Discovery through gossip

Abstract

We study randomized gossip-based processes in dynamic networks that are motivated by information discovery in large-scale distributed networks such as peer-to-peer and social networks. A well-studied problem in peer-to-peer networks is resource discovery, where the goal for nodes (hosts with IP addresses) is to discover the IP addresses of all other hosts. Also, some of the recent work on self-stabilization algorithms for P2P/overlay networks proceed via discovery of the complete network. In social networks, nodes (people) discover new nodes through exchanging contacts with their neighbors (friends). In both cases the discovery of new nodes changes the underlying networknew edges are added to the network -and the process continues in the changed network. Rigorously analyzing such dynamic (stochastic) processes in a continuously changing topology remains a challenging problem with obvious applications. 566HAEUPLER ET AL.This paper studies and analyzes two natural gossip-based discovery processes. In the push discovery or triangulation process, each node repeatedly chooses two random neighbors and connects them (i.e., "pushes" their mutual information to each other). In the pull discovery process or the two-hop walk, each node repeatedly requests or "pulls" a random contact from a random neighbor and connects itself to this two-hop neighbor. Both processes are lightweight in the sense that the amortized work done per node is constant per round, local, and naturally robust due to the inherent randomized nature of gossip.Our main result is an almost-tight analysis of the time taken for these two randomized processes to converge. We show that in any undirected n-node graph both processes take O(n log 2 n) rounds to connect every node to all other nodes with high probability, whereas (n log n) is a lower bound. We also study the two-hop walk in directed graphs, and show that it takes O(n 2 log n) time with high probability, and that the worst-case bound is tight for arbitrary directed graphs, whereas (n 2 ) is a lower bound for strongly connected directed graphs. A key technical challenge that we overcome in our work is the analysis of a randomized process that itself results in a constantly changing network leading to complicated dependencies in every round. We discuss implications of our results and their analysis to discovery problems in P2P networks as well as to evolution in social networks.

A deterministic worst-case message complexity optimal solution for resource discovery

“…al. in [28], in which they present two simple randomized algorithms based on gossiping that need Ω(n log n) time and Ω(n 2 log n) work per node on expectation. They only allow nodes to send a single message containing at most one address of size log n in each round.…”

“…Thus their model is more restrictive compared to the model used in [30,31] and leads to an increased runtime in the number of rounds. We present a deterministic solution that follows the idea of [28] and limits the number of messages each node has to send and the number of addresses transmitted in one message. Our goal is to reduce the number of messages sent and received by each node such that we avoid nodes to be overloaded.…”

“…We use the network model used in [30,31,28]. In the following we give a detailed description of the model.…”

“…Hence, the state of a node is fully determined by its local variables. Like in [30,31,28] we assume that a node can verify its neighborhood without extra work, such that there are no false identifiers in the network. Only local topology changes are allowed, i.e.…”

Optimal gossip with direct addressing

“…Recently, [11] gave an improved deterministic algorithm which achieves O(log n) rounds for the same setting. A similar setting but restricting a message to only contain one contact was considered in [8]. Lastly, direct addressing has also been extensively studied in general topologies.…”