Impact of traffic localization on communication rates in ad-hoc networks

Abstract

The effect of traffic distribution on communication rates and receiver complexity in ad-hoc networks is addressed, considering a network with constant density of users and a certain traffic model. Information theoretic upper bounds on communication rates are derived under an assumption that transmitting nodes as well as receiving nodes cooperate. It is shown that for the case of large signal attenuation the bounds hold even when the cooperation among users is limited to a certain region of the network domain. Furthermore, achievability bounds on communication rates are derived. The bounds rely on two proposed local cooperation strategies. A comparison shows that the upper bounds are tight and closely follow the achievability results. Finally, the impact of traffic localization on the receiver complexity is addressed.

Appendices

Appendix 1

1.1 Bound on tail probabilities

Consider k random variables X ₁,X ₂,…, X _k (possibly dependent) such that each X _i = ∑ ⁿ_l=1 Z _i,l where Z _i,1,Z _i,2,…, Z _i,n are independent Bernoulli r.v. with parameters p _i,1,p _i,2,…, p _i,n respectively. For any i and l: $\underline{p}\,\leq\,p_{i,l}\,\leq\,\bar{p}$. Then for any positive constant ε satisfying

$$ \lim\limits_{n \rightarrow \infty}(\varepsilon^2 n\underline{p}/4 - \hbox{ln}\, k)= +\infty $$

(23)

the following holds

$$ \begin{aligned} \lim\limits_{n \rightarrow \infty}\Pr[\forall i\,\,X_i \,\leq\,n\bar{p}(1+\varepsilon)] &=& 1 \\ \lim\limits_{n \rightarrow \infty}\Pr[\forall i\,\,n\underline{p}(1-\varepsilon)\,\leq\,X_i)] &=& 1 \end{aligned} $$

Proof

Each random variable X _i has expectation $\mu_i=\sum_{l=1}^{n}p_{i,l}$ and $n\underline{p} {\,\leq\,}{\mu_i}\,\leq\,n\bar{p}$. We use the Chernoff inequality to bound the lower and upper tail probabilities. Consider,

$$ \Pr \left(X_i \,\geq\,\mu_i (1+\varepsilon) \right)\,\leq\,2e^{-\varepsilon^2\mu_i/4} $$

for 0 ≤ ε < 2e−1. Now, applying the union bound gives

$$ \Pr \left(\exists i,\ X_i\,\geq\,\mu_i (1+\varepsilon) \right)\,\leq\,\sum\limits_{i=1}^k \Pr \left( X_i \,\geq\,\mu_i (1+\varepsilon) \right)\,\leq\,k e^{-\varepsilon^2 n\underline{p}/4}\ = e^{-(\varepsilon^2 n \underline{p}/4-\hbox{ln}\, k)} \mathop{\rightarrow}\limits^{{{n} \rightarrow \infty} } 0 $$

(24)

and the first statement of the proposition follows from

$$ \begin{aligned} \Pr \left( \forall i,\ X_i\,\leq\,n\overline{p}(1+\varepsilon) \right) &\,\geq\, 1 - \Pr \left(\exists i, X_i \,\geq\, \mu_i (1+\varepsilon) \right) \end{aligned} $$

(25)

$$ \ge \left(1- e^{-(\varepsilon^2 n \underline{p}/4-\hbox{ln}\, k)}\right) \mathop{\rightarrow}\limits^{{{n} \rightarrow \infty} } 1. $$

(26)

Similarly we obtain

$$ \begin{aligned} \Pr \left( \forall i, X_i \,\geq\, n\overline{p}(1-\varepsilon) \right) &= 1 - \Pr \left(\exists i, X_i\,\leq\,n \overline{p} (1-\varepsilon) \right) \end{aligned} $$

(27)

$$ \ge \left(1- e^{-(\varepsilon^2 n \underline{p}/4-\hbox{ln}\, k)} \right)\mathop{\rightarrow}\limits^{{{n} \rightarrow \infty} } 1. $$

(28)

