title: Distance distribution in random graphs and application to network exploration.
abstract: We consider the problem of determining the proportion of edges that are discovered in an Erdos-Rényi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of determining the proportion of edges connecting nodes that are at identical distance from the source node. The evolution of this quantity with the probability of existence of the edges exhibits intriguing oscillatory behavior. In order to perform our analysis, we introduce a different way of computing the distribution of distances between nodes. Our method outperforms previous similar analyses and leads to estimates that coincide remarkably well with numerical simulations. It allows us to characterize the phase transitions appearing when the connectivity probability varies.
authors.affiliation: Department of Mathematical Engineering, Université catholique de Louvain, 4 avenue Georges Lemaitre, B-1348 Louvain-la-Neuve, Belgium. vincent.blondel@uclouvain.be
authors.forename: Vincent D
authors.initials: VD
authors.lastname: Blondel
authors.forename: Jean-Loup
authors.initials: JL
authors.lastname: Guillaume
authors.forename: Julien M
authors.initials: JM
authors.lastname: Hendrickx
authors.forename: Raphaël M
authors.initials: RM
authors.lastname: Jungers