Expected Message Delivery Time for Small-world Networks in the Continuum Limit

TL;DR

This paper models the expected message delivery time in dense small-world networks on a plane, revealing linear growth at short distances and saturation at large distances, using iterative equations and greedy routing.

Contribution

It provides a novel analytical framework for calculating message delivery times in small-world networks with geographic routing, highlighting the transition from linear to saturated delay.

Findings

Message delivery time grows linearly at short distances.

Time saturates to a constant at large distances.

Routing depends mainly on short-range contacts.

Abstract

Small-world networks are networks in which the graphical diameter of the network is as small as the diameter of random graphs but whose nodes are highly clustered when compared with the ones in a random graph. Examples of small-world networks abound in sociology, biology, neuroscience and physics as well as in human-made networks. This paper analyzes the average delivery time of messages in dense small-world networks constructed on a plane. Iterative equations for the average message delivery time in these networks are provided for the situation in which nodes employ a simple greedy geographic routing algorithm. It is shown that two network nodes communicate with each other only through their short-range contacts, and that the average message delivery time rises linearly if the separation between them is small. On the other hand, if their separation increases, the average message delivery time rapidly saturates to a constant value and stays almost the same for all large values of their separation.