Appendix 2

2.1 Some important integrals

Let us define an integral over an intersection of two spheres of radius r ₁ and r ₂ centered at the origin of a d dimensional space by

$$ {\mathcal{I}}(r_1,r_2,m,\zeta) \mathop{=}\limits^{\rm def} \mathop{\int}\limits_{{\mathcal{S}}(r_2) \setminus {\mathcal{S}}(r_1)} \frac{\left[\sum_{j=1}^d x_j^2\right]^{\frac{m}{2}}}{1+\left[\sum_{j=1}^d x_j^2\right]^{\frac{\zeta}{2}}} \, \hbox{d} x_1 \, \ldots \hbox{d} x_d. $$

(29)

In order to bound it we make a coordinate exchange

$$ \begin{aligned} \,& x_1 = r \sin(\theta_1)\sin(\theta_2) \ldots \sin(\theta_{d-1}), x_2 = r \sin(\theta_1)\sin(\theta_2)\ldots \cos(\theta_{d-1}), \ldots, \\ \,& x_{d-1} =r\sin(\theta_1)\cos(\theta_2), x_d = r\cos(\theta_1) \end{aligned} $$

and the determinant of the transformation is

$$ \hbox{det}(J_d) = -r^{d-1} \sin^{d-2}(\theta_1)\sin^{d-3}(\theta_2) \ldots \sin(\theta_{d-2}). $$

(30)

Let us define a value

$$ S_d \mathop{\hbox{=}}\limits^{\rm def}\int_0^{\pi} \cdots \int_0^{\pi} \int_0^{2\pi} \frac{|\hbox{det}(J_d)|}{r^{d-1}}\, \\ \hbox{d} \theta_1 \,\ldots \\ \hbox{d} \theta_{d-1} = \left\{ \begin{array}{ll} \frac{2^{(d+1)/2} \pi^{(d-1)/2}}{(d-2)!!}, &\quad \hbox{ for } d \hbox{ odd } \\ \frac{2 \pi^{d/2}}{(\frac{1}{2}d-1)!}, & \quad \hbox{ for } d \hbox{ even }. \end{array} \right. $$

(31)

Then (29) can be written as

$$ {\mathcal{I}}(r_1,r_2,m,\zeta) = S_d \int\limits_{r_1}^{r_2} \frac{r^{m+d-1}}{1+r^\zeta} \\ \hbox{d} r $$

(32)

Easy bounding arguments lead to the following bounds

• Case 1 ζ < m + d:

  $$ \begin{aligned} &\underline{c}_{m,\zeta,d} \left( r_2^{m-\zeta+d} - r_1^{m-\zeta+d} \right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,\zeta,d} \left ( r_2^{m-\zeta+d} - r_1^{m-\zeta+d}\right) &\hbox{ for } r_1, r_2 \ge 1 \\ &\underline{c}_{m,d} \left( r_2^{m+d} - r_1^{m+d} \right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,d} \left( r_2^{m+d} - r_1^{m+d}\right) &\hbox{ for } r_1 < r_2 < 1 \\ &\underline{c}_{m,\zeta,d} r_2^{m-\zeta+d} - \underline{c}_{m,d}\left(1-r_1^{m+d} \right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,\zeta,d} r_2^{m-\zeta+d} - \overline{c}_{m,d} \left(1+r_1^{m+d}\right) & \hbox{ for } r_1 < 1 < r_2 \end{aligned} $$

• Case 2 ζ = m + d:

  $$ \begin{aligned} \,& \frac{S_d}{2} \left(\hbox{ln}\, r_2 - \hbox{ln}\, r_1 \right) < \mathcal{I}(r_1,r_2,m,\zeta) < S_d\left(\hbox{ln}\, r_2 - \hbox{ln}\, r_1 \right) & \hbox{ for } r_1, r_2 \ge 1 \\ \,&\underline{c}_{m,\zeta,d} \left( r_2^{m+d} - r_1^{m+d} \right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,\zeta,d} \left( r_2^{m+d} - r_1^{m+d}\right) & \hbox{ for } r_1 < r_2 < 1 \\ \,& \underline{c}_{m,d} \left(1-r_1^{m+d} \right)+ \frac{{\hbox{ln}}\, r_2}{2} < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,d} \left(1-r_1^{m+d}\right) + \hbox{ln}\, r_2 & \hbox{ for } r_1 < 1 < r_2 \end{aligned} $$

