Extended navigability of small world networks: exact results and new insights

TL;DR

This paper provides an exact analysis of greedy algorithms in small-world networks, revealing that optimal navigability depends on system parameters and can be achieved with more flexible network constructions than previously thought.

Contribution

It offers an exact asymptotic solution for greedy navigation in small-world networks and relaxes the notion that only specific constructions are optimal.

Findings

Exact asymptotic behavior of greedy algorithms derived

Optimal network design depends on additional criteria like cost

Flexibility in network construction for optimal navigability

Abstract

Navigability of networks, that is the ability to find any given destination vertex starting from any other vertex, is crucial to their usefulness. In 2000 Kleinberg showed that optimal navigability could be achieved in small-world networks provided that a special recipe was used to establish long range connections, and that a greedy algorithm, that ensures that the destination will be reached, is used. Here we provide an exact solution for the asymptotic behavior of such a greedy algorithm as a function of the system's parameters. Our solution enables us to show that the original claim that only a very special construction is optimal can be relaxed depending on further criteria, such as, for example, cost minimization, that must be satisfied.