Rumor Spreading with Bounded In-Degree

We consider a variant of the well-studied gossip-based model of communication for disseminating information in a network. Classically, in each time unit, every node u is allowed to contact a single random neighbor v. If u knows the data (rumor) to be disseminated, node v learns it (known as push) and if node v knows the rumor, u learns it (known as pull). While in the classic gossip model, each node is only allowed to contact a single neighbor in each time unit, each node can possibly be contacted by many neighboring nodes. If, for example, several nodes pull from the same common neighbor v, v manages to inform all these nodes in a single time unit. In the present paper, we consider a restricted model where at each node only one incoming request can be served. As long as only a single piece of information needs to be disseminated, this does not make a difference for push requests. It however has a significant effect on pull requests. If several nodes try to pull the information from the same common neighbor, only one of the requests can be served. In the paper, we therefore concentrate on this weaker pull version, which we call restricted pull. We distinguish two versions of the restricted pull protocol depending on whether the request to be served among a set of pull requests at a given node is chosen adversarially or uniformly at random. As a first result, we prove an exponential separation between the two variants. We show that there are instances where if an adversary picks the request to be served, the restricted pull protocol requires a polynomial number of rounds whereas if the winning request is chosen uniformly at random, the restricted pull protocol only requires a polylogarithmic number of rounds to inform the whole network. Further, as the main technical contribution, we show that if the request to be served is chosen randomly, the slowdown of using restricted pull versus using the classic pull protocol can w.h.p. be upper bounded by O(∆/δ · logn), where ∆ and δ are the largest and smallest degree of the network. ar X iv :1 50 6. 00 82 8v 1 [ cs .D C ] 2 J un 2 01 5