Finding a dominance order most consistent with a linear hierarchy: an improved algorithm for the I&SI method

• An improved algorithm for the I&SI method is presented.
• New matrix reordering techniques are described.
• Consequences of nonuniqueness of optimal orders are discussed.
• Excel workbooks performing the new algorithm and related analyses are provided.

To find a dominance order most consistent with a linear hierarchy, the I&SI method minimizes two criteria: first the number of inconsistencies I (i.e. number of dyads in which the lower-ranked individual dominates the higher-ranked one) and then (without increasing I) the total strength of inconsistencies SI (i.e. the sum of rank differences between individuals of inconsistent dyads or, in other words, the sum of distances of inconsistencies from the matrix diagonal). The original algorithm developed by de Vries (1998) to achieve this goal has two drawbacks. (1) It completely relies on the ‘generalized swapping rule’ (combined with partly random reordering) to decrease I, but moving single individuals' positions instead of pairwise switching them can complement the generalized swapping rule to make a more efficient algorithm. (2) It does not consider alternative orders, if more than one exists, with equally minimized I and SI values. In this paper, an improved algorithm for the I&SI method is presented. It includes: (1) more efficient searching for optimal orders; (2) consideration of all orders with equally minimized I and SI values; and (3) choice of one or more optimal orders based on correlation with a hierarchical combination of two dominance indices: primary ranking by the proportion of dominated individuals and secondary ranking by the difference between the number of dominated individuals and the number of individuals to which the focal individual is subordinate. Microsoft Excel workbooks are provided that perform the linearity test published by de Vries (1995) and the improved algorithm.