• Case 3 ζ > m + d:

  $$ \begin{aligned} & \underline{c}_{m,\zeta,d}\left( r_1^{m-\zeta+d} - r_2^{m-\zeta+d} \right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,\zeta,d}\left( r_1^{m-\zeta+d} - r_2^{m-\zeta+d} \right) &\hbox{ for } r_1, r_2 \ge 1 \\ & \underline{c}_{m,d}\left( r_2^{m+d} - r_1^{m+d} \right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,d}\left( r_2^{m+d} - r_1^{m+d}\right) &\hbox{ for } r_1 < r_2 < 1 \\ & \underline{c}_{m,\zeta,d} \left(1-r_2^{m-\zeta+d}\right) + \underline{c}_{m,d} \left(1-r_1^{m+d}\right) < \mathcal{I}(r_1,r_2,m,\zeta) < \overline{c}_{m,\zeta,d}\left(1-r_2^{m-\zeta+d} \right) + \\ &+ \overline{c}_{m,d}\left(1-r_1^{m+d}\right) & \hbox{ for } r_1 < 1 < r_2 \end{aligned} $$

  where $\underline{c}_{m,\zeta,d} = |S_d/(2m+2d-2\zeta)|$, $\underline{c}_{m,d} = S_d/(2m+2d)$, $\overline{c}_{m,d}= S_d/(m+d)$ and $\overline{c}_{m,\zeta,d} = |S_d/(m+d-\zeta)|$ are positive constants with respect to r₁ and r₂.

Appendix 3

3.1 Proof of Theorem 1

By $\Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i})$ we will denote the probability that ${\mathbf{x}}$ chooses its destination ${\mathbf{y}}$ inside the same cell $D_{{\mathbf{x}},i}$

$$ \Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i}) = \frac{\int\limits_{D_{{\mathbf{x}},i}} \frac{1}{1+\left[\sum_{j=1}^d(x_j-y_j)^2\right]^{\beta/4}} \\ \hbox{d} y_1 \ldots \\ \hbox{d} y_d }{\int\limits_{\left[0,n^{1/d}\right]^{d}} \frac{1}{1+\left[\sum_{j=1}^d(x_j-y_j)^2\right]^{\beta/4}} \\ \hbox{d} y_1 \ldots \\ \hbox{d} y_d } $$

(33)

To make an upper bound on this probability we integrate in the nominator over a sphere ${\mathcal{S}}_{{\mathbf{x}}}(\sqrt{d}\overline{b}(n)) \supset D_{{\mathbf{x}},i}$. In the denominator we integrate over 1/2^d part of the sphere ${\mathcal{S}}_{{\mathbf{x}}}(\frac{1}{2}n^{1/d})$ contained in [0,n ^1/d]^d. Thus

$$ \Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i})\,\leq\,2^d\frac{{\mathcal{I}}(0,\sqrt{d}\overline{b}(n),0,\frac{\beta}{2})}{{\mathcal{I}}(0,\frac{1 }{2}n^{1/d},0,\frac{\beta}{2})} $$

(34)

We use the fact that slightly more than half of the transmissions which go outside the cell, go to the other domain due to the chess board pattern. Therefore, $\Pr({\mathbf{x}} \rightarrow {\mathcal{D}}_{\mathbf{y}}) \ge (1-\Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i}))/2$. We consider the following cases.

Case 0 ≤ β < 2d. Applying the upper and lower bounds on ${\mathcal{I}}$ calculated in Appendix 2 to (34) we get

$$ \begin{aligned} \forall {\mathbf{x}} \in D_{{\mathbf{x}},i}, \Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i})\,\leq\,2^d \frac{\overline{c}_{0,\frac{\beta}{2},d}(\sqrt{d}\overline{b}(n))^{d-\frac{\beta}{2}}}{ \underline{c}_{0,\frac{\beta}{2},d}(\frac{1}{2}n^{1/d})^{d-\frac{\beta}{2}}} \end{aligned} $$

(35)

Substituting for $\overline{b}(n) \le \left[2^{2d+1-\beta/2}(\overline{c}_{0,\beta/2,d})(\underline{c}_{0,\beta/2,d}^{-1})d^{d/2- \beta/4}\right]^{\frac{1}{\beta/2-d}} n^{\frac{1}{d}}$, we get $\Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i}) \le \frac{1}{2}$ and therefore $\Pr({\mathbf{x}}\rightarrow {\mathcal{D}}_{\mathbf{y}}) \ge \frac{1}{4}$.

