Affinity in Concentric Circles ( ACC ): A Geometric Representation of Dendrogram and Interpretation

ABSTRACT Aim We introduce “Affinity in Concentric Circles” (ACC), a method that converts dendrograms to a geometric form using similarity scores to compare dendrograms and identify affinity change patterns. Location China for trilobites and global for nisusiid brachiopods. Taxon Trilobites (Ordovician of China) and brachiopods (Cambrian of worldwide). Methods ACC compares dendrograms derived from subset datasets against a dendrogram from an inclusive dataset that aggregates all subset datasets. It employs two similarity scores, local and global scores, for each cluster in the subset dendrogram. The local score is the original similarity value of the cluster calculated from subset datasets, whereas the global score for the same cluster is recalculated from the similarity values of the inclusive dataset. The local score is converted to an angle along a circle, and the global score to a diameter of the circle, respectively. This geometric conversion places two areas of a cluster at an angle along a circle. The procedure begins with a cluster with the highest local score and then sequentially adds area or merges clusters by centering and aligning the circles. As a result, areas in each subset dendrogram are placed along concentric circles at angles. Affinities in the ACC diagram are quantified using two metrics: pairwise distance (PD) between two members in a subset diagram and comprehensive travel distance (CTD) of a member across multiple subset dendrograms. Affinities among members are compared both within and across diagrams based on geometric distribution, and relative changes in the affinities are tracked by comparing average PD and CTD values. Results The ACC analyses of six trilobite‐occurring areas in China for four Ordovician epochs and three Ordovician evolutionary faunas reveal that Jiangnan slope, which is closest to Tianshan basin, was likely a center of diversification, Sino‐Korea platform was biogeographically isolated, Qaidam was closer to Sino‐Korea, and deep‐water areas were more strongly connected by active oceanic currents in depth. The average CTD values suggest that significant change in overall biogeographic stability occurred in the Arenig and Llanvirn. The analysis of seven nisusiid brachiopod‐occurring areas for four middle Cambrian ages reveals that Laurentia, the likely center of origin, became progressively isolated from other areas, Australia with the lowest average CTD underwent the most intense biogeographic changes, and overall biogeographic stability decreased toward the Drumian. These findings are largely inconsistent with the area distributions in the paleomaps. The distribution sector of ACC diagrams reflects β ‐diversity for each period and fauna. ACC represents and addresses Simpson's paradox in a unique manner and tends to reduce its manifestation. Main Conclusions ACC utilizes all clusters and their similarity scores of dendrograms from both subset and inclusive datasets for analysis, allowing us to compare dendrograms and identify patterns that are not apparent in dendrograms.