Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Congestion Due to Random Walk Routing

Published

Author(s)

Vladimir V. Marbukh, Iraj Saniee, Onuttom Narayan

Abstract

In this paper we derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the uniform multicommodity flow problem under random walk routing. We show the mean load is linearly dependent on the nodal degree with a common multiplier equal to the sum of the inverses of the non-zero eigenvalue of the graph Laplacian. Even though some aspects of the mean load value, such as linear dependence on the nodal degree, are intuitive and may be derived from the equilibrium distribution of the random walk on the undirected graph, the exact expression for the mean load in terms of the full spectrum of the graph has not been known before. Using the explicit expression for the mean load, we give asymptotic estimates for the load on a variety of graphs whose spectral density are well known. We conclude with numerical computation of the mean load for other well-known graphs without known spectral densities.
Citation
archive for electronic preprints of scientific papers

Keywords

multicommodity flow, network congestion, steady state, Laplacian of a graph, spectrum of a graph, random walk

Citation

Marbukh, V. , Saniee, I. and Narayan, O. (2013), Congestion Due to Random Walk Routing, archive for electronic preprints of scientific papers, [online], http://arxiv.org/ (Accessed May 23, 2024)

Issues

If you have any questions about this publication or are having problems accessing it, please contact reflib@nist.gov.

Created August 31, 2013, Updated February 19, 2017