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We consider the problem of determining the proportion of edges that arediscovered in an Erdos-Renyi graph when one constructs all shortest paths froma given source node to all other nodes. This problem is equivalent to the oneof determining the proportion of edges connecting nodes that are at identicaldistance from the source node. The evolution of this quantity with theprobability of existence of the edges exhibits intriguing oscillatory behavior.In order to perform our analysis, we introduce a new way of computing thedistribution of distances between nodes. Our method outperforms previoussimilar analyses and leads to estimates that coincide remarkably well withnumerical simulations. It allows us to characterize the phase transitionsappearing when the connectivity probability varies.

Distance distribution in random graphs and application to networks exploration