Peer counting and sampling in overlay networks: random walk methods

In this article we address the problem of counting the number of peers in a peer-to-peer system, and more generally of aggregating statistics of individual peers over the whole system. This functionality is useful in many applications, but hard to achieve when each node has only a limited, local knowledge of the whole system. We propose two generic techniques to solve this problem. The Random Tour method is based on the return time of a continuous time random walk to the node originating the query. The Sample and Collide method is based on counting the number of random samples gathered until a target number of redundant samples are obtained. It is inspired by the "birthday paradox" technique of [6], upon which it improves by achieving a target variance with fewer samples. The latter method relies on a sampling sub-routine which returns randomly chosen peers. Such a sampling algorithm is of independent interest. It can be used, for instance, for neighbour selection by new nodes joining the system. We use a continuous time random walk to obtain such samples. We analyse the complexity and accuracy of the two methods. We illustrate in particular how expansion properties of the overlay affect their performance.