# Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis
## 6 Conclusions  
The primary contribution of this article is the implementation over Neo4j of Chebyshev Walktrap, a community discovery algorithm designed for edge-fuzzy graphs, a class of fuzzy graphs used among others in [10], which is based on the Random Walker algorithmic principle. Additionally, Chebyshev Walktrap relies on the competitive factors of second order statistics though the Chebyshev inequality and on an optional relocation capability in order to bound unnecessarily costly walks and, thus, remaining inside a community and being trapped for too long within the boundaries of a community, respectively. The relocation aspect was also backported to the Markov Walktrap algorithm first proposed in [15]. The effect of relocation on the community coherence was evaluated  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Linear** & **CW** & **CW + I** & **CW + R** & **CW + IR** & **MW** & **MW + I** & **MW + R** & **MW + IR** & **FW** \\ \hline Poisson & 0.6199 & 0.6126 & 0.6616 & 0.6012 & 0.5881 & 0.5902 & 0.6211 & 0.5577 & 0.2742 \\ Binomial & 0.4323 & 0.4153 & 0.6059 & 0.5853 & 0.5648 & 0.5702 & 0.5814 & 0.5332 & 0.2131 \\ \hline
**Exponential** & **CW** & **CW + I** & **CW + R** & **CW + IR** & **MW** & **MW + I** & **MW + R** & **MW + IR** & **FW** \\ \hline Poisson & 0.5505 & 0.5345 & 0.7503 & 0.5462 & 0.5271 & 0.5383 & 0.7354 & 0.4811 & 0.1703 \\ Binomial & 0.5230 & 0.4841 & 0.6992 & 0.4737 & 0.5102 & 0.4890 & 0.6772 & 0.4283 & 0.1523 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Tversky index.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **CW + R** & **MW + R** \\ \hline Number of relocations & 18 & 14 \\ First relocation step & 101 & 44 \\ min between relocations & 17,812 & 32,991 \\ max between relocations & 27,099 & 64,818 \\ mean between relocations & 21,002 & 38,002 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Relocation summary.  
based on the asymmetric Tversky index using the Fuzzy Newman-Girvan algorithm from [15] as baseline, while its effect on the output distribution was assessed with the asymmetric Kullback-Leibler divergence. The latter was also the basis for evaluating the distance between the community size distribution generated by Fuzzy Newman-Girvan and the one computed by the Makrov Walktrap and the Chebyshev Walktrap. In these cases, the introduction of asymmetry resulted in the clear distinction between the baseline data and their variants.  
The test dataset was a large synthetic Kronecker graph whose edge fuzziness was controlled either by a binomial or by a Poisson distribution. In this dataset, our performance metrics showed that Chebyshev Walktrap yields more compact communities whose sizes are more clustered. Additionally, Markov Walktrap is, in many instances, slightly faster at the expense of a somewhat bigger memory footprint.  
The experimental results of Section 5 hint at some future research directions. More sophisticated from a probabilistic viewpoint, community discovery algorithms should be able to exploit the asymmetry of the edge fuzziness distribution through higher order concentration inequalities such as the Talagrand inequality, provided their computation is efficient. Moreover, new metrics for community matching, perhaps utilizing functional or semantic information should be developed. Additionally, methodologies for reliably assessing community coherence based on higher order structural or functional interactions should be sought. Finally, more experiments in larger graphs should be conducted in order to determine any inherent scalability limitations.  
[PERSON], [PERSON], and [PERSON] conceived of the idea, designed and performed the experiments, analyzed the results, drafted the initial manuscript and revised the final manuscript.  
The authors declare no conflict of interest.