Supplementary file S13. Concordance score between two clusterings

For the clusterings from two methods, first the cluster labels are adjusted. Denote $\mathbf{s_1}$ as the label vector for method 1, $\mathbf{s_2}$ as the label vector for method 2, and $\mathbf{s_1}$ as the reference labels, we apply clue::solve_LSAP() function to generate a mapping function $m()$ between the two sets of labels to maximize $\sum^n_i I(s_{1i}, m(s_{2i}))$ where $n$ is the length of $\mathbf{s_1}$ or $\mathbf{s_2}$. Denote the adjusted labels for the second method as $s'_{2i} = m(s_{2i})$, the concordance between the two clusterings are calculated as:

$$\frac{1}{n}\sum_i^n I(s_{1i}, s'_{2i})$$

where $I(x, y)$ is an indicator function with value 1 if $x = y$ and 0 if $x \ne y$. A concordance of 1 means the two clusterings are identical.