# An experimental approach: The graph of graphs

arXiv: https://arxiv.org/abs/2508.17520 · HTML: https://mlsanity.com/paper/2508.17520/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 24 Aug 2025.

## Builds on (most cited first, 8 of 8)

- [An application of incomplete pairwise comparison matrices for ranking top tennis players](https://mlsanity.com/paper/1611.00538/) (2016)
- [Inconsistency thresholds for incomplete pairwise comparison matrices](https://mlsanity.com/paper/2102.10558/) (2021)
- [The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices](https://mlsanity.com/paper/1701.04265/) (2017)
- [Filling in pattern designs for incomplete pairwise comparison matrices: (quasi-)regular graphs with minimal diameter](https://mlsanity.com/paper/2006.01127/) (2020)
- [On the Derivation of Weights from Incomplete Pairwise Comparisons Matrices via Spanning Trees with Crisp and Fuzzy Confidence Levels](https://mlsanity.com/paper/2112.10253/) (2021)
- [Optimal sequences for pairwise comparisons: the graph of graphs approach](https://mlsanity.com/paper/2205.08673/) (2022)
- [The incomplete Analytic Hierarchy Process and Bradley-Terry model: (in)consistency and information retrieval](https://mlsanity.com/paper/2210.03700/) (2022)
- [An experimental approach: Converting verbal expressions to numerical scales](https://mlsanity.com/paper/2507.04539/) (2025)

## Abstract (arXiv)

One of the essential issues in decision problems and preference modeling is the number of comparisons and their pattern to ask from the decision maker. We focus on the optimal patterns of pairwise comparisons and the sequence including the most (close to) optimal cases based on the results of a color selection experiment. In the test, six colors (red, green, blue, magenta, turquoise, yellow) were evaluated with pairwise comparisons as well as in a direct manner, on color-calibrated tablets in ISO standardized sensory test booths of a sensory laboratory. All the possible patterns of comparisons resulting in a connected representing graph were evaluated against the complete data based on 301 individual's pairwise comparison matrices (PCMs) using the logarithmic least squares weight calculation technique. It is shown that the empirical results, i.e., the empirical distributions of the elements of PCMs, are quite similar to the former simulated outcomes from the literature. The obtained empirically optimal patterns of comparisons were the best or the second best in the former simulations as well, while the sequence of comparisons that contains the most (close to) optimal patterns is exactly the same. In order to enhance the applicability of the results, besides the presentation of graph of graphs, and the representing graphs of the patterns that describe the proposed sequence of comparisons themselves, the recommendations are also detailed in a table format as well as in a Java application.