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ENZYMES emerges as the only dataset showing the same phenomenon. By looking at the magnitude of the accuracy boost, cliques seem to benefit more from a stratified granulation approach: indeed, the vast majority of the boosts in terms of accuracy are upper-bounded oy 10%, whereas the majority of path-based granulators reach at most a 3% accuracy boost Nith respect to the non-stratified counterpart. By looking at Figure 2, we can observe the number of symbols (i.e., the size of the alphabet) after the second genetic-driven feature selection stage. In principle, a clique- based approach is likely to yield a smaller alphabet: this is due to the fact that an n- vertex graph can have at most 0(3"/%) cliques, whereas the number of paths can grow as O(n!) in the worst case, so the set of prospective information granules (even without any subsampling) is smaller. Conversely to the accuracy case, a stratified approach does not necessarily yield benefits. As can be seen in the breakdown (Figure 4), as paths are concerned, the stratification yields a smaller alphabet for 4 datasets out of 10. As cliques are concerned, the same phenomenon happens for 3 out of 10 datasets. This phenomenon is particularly evident for binary classification problems: a clear sign that building two dedicate alphabets (one for the positive class, one for the negative class) is excessive and just one alphabet would be a smarter choice.

Figure 2 ENZYMES emerges as the only dataset showing the same phenomenon. By looking at the magnitude of the accuracy boost, cliques seem to benefit more from a stratified granulation approach: indeed, the vast majority of the boosts in terms of accuracy are upper-bounded oy 10%, whereas the majority of path-based granulators reach at most a 3% accuracy boost Nith respect to the non-stratified counterpart. By looking at Figure 2, we can observe the number of symbols (i.e., the size of the alphabet) after the second genetic-driven feature selection stage. In principle, a clique- based approach is likely to yield a smaller alphabet: this is due to the fact that an n- vertex graph can have at most 0(3"/%) cliques, whereas the number of paths can grow as O(n!) in the worst case, so the set of prospective information granules (even without any subsampling) is smaller. Conversely to the accuracy case, a stratified approach does not necessarily yield benefits. As can be seen in the breakdown (Figure 4), as paths are concerned, the stratification yields a smaller alphabet for 4 datasets out of 10. As cliques are concerned, the same phenomenon happens for 3 out of 10 datasets. This phenomenon is particularly evident for binary classification problems: a clear sign that building two dedicate alphabets (one for the positive class, one for the negative class) is excessive and just one alphabet would be a smarter choice.

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Table 1. Dataset statistics. Each dataset has been split into three disjoint sets: training, validation and test. For the six IAM datasets, we used the very same training /validation/test splits provided in the repository, whereas for the four TUDatasets, we had to perform our own splits with the following ratio: training set (50%), validation set (25%) and test set (25%). The splitting procedure has been performed in a stratified manner in order to preserve labels’ distribution across the three sets. Brief statistics about the datasets can be found in Table 1, along with the referenc paper in which each dataset has been originally presented, to which we refer the intereste reader for more information.
Table 1. Dataset statistics. Each dataset has been split into three disjoint sets: training, validation and test. For the six IAM datasets, we used the very same training /validation/test splits provided in the repository, whereas for the four TUDatasets, we had to perform our own splits with the following ratio: training set (50%), validation set (25%) and test set (25%). The splitting procedure has been performed in a stratified manner in order to preserve labels’ distribution across the three sets. Brief statistics about the datasets can be found in Table 1, along with the referenc paper in which each dataset has been originally presented, to which we refer the intereste reader for more information.
Figure 3. Distribution of the accuracy boosts for paths and cliques across the 10 datasets. A pos- itive shift means that the stratified approach outperforms the non-stratified approach. (a) Paths. (b) Cliques.
Figure 3. Distribution of the accuracy boosts for paths and cliques across the 10 datasets. A pos- itive shift means that the stratified approach outperforms the non-stratified approach. (a) Paths. (b) Cliques.
Figure 4. Distribution of the differences of the alphabet size for paths and cliques across the 10 datasets (after feature selection). A positive shift means that the stratified approach underperforms the non-stratified approach. (a) Paths. (b) Cliques.
Figure 4. Distribution of the differences of the alphabet size for paths and cliques across the 10 datasets (after feature selection). A positive shift means that the stratified approach underperforms the non-stratified approach. (a) Paths. (b) Cliques.
* Results refers to cross-validation rather than a separate test set. Table 3. Comparison against state-of-the-art graph classification system in terms of classification accuracy.
* Results refers to cross-validation rather than a separate test set. Table 3. Comparison against state-of-the-art graph classification system in terms of classification accuracy.
Related topics:
EngineeringComputer ScienceAlgorithmsInformation granulationInexact Graph MatchingData Mining and Knowledge DiscoveryStructural Pattern RecognitionGraph Classification
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