# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## Appendix A Relevant Range of the Resolution Parameter \(\alpha\)
### Lower Bound of \(\alpha\): Merging Complete Graphs  
Merging small and dense communities into larger but sparser ones, reflects the resolution limit problem at the other end of the resolution scale: small values of \(\alpha\) may cause this problem. Suppose we have a couple of complete graphs: \(\mathcal{M}\) consisting of \(m\) nodes and \(\mathcal{N}\) consisting of \(n\) nodes. If \(\mathcal{M}\) and \(\mathcal{N}\) are identified as two independent communities, it is straightforward to calculate their in-degrees: \(k_{in}^{\mathcal{M}}=m(m-1)\), \(k_{in}^{\mathcal{N}}=n(n-1)\). As for their out-degrees, for simplicity, we assume \(\mathcal{M}\) and \(\mathcal{N}\) are disconnected, i.e., \(k_{out}^{\mathcal{M}}=k_{out}^{\mathcal{N}}=0\)Then, the fitness of the network with \(\mathcal{M}\) and \(\mathcal{N}\) identified as separate communities can be calculated as  
\[\left(F_{\alpha}^{\beta}\right)^{\mathcal{M}}+\left(F_{\alpha}^{\beta}\right)^{ \mathcal{N}}=m^{\beta-\alpha}(m-1)^{\beta-\alpha}+n^{\beta-\alpha}(n-1)^{\beta -\alpha} \tag{11}\]  
For convenience, denote \(m(m-1)=a,n(n-1)=b\), \(\beta-\alpha=k\), then (11) becomes  
\[\left(F_{\alpha}^{\beta}\right)^{\mathcal{M}}+\left(F_{\alpha}^{\beta}\right)^ {\mathcal{N}}=a^{k}+b^{k} \tag{12}\]  
On the other hand, if \(\mathcal{M}\) and \(\mathcal{N}\) are merged into one large community (indicated by \(\mathcal{M}+\mathcal{N}\)), the in-degree, out-degree and fitness of this large community can be calculated as  
\[k_{in}^{\mathcal{M}+\mathcal{N}}=m(m-1)+n(n-1)=a+b,\]  
\[k_{out}^{\mathcal{M}+\mathcal{N}}=0,\]  
\[\left(F_{\alpha}^{\beta}\right)^{\mathcal{M}+\mathcal{N}}=\left[m(m-1)+n(n-1) \right]^{\beta-\alpha}=\left(a+b\right)^{k} \tag{13}\]  
We expect \(\mathcal{M}\) and \(\mathcal{N}\) be identified as two independent communities, which requires  
\[\left(F_{\alpha}^{\beta}\right)^{\mathcal{M}}+\left(F_{\alpha}^{\beta}\right) ^{\mathcal{N}}\geq\left(F_{\alpha}^{\beta}\right)^{\mathcal{M}+\mathcal{N}} \tag{14}\]  
\[\text{i.e., }a^{k}+b^{k}\geq\left(a+b\right)^{k} \tag{15}\]  
Consider a function of \(k\): \(f(k)=a^{k}+b^{k}-\left(a+b\right)^{k}\), (15) is equivalent to finding out a range of \(k\), within which \(f(k)\geq 0\).  
Since \(\frac{\mathrm{d}f}{\mathrm{d}k}=a^{k}\ln a+b^{k}\ln b-\left(a+b\right)^{k}\ln \left(a+b\right)\), without loss of generality, we assume \(m\leq n\), \(a\leq b\), then  
\[\frac{\mathrm{d}f}{\mathrm{d}k}\leq a^{k}\ln b+b^{k}\ln b-\left(a+b\right)^{k} \ln b=\left[a^{k}+b^{k}-\left(a+b\right)^{k}\right]\ln b=f(k)\ln b\]  
Since \(f(1)=0\), then \(\left.\frac{\mathrm{d}f}{\mathrm{d}k}\right|_{k=1}\leq f(1)\ln b=0\), indicating that \(f(k)\) is not increasing in the neighborhood of \(k=1\). Therefore, when \(k<1\), i.e., \(\alpha>\beta-1\), presumably \(f(k)\geq 0\), so that (15) would be satisfied. In this paper, we take \(\beta-1\) as the lower bound of \(\alpha\).  
Combining the above Appendices A.1 and A.2, we estimate a \"relevant range\" for the resolution parameter \(\alpha\) with a fixed value of the scaling factor \(\beta\): \(\beta-1<\alpha<2\beta-1\). It should be noted that this range of \(\alpha\) was roughly estimated through necessary conditions rather than sufficient or necessary-and-sufficient conditions: the \"real\" relevant scale of resolution can be expected to fall in this range, as a proper sub-region probably--but resolution limit problems can still exist in the rest of this range since a sufficient condition is not guaranteed here.