Interacting Urns Processes for Clustering of Large-Scale Networks of Tiny Artifacts

Pierre Leone,Elad M Schiller

doi:10.1155/2010/936195

Abstract

We analyze a distributed variation on the Pólya urn process in which a network of tiny artifacts manages the individual urns. Neighboring urns interact by repeatedly adding the same colored ball based on previous random choices. We discover that the process rapidly converges to a definitive random ratio between the colors in every urn. Moreover, the rate of convergence of the process at a given node depends on the global topology of the network. In particular, the same ratio appears for the case of complete communication graphs. Surprisingly, this effortless random process supports useful applications, such as clustering and computation of pseudo-geometric coordinate. We present numerical studies that validate our theoretical predictions.

Highlights

Designers of distributed algorithms often assume that each node is computationally powerful, capable of storing nontrivial amounts of data, and carrying out complex calculations
The time to stop the process was obtained while the clustered network stopped evolving visually, after about 20 draws per node; the global formation starts to emerge after 15 draws per node
Similar to the clustering application, the time to stop the process was obtained while the pseudogeometric coordinate stopped evolving visually, after about 20 draws per node; the global formation starts to emerge after 15 drawings per node

Summary

Introduction

Designers of distributed algorithms often assume that each node is computationally powerful, capable of storing nontrivial amounts of data, and carrying out complex calculations. Recent technological developments in wireless communications and microprocessors allow us to establish networks consisting of massive amounts of cheap and tiny artifacts that are tightly resource constrained. These networks of tiny artifacts are far more challenging than the traditional networks; on one hand, their relative scale is enormous, while on the other hand, each tiny artifact can only run a millicode that is aided by a miniature memory. Such limitations are not crippling if the system designer has a precise understanding of the tiny artifacts’ computational power and local interaction rules from which global formation emerges

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Results

Conclusion