# Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis
## 5 Analysis
### Community Coherence
On the contrary, with a symmetric distance metric such as, for instance, the Tanimoto similarity coefficient  
\[\tau_{T,V}\,\overset{\triangle}{=}\,\frac{|T\cap V|}{|T\cup V|}\,=\,\frac{|T \cap V|}{|T|+|V|-|T\cap V|}, \tag{45}\]  
no distinction can be made between the template and the variant, which can potentially lead to misleading results. At this point, it should be highlighted that Fuzzy Newman Girvan was executed only once since it is a deterministic algorithm. Returning to Label (44), the case \(w_{1}+w_{2}=1\) is of particular interest in data mining, as it confines the coefficients on the plane which maximizes the minimum distance of \(T\) from \(V\). Notice that algebraically this asymmetry stems from both the terms \(|T\setminus V|\) and \(|V\setminus T|\), which denote the number of elements of \(T\) not found in \(V\) and vice versa. Both terms signify in their own way how \(V\) is different from \(T\). The former corresponds to the part of \(V\) which is missing from \(T\), whereas the latter corresponds to any additions to \(V\). As a rule, \(|T\setminus V|\) is more important and, consequently, \(w_{1}>w_{2}\). As there is no standard rule for selecting \(w_{1}\) and \(w_{2}\), the following two schemes have been used, a linear  
\[w_{1}\,=\,\frac{s}{1+s}\,=\,\frac{1}{1+\frac{1}{s}},\quad w_{2}\,=\,\frac{1}{1 +s},\quad 1\,\leq\,s\,\leq\,5 \tag{46}\]  
and an exponential  
\[w_{1}\,=\,\frac{e^{s}}{1+e^{s}}\,=\,\frac{1}{1+e^{-s}},\quad w_{2}\,=\,\frac{1 }{1+e^{s}},\quad 0\,\leq\,s\,\leq\,2. \tag{47}\]  
Observe that in the first case \(\frac{w_{1}}{w_{2}}=s\), while in the second \(\frac{w_{1}}{w_{2}}=e^{s}\), which clearly represents a non-linear scaling of the first case. Furthermore, the second case is considerably biased in favor of \(|T\setminus V|\). Once for each possible pair of the \(m\) ground truth communities \(T_{i}\,1\leq i\leq m\) and the \(n\) estimated ones \(V_{j}\,1\leq j\leq n\) the \(mn\) Tversky indices have been computed, the similarity score \(J(s)\) for a given \(s\) is computed  
\[J(s)\,\overset{\triangle}{=}\,\frac{1}{mn}\,\sum_{i=1}^{m}\,\sum_{j=1}^{n}\, \
u_{T_{i},V_{j}}. \tag{48}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **CW** & **CW + I** & **CW + R** & **CW + IR** & **MW** & **MW + I** & **MW + R** & **MW + IR** & **FW** \\ \hline Poisson & 0.6409 & 0.6311 & 0.2766 & 0.4504 & 0.3826 & 0.3911 & 0.3281 & 0.4519 & 0.8012 \\ Binomial & 0.3885 & 0.3977 & 0.3519 & 0.3700 & 0.5804 & 0.5687 & 0.6140 & 0.5626 & 0.6748 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Kullback–Leibler divergence. Summing over the range of \(s\) and taking the average, the mean similarity score \(\bar{J}\) is obtained  
\[\bar{J}\stackrel{{\triangle}}{{=}}\frac{1}{|s|}\,\sum_{s}J(s). \tag{49}\]  
The overall similarity scores are shown in Table 7.  
Again, Chebyshev Walktrap with relocation outperforms the remaining algorithms as it has the highest similarity with the reference communities. Note that the exponential weighting scheme sharpens the difference between the algorithms by raising the maximum scores and lowering the minimum ones. For the experiments of the section, the termination criterion \(\tau_{0}\) was chosen to be a user supplied number of iterations, namely \(|V|\log|V|\). This number of iterations is sufficiently large for generating communities in a reliable way. Moreover, each iteration is very quick, so the overall execution time was kept at an acceptable level despite the large number of iterations.