# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 4 Discussion
### Scalability of the Community Fitness Function  
The scalability of our community fitness function (Formula (2)) originates from two aspects. First, its original form (Formula (1)) as introduced in [30] is, by design, more scalable than other community quality functions, for example, the standard modularity \(Q\) proposed by [PERSON] [27], which is generally reformulated as [3; 5; 8; 42]  
\[Q=\sum_{\mathcal{G}}Q^{\mathcal{G}}=\sum_{\mathcal{G}}\left[\frac{k^{\mathcal{ G}}_{in}}{2M}-\left(\frac{k^{\mathcal{G}}}{2M}\right)^{2}\right] \tag{6}\]  
Here, \(M\) is the total number of edges within the whole network; \(Q^{\mathcal{G}}\), \(k^{\mathcal{G}}_{in}\) and \(k^{\mathcal{G}}=k^{\mathcal{G}}_{in}+k^{\mathcal{G}}_{out}\) are the modularity, in-degree and total degree of community \(\mathcal{G}\). Obviously, the value of \(Q^{\mathcal{G}}\) depends heavily on the community size: in an LFR network, since \(k^{\mathcal{G}}_{in}/k^{\mathcal{G}}=1-\mu\) (\(\mu\) is the mixing parameter),  
\[Q^{\mathcal{G}}=\frac{k^{\mathcal{G}}_{in}}{2M}-\left(\frac{k^{\mathcal{G}}}{2M }\right)^{2}=\frac{k^{\mathcal{G}}_{in}}{2M}\bigg{[}1-\frac{k^{\mathcal{G}}}{2M (1-\mu)}\bigg{]}. \tag{7}\]  
In large networks, it can be expected that \(2M\gg k^{\mathcal{G}}\), so \(\frac{k^{\mathcal{G}}}{2M(1-\mu)}\approx 0\), thus \(Q^{\mathcal{G}}\approx\frac{k^{\mathcal{G}}_{in}}{2M}\propto k^{\mathcal{G}}_{in}\). This reflects a fact that the modularity \(Q^{\mathcal{G}}\) increases almost linearly with the community size (here, without loss of generality, we measure the community size by the in-degreerather than the number of nodes). Large gaps of \(Q^{\mathcal{G}}\) exist between communities of different sizes in a heterogeneous network, which inhibits simultaneous detections on communities of distinct sizes. In contrast, the community fitness function (Formula (1)) is  
\[f^{\mathcal{G}}=\frac{k_{in}^{\mathcal{G}}}{\left(k^{\mathcal{G}}\right)^{\alpha }}=(1-\mu)^{\alpha}(k_{in}^{\mathcal{G}})^{1-\alpha}\propto\left(k_{in}^{ \mathcal{G}}\right)^{1-\alpha} \tag{8}\]  
Since \(\alpha>0\), apparently \(f^{\mathcal{G}}\) increases not as fast as \(Q^{\mathcal{G}}\) with the community size: it tends to narrow the gap between large and small communities. As a result, in a heterogeneous network, communities having close densities of inner connections but far different sizes can be simultaneously identified by the community fitness function \(f^{\mathcal{G}}\), but not by the modularity function \(Q\). The multiresolution version of \(Q\) proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2008) does not solve the problem: it introduces a resolution parameter \(\gamma\) as in Formula (4):  
\[Q_{\gamma}=\sum_{\mathcal{G}}\left[\frac{k_{in}^{\mathcal{G}}}{2M}-\gamma \left(\frac{k^{\mathcal{G}}}{2M}\right)^{2}\right] \tag{9}\]  
However, since \(\left(\frac{k^{\mathcal{G}}}{2M}\right)^{2}\) is a minor item of the formula, \(\gamma\) is not effective enough to rescale \(Q_{\gamma}\) and overcome the resolution limit problem ([PERSON] and [PERSON], 2011). A similar problem also holds for a majority of previous multiresolution methods, including the popular Hamiltonian-based Potts models ([PERSON] and [PERSON], 2011; [PERSON] and [PERSON], 2011; [PERSON] and [PERSON], 2011; [PERSON] and [PERSON], 2011). That explains why previous multiresolution methods perform poorly on heterogeneous networks, as argued by ([PERSON] and [PERSON], 2011), while our method in this paper has outperformed all of them (see Section 3.2). The second origination of the scalability is the scaling factor \(\beta\): to some extent \(\beta\) makes the fitness function \"re-scalable\". As we have discussed in Section 2.1, the original community fitness function has \(\beta\) fixed to 1, and the varying range of the resolution parameter \(\alpha\) is between 0 and 1. According to our estimation in Appendix A, such a varying range is a \"relevant\" scale of resolution, within which both the merging of small communities and the splitting of large communities are restricted.