Case β = 2d.

$$ \begin{aligned} \forall {\mathbf{x}} \in D_{{\mathbf{x}},i}, \Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i})\,\leq\,2^d \frac{\overline{c}_{0,d}\hbox{ln\,}(\sqrt{d}\overline{b}(n))}{\underline{c}_{0,d}\hbox{ln\,}(\frac{1}{2}n^{1/d})}\,\leq\,\frac{2^{d+1}\left(\frac{1}{d} + \frac{1}{2} \hbox{ln\,} d + \hbox{ln\,} \overline{b}(n) \right)}{\frac{1}{d} \hbox{ln\,} n - \hbox{ln}\, 2} \end{aligned} $$

(36)

substituting for $\overline{b}(n) \le \frac{1}{4}n^{\frac{1}{d2^{d+2}}}$ we get $\Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i}) {\le}\frac{1}{2}$ and $\Pr({\mathbf{x}}\rightarrow {\mathcal{D}}_{{\mathbf{y}}}) \ge \frac{1}{4}$.

Case β > 2d.

$$ \begin{aligned} \forall {\mathbf{x}} \in D_{{\mathbf{x}},i}, \Pr({\mathbf{x}}\rightarrow D_{{\mathbf{x}},i})\,\leq\,2^d \frac{\frac{1}{d}(\sqrt{d}\overline{b}(n))^{d}}{\frac{1}{d}}\,\leq\,(2\sqrt{d}\overline{b}(n))^d < \frac{1}{2} \end{aligned} $$

(37)

for $\overline{b}(n) < \frac{1}{2\sqrt{d}2^{1/d}}$. Thus $\Pr({\mathbf{x}} \rightarrow {\mathcal{D}}_{\mathbf{y}}) \ge 1/4$.

Appendix 4

4.1 Proof of Theorem 2

Proof

We consider

$$ \begin{aligned} \frac{1}{n}{\mathbb{E}}\sum\limits_{i=1}^{n/2}\sum\limits_{j=1}^{n/2}\frac{1}{1+| {\mathbf{x}}_i-{\mathbf{y}}_j|^{\alpha}} &= \frac{1}{n}\int\limits_{{\mathcal{D}}_{{\mathbf{x}}}} \int\limits_{{\mathcal{D}}_{{\mathbf{y}}}} \frac{1}{1+\left[\sum\limits_{l=1}^d(x_l-y_l)^2\right]^{\frac{\alpha}{2}}} \, \hbox{d} x_1 \ldots \hbox{d} x_d \, \hbox{d} y_1 \ldots \hbox{d} y_d \\ & \le \frac{1}{n} \sum\limits_{i=1}^{\frac{n}{2\overline{b}(n)^d}} \int\limits_{D_{{\mathbf{x}},i}}\int\limits_{{\mathcal{D}}_{{\mathbf{y}}}} \frac{1}{1+\left[\sum\limits_{l=1}^d (x_l-y_l)^2\right]^{\frac{\alpha}{2}}} \, \hbox{d} x_1 \ldots \hbox{d} x_d \, \hbox{d} y_1 \ldots \hbox{d} y_d. \end{aligned} $$

(38)

Consider an arbitrary cell $D_{{\mathbf{x}},i} \subset {\mathcal{D}}_{{\mathbf{x}}}$. There are at most 3^d − 1 neighboring cells $D_{{\mathbf{y}},j}^{n} \subset {\mathcal{D}}_{{\mathbf{y}}},$ j = 1…3^d − 1 such that the intersection of $D_{{\mathbf{x}},i} \cap D_{{\mathbf{y}},j}^n$ is nonempty.

All other cells of ${\mathcal{D}}_{{\mathbf{y}}}$ are separated from $D_{{\mathbf{x}},i}$ by a distance of at least $\overline{b}(n)$. Let ${\mathcal{D}}_{{\mathbf{y}}}^* \mathop{=}\limits^{\rm def} {\mathcal{D}}_{{\mathbf{y}}}{\setminus}\cup_{j=1}^{3^d-1} D_{{\mathbf{y}},j}^n$. Then, (38) is equal to

