## 5 Conclusions  
Graphs are powerful structures that can capture topological and semantic information from data. However, in many contexts, graphs suffer from the major drawback of having different sizes, hence they cannot be easily compared (e.g., by means of their respective adjacency matrices) and designing a graph-based pattern recognition system is not trivial. In this paper, this problem has been addressed by moving towards an embedding space built on top of simplicial complexes extracted in a fully data-driven manner from the dataset at hand. The embedding procedure follows the symbolic histogram approach, where each pattern is described by the number of occurrences of a given meaningful symbol within the original pattern (graph). In the embedding space any Euclidean classifier can be used, either equipped or not with feature selection capabilities. Although not mandatory, performing feature selection either by properly choosing the classification system or with the help of optimisation techniques, benefits the model in a two-fold fashion: first, it reduces the embedding space dimension, speeding up the classification of new patterns; second, it improves the model interpretability. Indeed, a major strength of information granulation-based pattern recognition systems is that relevant, meaningful information granules (alphabet symbols) can be analysed by field-experts to derive insights for the problem at hand. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Hydrophilicity (avg)**} & \multicolumn{3}{c}{**Hydrophilicity (std)**} \\ \hline
**Class** & \(t\)**-Value** & \(p\)** & **Coefficient** & \(t\)**-Value** & \(p\)** & **Coefficient** \\ \hline
1 & 11.55 & \(<\)0.0001 & \(-\)4.17734 & 0.92 & 0.3563 & 0.24438 \\
2 & 10.52 & \(<\)0.0001 & \(-\)3.73211 & 0.0647 & 1.85 & 0.47999 \\
3 & 10.61 & \(<\)0.0001 & \(-\)3.38981 & 0.0651 & 1.84 & 0.43182 \\
4 & 11.08 & \(<\)0.0001 & \(-\)2.98596 & 2.11 & 0.0352 & 0.41574 \\
5 & 12.13 & \(<\)0.0001 & \(-\)2.43624 & 2.49 & 0.0127 & 0.36671 \\
6 & 10.73 & \(<\)0.0001 & \(-\)2.65512 & 2.57 & 0.01 & 0.46672 \\
7 & 11.55 & \(<\)0.0001 & \(-\)4.17734 & 0.92 & 0.3563 & 0.24438 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Hydrophilicity contribution to score for different classes. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Polarity (avg)**} & \multicolumn{3}{c}{**Polarity (std)**} \\ \hline
**Class** & \(t\)**-Value** & \(p\)** & **Coefficient** & \(t\)**-Value** & \(p\)** & **Coefficient** \\ \hline
1 & 11.27 & \(<\)0.0001 & 1.51515 & 1.77 & 0.0762 & \(-\)0.17376 \\
2 & 10.26 & \(<\)0.0001 & 1.35280 & 2.52 & 0.0118 & \(-\)0.24206 \\
3 & 10.43 & \(<\)0.0001 & 1.23898 & 2.62 & 0.0089 & \(-\)0.22655 \\
4 & 10.83 & \(<\)0.0001 & 1.08515 & 2.72 & 0.0066 & \(-\)0.19836 \\
5 & 11.84 & \(<\)0.0001 & 0.88388 & 3.16 & 0.0016 & \(-\)0.17190 \\
6 & 10.52 & \(<\)0.0001 & 0.96768 & 3.14 & 0.0017 & \(-\)0.21080 \\
7 & 11.27 & \(<\)0.0001 & 1.51515 & 1.77 & 0.0762 & \(-\)0.17376 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Polarity contribution to score for different classes. The proposed pattern recognition system has been tested on thirty open-access datasets and benchmarked against two suitable competitors: a kernel method (WJK) which works on simplicial complexes and (by definition) performs an implicit embedding towards an high-dimensional feature space and another information granulation-based classifier (GRALG) which performs explicit embedding but relies on simple paths rather than simplices. Computational results show that the proposed embedding technique outperforms GRALG in almost all of the tested datasets. Albeit WJK seems to be the best performing classification technique, it is noteworthy that no a-posteriori knowledge discovery phase can be performed, whereas the same is not true for information granulation-based classifiers. In order to stress this aspect, we faced two additional real-world problems: the prediction of proteins' enzymatic class and their solubility.