# Minimizing Interference-to-Signal Ratios in Multi-Cell Telecommunication Networks
## 4 Experiments
### Observations (Scenario 2)  
To cover a wider spectrum of parameters, for Scenario 2, we chose a non-uniform distribution for the base stations and set \(\alpha=3\) to model a rural environment (see ([1], Section 2)). The measured running times were not plotted because they mostly depend on the size of the matrix \(W\) (and very weakly on \(M\)). In fact, it turns out that the overall picture in Scenario 2 is very similar to Scenario 1. Figures 4a and 5a show that as the number of clusters grow, the higher the probability that spectral clustering generates an infeasible solution (containing a cluster without a base station). Even when a feasible solution is found by spectral clustering, most of them have a higher tinf value than the solution produced by Algorithm 2; see Figures 4b and 5b. An example problem generated in the setting of Scenario 2 is visualized in Figure 6, along with the solutions provided by similarity clustering and spectral clustering.  
Figure 3: Scenario 1: Analyzing the output of DP-Similarity Clustering and Spectral Clustering for \(b=200\) BSs and \(u=500\) users; samples obtained over 100 randomly generated problems. Although Spectral Clustering performs slightly better then DP-Similarity Clustering for \(M\leq 26\), its running time is prohibitive in this regime of the total numbers of BSs and users. (**a**) For \(M\geq 65\), more than half of the solutions provided by Spectral Clustering have a cluster that only contains users but zero BSs. (**b**) Mean tinf values as a function of the number of clusters \(M\); the ribbons show the standard deviation over the successfully solved samples. (**c**) For each \(M=1,\ldots,66\), the graph shows the ratios of those samples where DP-Similarity Clustering and where Spectral Clustering provided the solution with the smaller tinf value. Although DP-Similarity Clustering provides the better solution in the majority of cases, it should be noted that the mean tinf values are about \(30\%-50\%\) larger for the Spectral Clustering algorithm. (**d**) Running times for \(b=200\) BSs and \(M=40\) clusters, as the number of users grows; the \(y\)-axis is logarithmic. Note that even for a moderate \(u\approx 100\) users, the running time of Spectral Clustering grows to \(50\) ms, which may be prohibitive in applications.  
Figure 4: Scenario 2: Analyzing the output of DP-Similarity Clustering and Spectral Clustering for \(b=50\) BSs and \(u=200\) users; samples obtained over 100 randomly generated problems. (**a**) For \(M\geq 33\), more than half of the solutions provided by Spectral Clustering have a cluster that only contains users but zero BSs. (**b**) Mean tinf values as a function of the number of clusters \(M\); the ribbons show the standard deviation over the successfully solved samples.  
Figure 5: Scenario 2: Analyzing the output of DP-Similarity Clustering and Spectral Clustering for \(b=200\) BSs and \(u=500\) users; samples obtained over 100 randomly generated problems. (**a**) For \(M\geq 48\), more than half of the solutions provided by Spectral Clustering have a cluster that only contains users but zero BSs. (**b**) Mean tinf values as a function of the number of clusters. \(M\); the ribbons show the standard deviation over the successfully solved samples.