$$ \begin{aligned} \,& \frac{1}{n} \sum_{i=1}^{\frac{n}{2\overline{b}(n)^d}} \Bigg( \sum_{j=1}^{3^d-1} \int \limits_{D_{{\mathbf{x}},i}} \int \limits_{D_{{\mathbf{y}},j}^n} \frac{1}{1+\left[\sum \limits_{l=1}^d (x_l-y_l)^2\right]^{\frac{\alpha}{2}}} \hbox{d} x_1 \ldots \hbox{d} x_d \cdot \hbox{d} y_1 \ldots \hbox{d} y_d \\ \,&+ \int \limits_{D_{{\mathbf{x}},i}} \int \limits_{{\mathcal{D}}_{{\mathbf{y}}}^*} \frac{1}{1+\left[\sum \limits_{l=1}^d (x_l-y_l)^2\right]^{\frac{\alpha}{2}}} \hbox{d} x_1 \ldots \hbox{d} x_d \hbox{d} y_1 \ldots \hbox{d} y_d \Bigg) \\ \,&\,\leq\,\frac{1}{n} \sum_{i=1}^{\frac{n}{2\overline{b}(n)^d}} \left( (3^d-1) K(\overline{b}(n),\alpha,d)+\overline{b}(n)^d{\mathcal{I}}(\overline{b}(n), \sqrt{d}n^{1/d},0,\alpha)\right) \end{aligned} $$

(39)

where in the second term for any ${\mathbf{x}} \in D_{{\mathbf{x}},i}$ we integrate over $({\mathcal{S}}_{{\mathbf{x}}}(\sqrt{d}n^{1/d})\setminus{\mathcal{S}}_{{\mathbf{x}}}(\overline{b}( n)))\supset {\mathcal{D}}_{{\mathbf{y}}}^*$. The first term is equal to

$$ \begin{aligned} K(\overline{b}(n),\alpha,d) &\mathop{=}\limits^{\rm def} \int\limits_{[-\overline{b}(n),0]\times[0,\overline{b}(n)]^{d-1}} \int\limits_{[0,\overline{b}(n)]^d} \frac{1}{1+\left[\sum\limits_{l=1}^d (x_l-y_l)^2\right]^{\frac{\alpha}{2}}} \hbox{d} y_1 \ldots \hbox{d} y_d \hbox{d} x_1 \ldots \hbox{d} x_d \end{aligned} $$

(40)

With exchange of variables $\tilde{x}_1 = -x_1$, $\tilde{x}_2 = x_2$,…, $\tilde{x}_d = x_d$, $\tilde{y}_1 = y_1$, and $\tilde{y}_2 = y_2-x_2$,…, $\tilde{y}_d = y_d-x_d$ we get

$$ \begin{aligned} \,&K(\overline{b}(n),\alpha,d) = \int \limits_0^{\overline{b}(n)} \ldots \int \limits_0^{\overline{b}(n)} \int\limits_{-\tilde{x}_d}^{\overline{b}(n)-\tilde{x}_d} \ldots \int\limits_{-\tilde{x}_2}^{\overline{b}(n)-\tilde{x}_2} \frac{1}{1+\biggl[(\tilde{x}_1+\tilde{y}_1)^2+\sum\limits_{j=2}^d \tilde{y}_j^2\biggr]^{ \frac{\alpha}{2}}} \hbox{d} \tilde{y}_1 \ldots \, \hbox{d} \tilde{y}_d \, \hbox{d} \tilde{x}_1 \ldots \hbox{d} \tilde{x}_d \\ \,&\le \int \limits_0^{\overline{b}(n)} \ldots \int \limits_0^{\overline{b}(n)} \int \limits_{-\overline{b}(n)}^{\overline{b}(n)} \ldots \int \limits_{-\overline{b}(n)}^{\overline{b}(n)} \frac{1}{1+\left[\tilde{x}_1^2+\tilde{y}_1^2+ \sum \limits_{j=2}^d \tilde{y}_j^2\right]^{\frac{\alpha}{2}}} \hbox{d} \tilde{y}_1 \ldots \, \hbox{d} \tilde{y}_d \, \hbox{d} \tilde{x}_1 \ldots \hbox{d} \tilde{x}_d\\ \,&= \overline{b}(n)^{d-1} \int \limits_0^{\overline{b}(n)} \int \limits_{-\overline{b}(n)}^{\overline{b}(n)}\ldots \int \limits_{-\overline{b}(n)}^{\overline{b}(n)} \frac{1}{1+\left[\tilde{x}_1^2+\tilde{y}_1^2+ \sum \limits_{j=2}^d\tilde{y}_j^2\right]^{\frac{\alpha}{2}}} \hbox{d} \tilde{y}_1 \ldots \, \hbox{d} \tilde{y}_d \, \hbox{d} \tilde{x}_1\,\leq\,\overline{b}(n)^{d-1}\mathcal{I}(0,\sqrt{d+1}\overline{b}(n),1,\alpha) \end{aligned} $$

