Blockmodelling is the process of determining community structure in a graph. Real graphs contain noise and so it is up to the blockmodelling method to allow for this noise and reconstruct the most likely role memberships and role relationships. Relationships are encoded in a graph using the absence and presence of edges. Two objects are considered similar if they each have edges to a third object. However, the information provided by missing edges is ambiguous and therefore can be measured in different ways. In this article, we examine the effect of the choice of block metric on blockmodelling accuracy and find that data relationships can be position based or set based. We hypothesise that this is due to the data containing either Hamming noise or Jaccard noise. Experiments performed on simulated data show that when no noise is present, the accuracy is independent of the choice of metric. But when noise is introduced, high accuracy results are obtained when the choice of metric matches the type of noise.

Decision tree learning is a widely used approach in machine learning, favoured in applications that require concise and interpretable models. Heuristic methods are traditionally used to quickly produce models with reasonably high accuracy. A commonly criticised point, however, is that the resulting trees may not necessarily be the best representation of the data in terms of accuracy and size. In recent years, this motivated the development of optimal classification tree algorithms that globally optimise the decision tree in contrast to heuristic methods that perform a sequence of locally optimal decisions. We follow this line of work and provide a novel algorithm for learning optimal classification trees based on dynamic programming and search. Our algorithm supports constraints on the depth of the tree and number of nodes. The success of our approach is attributed to a series of specialised techniques that exploit properties unique to classification trees. Whereas algorithms for optimal classification trees have traditionally been plagued by high runtimes and limited scalability, we show in a detailed experimental study that our approach uses only a fraction of the time required by the state-of-the-art and can handle datasets with tens of thousands of instances, providing several orders of magnitude improvements and notably contributing towards the practical use of optimal decision trees.