Order-Optimal Consensus Through Randomized Path Averaging

• Published in: IEEE transactions on information theory Vol. 56; no. 10; pp. 5150 - 5167
• Main Authors: Bénézit, Florence, Dimakis, Alexandros G, Thiran, Patrick, Vetterli, Martin
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Average consensus; Convergence; Delay; distributed algorithms; Distributed processing; Exact sciences and technology; Flooding; gossip algorithms; Graphs; Inequalities; Information processing; Information theory; Information, signal and communications theory; Markov analysis; Markov processes; Messages; Network topology; Networks; Optimization; Routing; sensor networks; Switching and signalling; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Teletraffic; Topological manifolds; Topology; Transmission and modulation (techniques and equipments); Performance evaluation; Averaging method; Nearest neighbour; Packet switching; Wireless telecommunication; Teletraffic; gossip algorithms; sensor networks; Random graph; Markov chain; Distributed processing; Multihop network; Message passing; Information network; Average consensus; distributed algorithms; Traffic management; Distributed algorithm; Information processing; Long distance transmission; Sensor array; Traffic congestion; Signal detection
• ISSN: 0018-9448; 1557-9654; 1557-9654
• DOI: 10.1109/TIT.2010.2060050

Gossip algorithms have recently received significant attention, mainly because they constitute simple and robust message-passing schemes for distributed information processing over networks. However, for many topologies that are realistic for wireless ad-hoc and sensor networks (like grids and random geometric graphs), the standard nearest-neighbor gossip converges as slowly as flooding (O(n 2 ) messages). A recently proposed algorithm called geographic gossip improves gossip efficiency by a √n factor, by exploiting geographic information to enable multihop long-distance communications. This paper proves that a variation of geographic gossip that averages along routed paths, improves efficiency by an additional √n factor, and is order optimal (O(n) messages) for grids and random geometric graphs with high probability. We develop a general technique (travel agency method) based on Markov chain mixing time inequalities which can give bounds on the performance of randomized message-passing algorithms operating over various graph topologies.