Beyond the Clique: Comparing Clique and Dowker Complexes for Co-occurrence Data in Learning Analytics

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.