Scaling Laws for One- and Two-Dimensional Random Wireless Networks in the Low-Attenuation Regime

Abstract

The capacity scaling of extended two-dimensional wireless networks is known in the high attenuation regime, i.e. when the power path loss exponent α is greater than 4. This has been accomplished by deriving information theoretic upper bounds for this regime that match the corresponding lower bounds. On the contrary, not much is known in the so-called low attenuation regime when 2 ≤ α ≤ 4. (For one-dimensional networks, the uncharacterized regime is 1 ≤ α ≤ 2.5.) The dichotomy is due to the fact that while communication is highly power-limited in the first case and power-based arguments suffice to get tight upper bounds, the study of the low attenuation regime requires a more precise analysis of the degrees of freedom involved. In this paper, we study the capacity scaling of extended wireless networks with an emphasis on the low attenuation regime and show that in the absence of small scale fading, the low attenuation regime does not behave significantly different from the high attenuation regime.

Exact Capacity Scaling of Extended Wireless Networks

“…less th quantity computed for a regular network with lol each left-hand side vertex and 2 log n nodes at eac side vertex. The most convenient way to index the node po; resulting regular network is to use double indic( In the sequel, we are going to prove that the following upper bound holds with high probability: E(Tr((H*H)t)) < t1 n (Ki log n)31 (8) where tl = 1(+1)! are the Catalan numbers and K1 > 0 is a constant independent of n. By Chebyshev's inequality, this allows to conclude that for any m, P(Bn,E) < <(H m) 1_ lim (t1n (K1 logn) Recall that the random variables Hik are independent and zeromean, so the expectation is only non-zero when the terms in the product form conjugate pairs.…”

“…However, a binning argument using parts (a) and (b) of Lemma 1 provides again a connection to regular networks. Skipping the binning argument itself, let us directly concentrate on the regular case when the matrix elements of H are given by ei Oik and dk ,,,k is given in (6 (8) in the case 1 = 2. For 1 > 2, the non-vanishing terms in (9) can be associated to rooted planted planar trees that have 1 branches.…”

“…A recent work [2] shows that the answer is the latter: authors exhibit a hierarchichal cooperation scheme which yields a throughput scaling of E)(n2-a/2-2 ) bits/second for any c > 0, when the signals transmitted from one user to another at distance r apart are subject to a power attenuation of r-and to random phase changes. Therefore for av < [3], [4], [5], [6], [7], [8]. For the deterministic channel model with no random phases, [8] establishes a O(n I8 n)-scaling for the aggregate throughput, which is not much larger than the E)(\n)-scaling achieved by multihop, even for small values of a.…”

“…Therefore for av < [3], [4], [5], [6], [7], [8]. For the deterministic channel model with no random phases, [8] establishes a O(n I8 n)-scaling for the aggregate throughput, which is not much larger than the E)(\n)-scaling achieved by multihop, even for small values of a. For the more general channel model with random phases considered in this paper, the best result to date [7] shows that whenever av > 4 (av > 4.5 in [6], av > 5 in [4]), nearestneighbour multihop is order-optimal.…”

Achievable rates for Gaussian relay networks with a supplementary non-broadcast link

“…The results have been largely pessimistic, indicating that per-node throughout goes to zero as the network size increases [ 1], [2]. Even worse, the problem appears to be physical in nature and cannot be over come by clever protocols, codes, and modulations, i.e., it appears space itself is a capacity bearing object [3].…”