Searching for Nodes in Random Graphs

TL;DR

This paper investigates the efficiency of greedy search algorithms on specially designed random graphs, analyzing success probabilities and hop counts, and identifying phase transitions related to link density.

Contribution

It introduces two types of random graphs tailored for greedy search and derives equations for success probability and hop count, revealing phase transition phenomena.

Findings

Success probability equations derived for the search process

Identification of a phase transition as link number varies

Numerical and analytical evidence of transition phenomena

Abstract

We consider the problem of searching for a node on a labelled random graph according to a greedy algorithm that selects a route to the desired node using metric information on the graph. Motivated by peer-to-peer networks two types of random graph are proposed with properties particularly amenable to this kind of algorithm. We derive equations for the probability that the search is successful and also study the number of hops required, finding both numerical and analytic evidence of a transition as the number of links is varied.