# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### On the Heterogeneous Lancichinetti-Fortunato-Radicchi (LFR) Benchmark Networks
We use \(\mu_{1}\) in each figure to indicate the threshold at which the NMIs of our strongest plateaus start to deviate from perfection: when \(\mu\leq\mu_{1}\), the NMIs always equal to 1. When \(\mu>\mu_{1}\), our strongest plateaus detect the whole network as one community, thus, the NMIs suddenly decrease to 0. However, our second strongest plateaus still recover the implanted communities perfectly and retain high levels of NMIs, until \(\mu\) becomes really large. Similarly, we use \(\mu_{2}\) to indicate the same threshold for Infomap: when \(\mu>\mu_{2}\), the whole network is detected as one single community. Since for Infomap, there is no \"secondary solutions,\" thus, when \(\mu>\mu_{2}\), the NMI always equals to 0. Previous multiresolution methods tested in [42] mostly perform well on the \"small\" networks in Figure 4--but they all perform much worse on the large and heterogeneous networks in Figure 5. In contrast, our method, being also a multiresolution method, recreates the implanted communities for all networks in Figures 4 and 5 as perfectly as Infomap within the range \(\mu\leq\mu_{2}\); their NMI curves roughly overlap with each other. More specifically, when \(\mu\leq\mu_{1}\) or \(\mu>\mu_{2}\), our strongest plateaus yield exactly the same results as Infomap; when \(\mu_{1}<\mu<\mu_{2}\), our second strongest plateaus and Infomap recover the implanted communities to the same extent of perfection. Compared to previous methods of multiresolution that are tested in [42], our method performs more robustly on large heterogeneous networks. It seems to have overcome the resolution limit problem causedby network heterogeneity. As we will suggest in the Discussion, such an outperformance, too, can be attributed to the scalability of our community fitness function. In addition, our best-fit NMIs shown in Figures 4 and 5 are not artificially selected. For a given network, usually it is our strongest plateau that suggests the most significant result. In case the strongest plateau suggests to trivially merge the whole network into one community, the second strongest plateau is the most informative. This means that our best-fit solutions are detected _automatically_, without the need of knowing any information about the implanted communities. In contrast, previous methods including that in [35], rely on the a priori knowledge to judge from a mess of detection results which ones are the best--without knowing the implanted communities, there is no clue to pick out the results with the \"strongest correlations\" [35]. The automaticity of our method should be attributed to the filtering strategy suggested in Section 2.3, which guarantees both the stability, and the significance, of all our final outputs. Figure 4: Normalized mutual information (NMI) between the communities detected in relatively small LFR networks by our method and Infomap against the implanted communities with varying \(\mu\). The following network parameters are shared among all subfigures: average node degree \(\langle k\rangle=20\), maximum node degree \(k_{\max}=50\), power-law exponent of the degree distribution of nodes \(\tau_{1}=-2\), and that of the community size \(\tau_{2}=-1\). Other parameters including the network size \(N\) and the range of community size \(s\) are labelled in each subfigure: (**a**) \(N=1000\), \(10\leq\mathrm{s}\leq 50\); (**b**) \(N=1000\), \(20\leq\mathrm{s}\leq 100\); (**c**) \(N=5000\), \(10\leq\mathrm{s}\leq 50\); (**d**) \(N=1000\), \(20\leq\mathrm{s}\leq 100\).