.
abstract: We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite-size quenched or random multitype graphs. (ii) We define a very general random graph ensemble that encompasses most of the models published to this day and also makes it possible to model structural properties not yet included in a theoretical framework. Site and bond percolation on this ensemble is solved exactly in the infinite-size limit using probability generating functions [i.e., the percolation threshold, the size, and the composition of the giant (extensive) and small components]. Several examples and applications are also provided. (iii) Our approach can be adapted to model interdependent graphs-whose most striking feature is the emergence of an extensive component via a discontinuous phase transition-in an equally general fashion. We show how a graph can successively undergo a continuous then a discontinuous phase transition, and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition.
authors.affiliation: Département de physique, de génie physique, et d'optique, Université Laval, Québec, Québec, Canada G1V 0A6.
authors.forename: Antoine
authors.initials: A
authors.lastname: Allard
authors.affiliation: Département de physique, de génie physique, et d'optique, Université Laval, Québec, Québec, Canada G1V 0A6.
authors.forename: Laurent
authors.initials: L
authors.lastname: Hébert-Dufresne
authors.affiliation: Département de physique, de génie physique, et d'optique, Université Laval, Québec, Québec, Canada G1V 0A6.
authors.forename: Jean-Gabriel
authors.initials: JG
authors.lastname: Young
authors.affiliation: Département de physique, de génie physique, et d'optique, Université Laval, Québec, Québec, Canada G1V 0A6.
authors.forename: Louis J
authors.initials: LJ
authors.lastname: Dubé