A note on uniform power connectivity in the SINR model

In this paper we study the connectivity problem for wireless networks under the Signal to Interference plus Noise Ratio (SINR) model. Given a set of radio transmitters distributed in some area, we seek to build a directed strongly connected communication graph, and compute an edge coloring of this graph such that the transmitter-receiver pairs in each color class can communicate simultaneously. Depending on the interference model, more or less colors, corresponding to the number of frequencies or time slots, are necessary. We consider the SINR model that compares the received power of a signal at a receiver to the sum of the strength of other signals plus ambient noise . The strength of a signal is assumed to fade polynomially with the distance from the sender, depending on the so-called path-loss exponent $\alpha$. We show that, when all transmitters use the same power, the number of colors needed is constant in one-dimensional grids if $\alpha>1$ as well as in two-dimensional grids if $\alpha>2$. For smaller path-loss exponents and two-dimensional grids we prove upper and lower bounds in the order of $\mathcal{O}(\log n)$ and $\Omega(\log n/\log\log n)$ for $\alpha=2$ and $\Theta(n^{2/\alpha-1})$ for $\alpha<2$ respectively. If nodes are distributed uniformly at random on the interval $[0,1]$, a \emph{regular} coloring of $\mathcal{O}(\log n)$ colors guarantees connectivity, while $\Omega(\log \log n)$ colors are required for any coloring.

💡 Research Summary

The paper investigates the fundamental connectivity problem for wireless networks under the Signal‑to‑Interference‑plus‑Noise Ratio (SINR) model when all transmitters operate at the same power level. The authors ask: how many frequency or time slots (colors) are required to guarantee that a directed strongly‑connected communication graph can be realized, i.e., every transmitter–receiver pair can communicate simultaneously with a feasible SINR? Their analysis is carried out for three canonical node placements: (i) one‑dimensional (1‑D) regular grids, (ii) two‑dimensional (2‑D) regular grids, and (iii) nodes drawn uniformly at random on the unit interval.

Key technical framework. The received power of a signal decays as d^‑α, where α>0 is the path‑loss exponent. The SINR condition at a receiver r for a sender s is
 P·d(s,r)^‑α / (∑_{t≠s} P·d(t,r)^‑α + N) ≥ β,
with common transmit power P, ambient noise N, and threshold β. Because all nodes use the same P, the problem reduces to controlling the sum of interfering terms by appropriately spacing simultaneously active transmitters (the color classes).

1‑D regular grid (spacing 1). When α>1, the interference from a node at distance k decays faster than 1/k, making the total interference from an infinite line convergent. By assigning colors in a periodic pattern with a constant period, the authors prove that a constant number of colors (independent of n, the number of nodes) suffices to keep the SINR above β for every link. The proof uses a simple geometric series bound on the interference contributed by nodes of the same color.

2‑D regular grid (n × n). For α>2 the interference from a node at Euclidean distance r decays faster than 1/r², and the double series ∑_{(i,j)≠(0,0)} 1/(i²+j²)^{α/2} converges. Consequently, a constant‑size coloring (again periodic in both dimensions) guarantees connectivity. When α=2, the series diverges logarithmically; the authors construct an O(log n) upper bound by a hierarchical coloring scheme and a matching lower bound of Ω(log n / log log n) via a packing argument that forces any coloring with fewer colors to create a “bottleneck” link whose SINR falls below β. For α<2 the interference grows polynomially with n, and they show that Θ(n^{2/α‑1}) colors are both necessary and sufficient. The proofs combine potential‑function arguments with careful counting of lattice points inside circles of radius r.

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