# Double-Constrained Consensus Clustering with Application to Online Anti-Counterfeiting
## 6 Conclusions and Future Work  
We have presented a novel approach to semi-supervised ensemble clustering that makes double use of pairwise constraints on the grounds that they can be exploited both to generate better base partitions and improve the combination of such partitions. The first step is carried out through constraint-enriched single clustering algorithms, whereas the merging is achieved by incorporating compliance with constraints into an objective function that maximizes the PRI similarity between the consensus partition and the constrained base partitions. The compliance with constraints also takes into account their logical properties, thus helping optimize their use. The effectiveness of the proposed framework is indicated by the fact that reusing the constraints in the consensus function produced much better results than combining the constrained base partitions with the unenhanced consensus function. We found that under a suitable set of experimental conditions, the average improvement was 5.54%. We have also argued that the proposed framework can be used to combat web counterfeiting because constraints can be partly acquired automatically, and we have shown its \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Feature** & **Min.** & **Max.** & **Avg.** & **0 (CC)** & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** \\ \hline _Dom_ & 0.47 & 0.85 & 0.65 & 0.70 & 0.65 & 0.72 & 0.72 & 0.77 & 0.78 & 0.79 & 0.79 & 0.91 \\  & & & & (8\%) & (0\%) & (11\%) & (11\%) & (18\%) & (20\%) & (22\%) & (40\%) \\ \hline _Header30_ & 0.30 & 0.69 & 0.55 & 0.69 & 0.68 & 0.62 & 0.71 & 0.72 & 0.76 & 0.81 & 0.82 & 0.87 \\  & & & & (25\%) & (24\%) & (13\%) & (29\%) & (31\%) & (38\%) & (47\%) & (49\%) & (58\%) \\ \hline _Header60_ & 0.30 & 0.69 & 0.54 & 0.68 & 0.68 & 0.68 & 0.71 & 0.72 & 0.75 & 0.80 & 0.81 & 0.86 \\  & & & & (26\%) & (26\%) & (26\%) & (31\%) & (33\%) & (39\%) & (48\%) & (50\%) & (59\%) \\ \hline _Header90_ & 0.30 & 0.69 & 0.58 & 0.69 & 0.73 & 0.71 & 0.70 & 0.72 & 0.75 & 0.81 & 0.82 & 0.85 \\  & & & & (19\%) & (26\%) & (22\%) & (21\%) & (24\%) & (29\%) & (40\%) & (41\%) & (47\%) \\ \hline _Header120_ & 0.29 & 0.69 & 0.56 & 0.66 & 0.67 & 0.65 & 0.72 & 0.70 & 0.73 & 0.79 & 0.81 & 0.85 \\  & & & & (18\%) & (20\%) & (16\%) & (29\%) & (25\%) & (30\%) & (41\%) & (45\%) & (52\%) \\ \hline _Header150_ & 0.33 & 0.69 & 0.57 & 0.68 & 0.66 & 0.62 & 0.72 & 0.72 & 0.75 & 0.80 & 0.81 & 0.86 \\  & & & & (19\%) & (16\%) & (9\%) & (26\%) & (26\%) & (32\%) & (40\%) & (42\%) & (51\%) \\ \hline _Privacy_ & 0.61 & 0.76 & 0.72 & 0.76 & 0.74 & 0.76 & 0.80 & 0.80 & 0.84 & 0.85 & 0.87 \\  & & & & (6\%) & (3\%) & (6\%) & (11\%) & (11\%) & (11\%) & (17\%) & (18\%) & (21\%) \\ \hline _Shipping_ & 0.47 & 0.72 & 0.65 & 0.72 & 0.72 & 0.73 & 0.75 & 0.77 & 0.76 & 0.79 & 0.80 & 0.87 \\  & & & & (11\%) & (11\%) & (12\%) & (15\%) & (18\%) & (17\%) & (22\%) & (23\%) & (34\%) \\ \hline _avg_ & 0.38 & 0.72 & 0.60 & 0.70 & 0.69 & 0.69 & 0.73 & 0.74 & 0.76 & 0.80 & 0.81 & 0.87 \\  & & & & (17\%) & (15\%) & (15\%) & (22\%) & (23\%) & (27\%) & (33\%) & (35\%) & (45\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of CCC on the eight _hclust_ algorithms for each clustering feature (first column) as the number of constraints grows from 0 to 80.  
utility for detecting affiliate marketing programs behind fraudulent e-commerce websites present in search engine results. From a practical point of view, our research confirms that semi-supervised ensemble clustering is generally a viable strategy to address the inherent limitations of single unsupervised clustering algorithms while offering new insights into the increased robustness that can be achieved when changing the dataset to be clustered, the clustering technique, and the clustering features. There are several main directions for future research. First, we plan to extend the objective function from constraint counting to constraint weighting, as hinted at in the article.