where we integrate over a d + 1-dimensional sphere of radius $\overline{b}(n)\sqrt{d+1}$ covering the domain $[-\overline{b}(n),\overline{b}(n)]^{d+1}$. Substituting this result into (39) and using the upper bounds from Appendix 2 leads to

$$ \begin{aligned} \frac{1}{n}{\mathbb{E}}\sum\limits_{i=1}^{n/2}\sum\limits_{j=1}^{n/2}\frac{1}{1+| {\mathbf{x}}_i-{\mathbf{y}}_j|^{\alpha}}\,\leq\,c(d,\alpha) \times\left\{ \begin{array}{ll} n^{\frac{d-\alpha}{d}}, & 0 \le \alpha < d \\ \hbox{ln}\, n, & \alpha = d \\ \overline{b}(n)^{d-\alpha}, & d < \alpha < d+1 \\ \frac{\hbox{ln}\, \overline{b}(n)}{\overline{b}(n)}, & \alpha = d+1 \\ \frac{1}{\overline{b}(n)}, & \alpha > d+1,\, \end{array} \right. \end{aligned} $$

(41)

where c(d,α) is independent of $\overline{b}(n)$ and n. The upper bound on transmission rate per communicating pair is obtained by substituting (4) into (8)

$$ R \le {\mathbb{E}}C(\user2{G},P) \le c(d,\alpha) \times \left\{ \begin{array}{ll} \hbox{ln}\, n, & 0 \le \alpha < d \\ \hbox{ln}\,(\hbox{ln}\, n), & \alpha = d \\ \overline{b}(n)^{d-\alpha}, & d < \alpha < d+1 \\ \frac{\hbox{ln}\, \overline{b}(n)}{\overline{b}(n)}, & \alpha = d+1 \\ \frac{1}{\overline{b}(n)}, & \alpha > d+1 \end{array} \right. $$

(42)

.

Finally, we use the bound on $\overline{b}(n)$ from Theorem 1 to obtain the upper bounds on the rate per communicating pair for different cases of β. The results are tabulated in Table 2.

Appendix 5

5.1 Proof of Lemma 5

The interference experienced by a particular node ${\mathbf{x}} =(x_1, x_2, \ldots, x_d) \in D_i$ can be upper bounded as

$$ I\,\leq\,\int \limits_{{\mathcal{D}}\setminus (D_i \bigcup \limits_{j} D_j^{{\rm n}}) } \frac{P}{1+\left[ \sum \limits_{i=1}^{d}(x_i - x^{\prime}_i)^2\right]^{\frac{\alpha}{2}}} \\ \hbox{d} x^{\prime}_1 \ldots \\ \hbox{d} x^{\prime}_d. $$

(43)

assuming that all the nodes outside D _i ’s neighbors transmit. We integrate over ${\mathcal{S}}_{{\mathbf{x}}}(\sqrt{d}n^{1/d}){\setminus}{\mathcal{S}}_{{\mathbf{x}}}(\underline{b}(n)) \supset {\mathcal{D}}{\setminus}(D_i \bigcup \limits_j D_j^{\rm n})$ and get

$$ I\,\leq\,P{\mathcal{I}}(\underline{b}(n),\sqrt{d}n^{1/d},0,\alpha) \rightarrow 0 $$

(44)

for α > d and $n \rightarrow \infty$.

For α > d the fraction inside the logarithm tends to zero when n goes to infinity. Hence we can use the inequality ln(1 + x) ≥ x/2 (for 0 < x < 1) and obtain

$$ R_{\rm cell}\,\geq\,\frac{WP\underline{b}(n)^d}{4\hbox{ln}\, 2(2L+1)^d 2^{\alpha+1} d^{\frac{\alpha}{2}}\underline{b}(n)^{\alpha}(I+N_0)}. $$

(45)

Substituting (44) in (45) gives the result.

