**Abstract** : Random gossip (push and pull) is one of the most studied protocols for dis- seminating information in a network, e.g., [1, 3]. 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 successfully contacted by and thus interact with many neighboring nodes. As an extreme case, consider the behavior of random pull in a star network where a single center node is connected to n − 1 leaf nodes. In fact, all recent papers which study the time complexity of the random push-pull protocol critically rely on the fact that a node can be contacted by many nodes in a single round, e.g., [2]. However, in order to obtain applicable and scalable protocols, ideally, we would like to not only limit the number of interactions each node initiates, but also the number of interactions each node participates in.
We therefore study a weaker variant of the described random pull algorithm, which we call rpull (stands for restricted pull). In each round, every node can still initiate a connection to one uniformly random neighbor.