exponential tail.

Fig. \[fig:loyaltyparam\]-a depicts as an example the results for the $N=6$ case obtained with different $p$. At $p=1.2$ (black curve) the algorithm detects two big clusters with 74.6% and 25.1% of the population plus two satellites due to noise with 0.3% of cumulative population. With $p=1.4$ (dark grey curve) the second cluster at $p=1.2$ splits into eight clusters with smaller relative populations ranging from 1.0% to 7.8%, while the noise is composed by the remaining 20 clusters (nicely fitted by an exponential function in Fig. \[fig:loyaltyparam\]-a). At $p=1.6$ (light grey curve) only four significant clusters are detected. Their populations are 41.5%, 14.9%, 8.5% and 7.1% of the total population (cumulatively the 72.0%), while the remaining 80 clusters have a cumulative relative population of 28.0% and constitute noise. With even higher values of $p$ the network breaks more and more into small noisy clusters.

These three scenarios show how changing the granularity parameter $p$ leads to different signal to noise ratio. This behaviour is not necessarily monotonic: incrementing $p$ at first increases the number of significative clusters up to a maximum after which the noise grows and becomes dominant. However, similarly to the choice of $S^{*}$, our priority is 