Dowker complexes better capture group co-occurrences than clique complexes
Beyond the Clique: Comparing Clique and Dowker Complexes for Co-occurrence Data in Learning Analytics
Social and Information Networks
Summary
People often analyze data where groups of items appear together, but the common method can mistake three items appearing separately as a real group happening together. The authors compare two ways of building shapes from co-occurrence data: one that looks only at pairs (clique complexes) and another that looks at actual groups that appear together (Dowker complexes). They find that Dowker complexes detect real group patterns better and avoid false assumptions that clique complexes make. This improved approach helps reveal more accurate structure in data like online discussions or tags.
What this means in practice
- •For learning analytics teams: Use Dowker complexes to more accurately identify real group interactions in co-occurrence data from online learning platforms and forums.
- •For social network analysts: Improve detection of genuine group formations in social media interactions by applying Dowker complex analysis instead of relying on pairwise links only.
Authors
Koichi Yasutake, Wakana Tsuji, Hitoshi Inoue
Abstract
Learning analytics increasingly represents the relations in co-occurrence data---codes in a window of discourse, participants in a thread, tags on a post---as simplicial complexes and analyses them with persistent homology. The standard approach uses the clique complex. Because its simplices are determined by the pairwise network alone, it cannot distinguish "three elements co-occurred together" from "each of the three pairs co-occurred separately". The Dowker complex, in contrast, takes as simplices the sets of entities actually observed to co-occur. We compare analyses based on these two complexes. First, we show that, on the same 1-skeleton, the Dowker complex is a subcomplex of the clique complex, the map on first homology induced by the inclusion is surjective, and its kernel is generated by phantom triangles that never co-occurred; that is, the clique construction can only erase holes. We then test this on real data. On four Stack Exchange data sets, 46--91% of clique triangles are phantom and 522--1,970 holes are erased. On the example data of the learning analytics R packages tna/Nestimate, the first Betti number $β_1$ of the clique complex is 0 at every threshold, whereas the Dowker complex detects holes. Against a degree-preserving null model, the Dowker complex departs strongly in five of the six data sets. Moreover, this difference does not appear in the fixed-threshold analyses used in practice. We conclude that the construction should follow the data type and that, for observed groups, the Dowker complex is the appropriate choice; it fills a gap in current tooling. Code is available at https://github.com/igu-lab/beyond-the-clique.