Appendix 6

6.1 Proof of Lemma 6

Node ${\mathbf{x}} = (x_1, x_2, \ldots x_d)$ in the network domain chooses its destination ${\mathbf{x}}^{\prime}$ randomly according to the probability distribution with density function given by

$$ \pi_{{\mathbf{x}}}({\mathbf{x}}^{\prime})= \frac{c_{\mathbf{x}}(\beta)}{1+\left[\sum\limits_{j=1}^d(x_ j - x^{\prime}_j)^2 \right]^{\frac{\beta}{4}}}, \quad\hbox{where}, c_{\mathbf{x}}(\beta) = \left[\int\limits_{{\mathcal{D}}} \frac{1}{1+\left[\sum_{j=1}^d(x_j-x^{\prime}_j)^2 \right]^{\beta/4}} \hbox{d} x^{\prime}_1 \hbox{d} x^{\prime}_2 \ldots \hbox{d} x^{\prime}_d \right]^{-1} $$

(46)

is a normalization constant. We upper bound the average distance from ${\mathbf{x}}$ to its destination as follows

$$ \bar{s} = {\mathbb{E}}|{\mathbf{x}}-{\mathbf{x}}^{\prime}|\,\leq\,c_{\mathbf{x}}(\beta) \int_{{\mathcal{D}}} \frac{\left[\sum_{j=1}^d x^{'2}_j\right]^{\frac{1}{2}}}{1+\left[\sum_{j=1}^d x^{'2}_j\right]^{\frac{\beta}{4}}} \hbox{d} x^{\prime}_1 \hbox{d} x^{\prime}_2 \ldots \hbox{d} x^{\prime}_d\,\leq\,\frac{{\mathcal{I}}(0,\sqrt{d}n^{1/d},1,\beta/2)}{{\mathcal{I}}(0,\frac{1}{2}n^{1/d}, 0,\beta/2)}. $$

(47)

Simplifying (47), we obtain (19) for different values of d and β.

Appendix 7

7.1 Proof of Lemma 7

We numerate the routes between SD pairs with index l = 1…n/2 and denote the probability that route l passes through cell D _i by p _l,i. We observe that this probability does not depend on l but may depend on i. Hence, ${\forall}l, p_{l,i} =p_i$. Consider

$$ \begin{aligned} \sum\limits_{i=1}^{n/\underline{b}(n){^d}} {\mathbb{E}} {\hbox{{\{Number of paths through }}} D_i\} = {\mathbb{E}} {\hbox{\{Number of cells crossed by all}}\, n\, {\hbox{paths}}\}\\\,\leq\,{\mathbb{E}} {\hbox{\{Length of all}}\,n\, {\hbox{paths}}\}/\underline{b}(n) = n \bar{s}/\underline{b}(n) \end{aligned}$$

(48)

On the other hand

$$ {\mathbb{E}} \{\hbox{Number of paths through $D_i$}\} = \sum_{l=1}^n p_{l,i} = n p_{i}. $$

(49)

Now consider cells in the center of the domain i.e. in a sub-domain $\tilde{{\mathcal{D}}} = [1/4n^{1/d},3/4n^{1/d}]^d$. The probability p _i is the largest for the central cell $(\forall i, p_i \le p_{\rm c})$. From geometric reasons it follows that p _i /p _c ≥1/2^d. Therefore, from (48) we have

$$ \frac{n\bar{s}}{\underline{b}(n)} = \sum_{i=1}^{n/\underline{b}(n)^d} {\mathbb{E}} \{\hbox{Number of paths through $D_i$}\}\,\geq\,\frac{1}{2^{d}}\sum_{D_i \in \tilde{{\mathcal{D}}}} n p_{\rm c}\,\geq\,\frac{1}{2^{2d}} \frac{n^2}{\underline{b}(n)^d} p_{\rm c} $$

(50)

Thus

$$ p_{l,i} = p_{i} \,\leq\,p_{\rm c}\,\leq\,2^{2d} \frac{\bar{s} \underline{b}(n)^{d-1}}{n}. $$

(51)

The proof of statement (20) now follows from the Appendix 1 for m = n, $\underline{p} = \underline{b}(n)^d /n$, $\overline{p}= 2^{2d} \frac{\bar{s} \underline{b}(n)^{d-1}}{n}$, $k=n/\underline{b}(n)^d$ and ε = 1/2.