# A Light Signaling Approach to Node Grouping for Massive MIMO IoT Networks
## 7 Numerical Evaluation
The second comparison algorithm, which we will refer to as _clumped_ node partitioning, is intended to represent a worst-case scenario in which nodes whose channels have similar directional properties are grouped together. In this algorithm, the nodes are first sorted according to their dominant directions. Then the first \(N=\frac{|\mathcal{K}|}{|\mathcal{G}|}\) nodes are placed in the first group, the next \(N\) nodes in the next group, and so on, until all nodes have been assigned a group. The computational complexity of this algorithm is \(O(|\mathcal{K}|log(|\mathcal{K}|)\), since it first sorts the nodes and then performs a sequential group assignment for each node. In our test instances, the number of nodes is always divisible by the number of groups, so there will be no nodes left over with this procedure, and all groups will contain the same number of nodes. Clumped partitioning gives an indication of the detriment to performance that might be caused by a poor node partition with respect to the directional channel properties we consider. The third comparison algorithm, called _power_ partitioning, uses the entire channel matrix as an input, rather than only the directional properties of the node channels, and thus is used as a comparison against the case where the full channel information is obtained before partitioning. To the best of our knowledge, no existing methods from the literature take a similar approach to ours in terms of avoiding the collection of CSI and instead using other properties. Collection of the full CSI is not feasible in practice. This is firstly because it would require partitioning nodes after they have transmitted pilots in the coherence block in which they are to transmit, whereas in reality pilots can only be allocated to nodes after partitioning has been performed. Secondly, in an IoT scenario, the signaling cost for collecting channel information in order to perform partitioning is too high; our aim is to provide a partitioning strategy with low energy usage for the end devices. \begin{table}
\begin{tabular}{l l} \hline \hline
**Node Partitioning Method** & **Description** \\ \hline Optimal Solution & Solving MMAD to optimality. \\ Approximation algorithm & Algorithm 1 \\ \multirow{2}{*}{Clumped partitioning} & Groups nodes with similar directional properties together. Worst-case scenario using directional properties. \\ \multirow{2}{*}{Power partitioning} & Groups nodes with similar received power at the base station together. Partitioning suitable for use with max-min fair power control. Requires full CSI. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Partitioning Algorithms Tested. Power partitioning sorts the nodes according to the received power of their channels at the base station, and then groups those nodes with similar power levels. Similarly to clumped partitioning, this is achieved by assigning the \(N=\frac{|\mathcal{K}|}{|\mathcal{G}|}\) nodes with the highest received power to the first group, the next \(N\) highest power nodes to the next group, and so on. As for clumped partitioning, the computational complexity is \(O(|\mathcal{K}|log(|\mathcal{K}|))\), as power partitioning works in a similar way. This does not include the complexity for collecting and processing the CSI in order to find the received power for each node, since this will depend on the specific implementation for downlink precoding and uplink combining. For MRC precoding, the interference scales with the interfering nodes' received power [3]. Power partitioning thus provides a reasonably fair partition, in which nodes experience interference only from other nodes with similar received power. Power partitioning is thus similar to max-min fair power control [40], in which a common SINR is achieved for all users by adjusting their transmission power such that the received power from each user at the base station is equal.