The diameter of protean graphs

Abstract

The web graph is a real-world self-organizing network whose vertices correspond to web pages, and whose edges correspond to links between pages. Many stochastic models for the web graph have been recently proposed, with the aim of reproducing one or more of its observed properties and parameters. Some of the most intensely studied parameters for the web graph are the degree distribution and diameter. A recent stochastic model of the web graph is the protean graph P n ( d , η ) . In this model, vertices are renewed over time, and older vertices are more likely to receive edges than younger ones. While previous work on the model focussed on the power law degree distribution of protean graphs, in this note we study its diameter. Since the protean graphs may be disconnected, we focus on the diameter of the giant component. Our main result is that diameter of the giant component of P n ( d , η ) is equal to Θ ( log n ) , which supports experimental data observed in the actual web graph.