On the critical transmission range in one-dimensional sensor networks under non-uniform node placement

We consider $n$ sensors independently placed at points $X_1, … , X_n$ on the unit interval $[0,1]$ according to some probability distribution function $F$.
Two sensor nodes communicate with each other if their distance is less than some given transmission range $\rho >0$. We define the critical transmission range $R_n$ as the smallest transmission range such that the nodes $X_1, \ldots , X_n$ form a connected graph (under the notion of adjacency implied by the ability of nodes to communicate). Since the distribution of $R_n$ is usually not tractable, we are interested in developing an asymptotic theory for $R_n$ as $n$ becomes large: We seek a deterministic sequence $\rho^\star: \mathbb{N}_0 \rightarrow \mathbb{R}_+$ such that the ratio $R_n / \rho^\star_n$ converges to some non-trivial limit $L$ in an appropriate sense. When available, such results suggest $\rho^\star_n L$ as a proxy or approximation for $R_n$.

We assume that $F$ admits a continuous density $f$, and two qualitatively different cases are identified, namely $f_\star > 0$ and $f_\star = 0$ with $f_\star = \inf \left ( f(x), \ x \in [0,1] \right )$. In each case, we present results on the form of $\rho^\star_n$ and $L$. In the process we make contact with the existence and nature of critical thresholds for the property of graph connectivity in the underlying geometric random graph. Engineering implications for power allocation are discussed.

This is joint work with former Ph.D. student Guang Han (now at Motorola).