## 6 Conclusions  
Mining frequent approximate subgraphs with structural and label variations in an undirected graph collection has been scarcely studied in the literature. In this paper, we have proposed AGCM-SLV, an algorithm for mining frequent approximate subgraphs with structural and label variations in graph collections. Our algorithm follows a depth-first traversal strategy starting from frequent nodes (which are single-node subgraphs) and expands them, verifying both the support and dissimilarity thresholds while discarding those that do not meet the defined thresholds. To assess the proposed algorithm, we conducted experiments comparing it with REATUM, the most similar algorithm to ours reported in the literature. From these experiments, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Noise Level** & \(\beta=1\) & \(\beta=2\) \\ \hline \multicolumn{3}{c}{PTC- FM} \\ \hline
5\% & 100\% & 100\% \\
10\% & 100\% & 100\% \\
15\% & 100\% & 100\% \\
20\% & 100\% & 98.18\% \\ \hline \multicolumn{3}{c}{PTC- FR} \\ \hline
5\% & 100\% & 96.36\% \\
10\% & 100\% & 98.18\% \\
15\% & 100\% & 100\% \\
20\% & 100\% & 100\% \\ \hline \multicolumn{3}{c}{PTC- MM} \\ \hline
5\% & 100\% & 100\% \\
10\% & 100\% & 97.96\% \\
15\% & 100\% & 100\% \\
20\% & 100\% & 100\% \\ \hline \multicolumn{3}{c}{PTC- MR} \\ \hline
5\% & 100\% & 100\% \\
10\% & 100\% & 100\% \\
15\% & 100\% & 94.55\% \\
20\% & 100\% & 94.55\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Percentage of frequent approximate subgraphs preserved across different noise levels, compared to those mined in the noise-free datasets, employing a support threshold \(\alpha=90\%\) and dissimilarity thresholds \(\beta=1\) and \(\beta=2\). we can conclude that, due to AGCM-SLV expanding frequent subgraphs through their neighboring edges instead of only their neighboring nodes (as REAFUM does), our proposed algorithm mines frequent subgraphs that REAFUM cannot mine in a shorter runtime. On the other hand, we also performed experiments to analyze the runtime behavior of AGCM-SVL, varying dissimilarity and support thresholds, on real-world datasets. These experiments reveal that AGCM-SLV's runtime increases at a higher rate when the dissimilarity threshold is increased compared with the support threshold. Our scalability experiments allow us to conclude that although AGCM-SLV exhibits fast growth in terms of runtime and memory usage as the collection grows. The increase in runtime is faster than the increase in memory usage, especially when increasing the dissimilarity threshold. Finally, we assessed the impact of random white noise in the graph collection when our proposed algorithm is applied for mining frequent approximate subgraphs with structural and label variations. This experiment allows us to conclude that AGCM-SLV is tolerant to white noise, since more than 99% of the frequent approximate subgraphs mined without noise are preserved in noise conditions. In our experiments, we address approximate label similarities through dictionaries, as this approach enables the easy definition of similarities among labels across different domains. However, the algorithm can be easily adapted to include user-defined similarity functions for more complex or domain-specific scenarios. The main limitation of the proposed algorithm is its time complexity, as the algorithm can only be used on relatively small collections of graphs. However, to our knowledge, it is the first algorithm to address the mining of all frequent approximate subgraphs with structural and label variations in graph collections allowing user-defined partial similarities between node and edge labels. This limitation presents an area of opportunity for the development of more efficient algorithms that can operate with larger collections of graphs. Another interesting problem to address as future work is the development of algorithms for mining frequent subgraphs with structural and label variations in dynamic collections of graphs or for streaming data problems, where the graphs change over time.