## 2 Materials and Methods
### Function to Find Mutually Acceptable Lines
One of the most comprehensive ways to estimate the bit error rate for multiple mutually interfering lines is to use baseband equivalent simulation for the signals, propagation effects, and transceiver digital signal processing (DSP) pipelines [51]. However, this approach is computationally intensive. Additive white Gaussian noise (AWGN) interference is the simplest assumption and does not require modeling the intrinsics of transceiver DSP stages. One of the main requirements is signal statistical independence. Line signals usually can be considered statistically independent ones due to random transmitted information, random signal carrier phase offset, random carrier frequency fluctuation, random channel propagation conditions, random clock drifts in the case of time division multiple access, etc. Having found the SINR for the line under the AWGN assumption, we can calculate the bit error rate based on the known formulas for various signal types. We can derive the formula which depends only on distances between nodes and signal-to-thermal-noise ratio (SNR) for the signal-to-interference-and-noise ratio (SINR) for every line in a network:  
\[SINR=1/((1/SNR)+(1/SIR))=1/((1/SNR)+\sum(D_{interfering\ line}/D_{interfering\ tx\ to\ rx})), \tag{5}\]  
using the following assumptions:  
* AWGN independent interference. * Free space propagation signal attenuation. * Orthogonal channel resources are time slots, close frequency channels, with which we can neglect the free space propagation attenuation difference between different frequencies, set of mutually orthogonal code sequences using time or close frequencies, or both, and unique permutations of equal size of the aforementioned orthogonal resources. * Every transceiver has the same uniform omnidirectional antenna with isotropic gain equal to one. * Every line has an equal target SNR, and every transceiver uses power control to reach that target. If the target SNR is equal to the minimal necessary to reach the target bit error rate, we cannot have any interference from other lines, and we have to assign different orthogonal resources for each line. If we have some line budget, we can allow line interference. Let us elaborate on the distances in the equation with the help of Figure 5. We have the line from the node 1 transmitter to the node 2 receiver. Active lines between nodes 5 and 6 and nodes 3 and 4 are causing interference to the receiver of node 2. The inverse signal-to-interference ratio (SIR) for the line between nodes 1 and 2 is equal to the following:  
\[\frac{1}{SIR_{line\ 1\to 2}}=\frac{distance_{5\to 6}}{ distance_{5\to 2}}+\frac{distance_{3\to 4}}{distance_{3\to 2}}. \tag{6}\]  
We now have all the components to compute the mutually acceptable lines. The set of unilaterally acceptable lines is as follows:  
\[A_{l}^{unilateral}=\{A_{l}^{simplex}\cap A_{l}^{spatial}\cap A_{l}^{interference }\}. \tag{7}\]  
And the mutually acceptable lines for the line \(l\) are as follows:  
\[A_{(\alpha,n_{a})}^{simplex}=L\setminus\{\{(\alpha,n_{a})\}\cup\{(n,\alpha):n \in N_{a}\}\cup\{(n_{a},n):n\in N_{n_{a}}\}\}. \tag{1}\]0