Connectivity in one-dimensional ad hoc networks with an access point

In this paper, we study the connectivity in one-dimensional ad hoc wireless networks with an fixed access point. In recent years, various closed expressions for the probability of connectivity on one-dimensional networks (interval graphs) have been derived by many researchers. We will provide some numerical validation for them by means of extensive simulations.

💡 Research Summary

The paper investigates the probability that a one‑dimensional ad‑hoc wireless network becomes fully connected when a fixed access point (AP) is present somewhere along the line. The authors build on a substantial body of prior work that derived closed‑form expressions for the connectivity probability of pure one‑dimensional networks (often modeled as interval graphs). In those classic models, (n) mobile nodes are placed independently and uniformly on a line segment of length (L); two nodes can communicate if their Euclidean distance does not exceed a common transmission radius (r). The probability that the resulting random geometric graph is connected, (P_{\text{conn}}^{(0)}(n,r,L)), has been expressed in several equivalent closed forms, for example as a finite alternating sum involving binomial coefficients.

The novelty of this study lies in incorporating a stationary AP at a deterministic location (x_{\text{AP}}\in