# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### Tests on Extremely Large Networks with Ground Truth Communities
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Network**} & \multirow{3}{*}{**Size**} & \multirow{3}{*}{**Number**} & \multicolumn{3}{c}{**Numbers of Communities Suggested by**} \\ \cline{3-6}  & & & **of Edges** & **“Ground Truth” 1** & \(Q\) & **Infomap** & \(F_{\alpha}^{\beta}\) **(\(\beta\)=1) 2** \\ \hline Amazon & 334,863 & 925,872 & 4097 & 243 & 141,557 & 3373 \\ DBLP & 317,080 & 1,049,866 & 12,878 & 213 & 302,402 & 114,049 \\ \hline \hline \end{tabular}  
* The “ground truth” communities suggested in [67] contain nesting communities, i.e., communities that are proper subsets of other communities. Numbers of the “ground truth” communities shown in this table have excluded all such properly nested communities. Figure S8 exhibits the distributions of sizes of the ground truth communities for these two networks. 2 Our method (\(F_{\alpha}^{\beta}\) with \(\beta\) = 1) detects different numbers of communities with different resolution parameter \(\alpha\). Numbers of communities shown here are detected at the resolutions having the highest values of normalized mutual information (NMI) to the ground truth communities.  
\end{table} 
Table 2: Network sizes (numbers of nodes), numbers of edges and numbers of communities contained in the Amazon and DBLP networks. and Infomap keeps increasing with the increase of \(\alpha\), until it exceeds the upper bound \(2\beta-1\). To show this trend, we expand the varying range of \(\alpha\) from \((0,1)\) to \((0,1.5)\) for \(\beta=1\) in Figures 9b and 10b. When \(\alpha\) approaches \(1.5\), the NMI between communities detected by our method and Infomap comes up to \(0.81\) for the Amazon network, and \(0.96\) for the DBLP network. Therefore, within proper scales of resolution, our detection results for both networks are highly consistent with those detected by Infomap and \(Q\). Considering the metadata (i.e., \"ground truths\") suggested by [67], our best recovery to the ground truth of the Amazon network is achieved at \(\alpha=0.021\), \(\beta=1\); the corresponding NMI = \(0.47\), which is greater than the NMIs achieved by all other three methods; see Figure 9a,c. On the other hand, our best recovery to the ground truth of the DBLP network emerges at an \(\alpha\) that slightly exceeds the upper bound \(2\beta-1\): when \(\beta=1\) and \(\alpha=1.06\), our best NMI = \(0.69\). This implies that the ground truth of the DBLP network somewhat tends to over-split the network into more communities, just as Infomap does. Our method recovers the ground truth of the DBLP network better than \(Q\), nearly to the same level as Infomap; see Figure 10a. Figure 10: Normalized mutual information (NMI) among the communities detected in the DBLP collaboration network by different methods. (**a**) \(Q\) (blue dashed line), Infomap (red dashed line) and our method (\(F_{\alpha}^{\beta}\) with \(\beta=1\) and varying \(\alpha\)) versus the ground truth suggested in [67]. (**b**) Our method (\(F_{\alpha}^{\beta}\) ) versus \(Q\) and Infomap. (**c**) \(Q_{\gamma}\) versus the ground truth at varying \(\gamma\), where the red horizontal dashed line signs the level of NMI of the best fit between our result and the ground truth, and the vertical dashed line signs the resolution \(\gamma\) at which \(Q_{\gamma}\) starts to outperform our method. Figure 9: Normalized mutual information (NMI) among the communities detected in the Amazon product co-purchasing network by different methods. (**a**) \(Q\) (blue dashed line), Infomap (red dashed line) and our method (\(F_{\alpha}^{\beta}\) with \(\beta=1\) and varying \(\alpha\)) versus the ground truth suggested in [67]. (**b**) Our method (\(F_{\alpha}^{\beta}\) ) versus \(Q\) and Infomap. (**c**) \(Q_{\gamma}\) versus the ground truth at varying \(\gamma\), where the red horizontal dashed line signs the level of NMI of the best fit between our result and the ground truth. As for the multiresolution modularity \(Q_{\gamma}\) on the Amazon network, its performance is not as good as, but still to the same level as, our method.