# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### Strategy to Filter the Output  
We adopt a strict filtering strategy to ensure the selectivity of our method. As a heuristic approach, the Louvain algorithm optimizing our community fitness function often converges to different solutions in different realizations. Many of these solutions are \"local maxima\", which emerge only by chance. To retrieve from messy outputs the most relevant solutions that can be persistently detected, previous methods customarily rely on the stability of plateaus [5; 30; 35; 44; 45; 46], only the term \"plateau\" was, so far, everywhere loosely defined (see our arguments in the Introduction). In this section, we suggest a much stricter definition for plateaus and, correspondingly, a strict strategy to identify them. By our definition, a plateau is a continuous scale of resolution within which the best solutions found by a heuristic optimization algorithm uniformly converges to an identical topological structure of community. To detect such plateaus, it is required to compare not only the topologies of solutions obtained at each fixed resolution, but also those at different resolutions. We suggest the following strategy to discover our plateaus:  
1. At each fixed resolution (i.e., with fixed values of parameters \(\alpha\) and \(\beta\)), we implement the Louvain algorithm on the same network in multiple realizations. Among the outputs of all realizations, we adopt the ones with the highest value of \(F_{\alpha}^{\beta}\) as our best solutions obtained at this resolution. In addition, we require the topology of these best solutions to be unique: in case two or more solutions have equally highest values of \(F_{\alpha}^{\beta}\), but represent different topological structures of community, all solutions obtained at this resolution will be abandoned, and the corresponding resolution will be considered \"irrelevant\" and not contributing to any potential plateau. 2. At different resolutions, with varying values of \(\alpha\) (during which the value of \(\beta\) is still fixed), we run the above step 1 and obtain the best-and-unique solutions at all relevant resolutions. Then we compare the topologies of these best-and-unique solutions, and classify them into different plateaus; solutions classified to the same plateau must represent exactly the same topological structure of community. The above step 1 compares the topologies of communities detected at each fixed resolution. We require topological uniqueness of all our best solutions because otherwise they might be the results of randomness due to the Louvain heuristic; below, we will show some examples. Step 2 compares the topologies of the best-and-unique solutions obtained at varying resolutions. Plateaus defined and identified as above strictly fulfil \"one plateau,one topological structure of community\". Our strict strategy guarantees such plateaus are truly stable, since random outputs of the algorithm have all been filtered out. Only then can the stability (or say, \"robustness\") of solutions be measured by the lengths of their corresponding plateaus. It is common in previous literature [5; 30; 35; 44; 30; 45] to either take the communities detected by the longest plateaus as the most relevant results, or artificially select favored structures that best fit the a priori knowledge about the network. To our viewpoint, all best-and-unique solutions represented by the our strictly-defined plateaus should have their own particular information: they discover different structures that represent different aspects of the given network. Thus in this paper, we do not select plateaus simply by their lengths. Instead, we output and interpret \"only the stable plateaus, and all the stable plateaus\". Figure 1 shows the whole framework of our proposed approach, with all the three components discussed above settled in place. **Step 2. At varying resolutions, classify the \"best-and-unique\" solutions into plateaus.