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The performance of similarity measures for search, indexing, and data mining applications tends to degrade rapidly as the dimensionality of the data increases. The effects of the so-called 'curse of dimensionality' have been studied by researchers for data sets generated according to a single data distribution. In this paper, we study the effects of this phenomenon on different similarity measures for multiply-distributed data. In particular, we assess the performance of shared-neighbor similarity measures, which are secondary similarity measures based on the rankings of data objects induced by some primary distance measure. We find that rank-based similarity measures can result in more stable performance than their associated primary distance measures.