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. 2019 Jun 12;9:8574. doi: 10.1038/s41598-019-44892-y

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Two examples of counter-intuitive bias in clustering comparisons. Four clusterings are considered over 9 elements, and compared using the Fowlkes-Mallows index (FM), normalized mutual information (NMI), and our element-centric similarity measure. We argue that the comparison between the clusterings on the left is more similar than the comparison between clusterings on the right. (a) Both NMI and the element-centric similarity follow this intuition, but FM is biased towards large clusters and suggests the comparison on the right is more similar. (b) Both FM and the element-centric similarity follow this intuition, but NMI is biased towards many clusters and suggests the comparison on the right is more similar.