Counting k-Hop Paths in the Random Connection Model

TL;DR

This paper analyzes the probability and variance of k-hop paths in a random geometric graph model of wireless networks, providing detailed combinatorial insights and potential applications in broadcast time estimation.

Contribution

It introduces a combinatorial enumeration approach to precisely analyze k-hop connection probabilities and variances in the random connection model, advancing beyond mean field approximations.

Findings

Variance of 3-hop paths decomposes into four exponential terms.

Each term corresponds to a distinct sub-structure of intersecting paths.

Results can be applied to bound broadcast times in wireless networks.

Abstract

We study, via combinatorial enumeration, the probability of k-hop connection between two nodes in a wireless multi-hop network. This addresses the difficulty of providing an exact formula for the scaling of hop counts with Euclidean distance without first making a sort of mean field approximation, which in this case assumes all nodes in the network have uncorrelated degrees. We therefore study the mean and variance of the number of k-hop paths between two vertices x,y in the random connection model, which is a random geometric graph where nodes connect probabilistically rather than deterministically according to a critical connection range. In the example case where Rayleigh fading is modelled, the variance of the number of three hop paths is in fact composed of four separate decaying exponentials, one of which is the mean, which decays slowest as the Euclidean distance between the endpoints goes to infinity. These terms each correspond to one of exactly four distinct sub-structures with can form when pairs of paths intersect in a specific way, for example at exactly one node. Using a sum of factorial moments, this relates to the path existence probability. We also discuss a potential application of our results in bounding the broadcast time.