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

• The Dowker complex distinguishes a group that co-occurred from three separate pairs, while the clique complex cannot distinguish them from the pairwise network alone.Because the Dowker complex is built from observed groups, homology and persistent homology apply to it directly.Source: §1 · p. 2, Figure 1 · p. 2
• Learning-analytics simplicial pipelines build clique complexes from pairwise matrices, and Nestimate's simplicial and persistent-homology functions cannot take its hypergraph representation as input.The cited package version is 0.8.5.Source: §1 · p. 2
• The paper states that clique construction can only erase first-homology classes of the Dowker complex and identifies erased classes with phantom triangles.It proposes the phantom rate and kernel dimension as measures, noting that count filtrations lack the metric setting's approximation guarantee.Source: §1 · p. 3
• The authors measure the loss on four Stack Exchange datasets and the two datasets distributed with tna/Nestimate, reproducing the tools' output exactly and comparing both constructions with a degree-preserving null model.Source: §1 · p. 3

• Observed groups are sets of entities, and w(σ) counts how many observed groups contain every entity in σ.The groups can also be represented as a bipartite relation between entities and groups.Source: §2 · p. 3
• A clique complex contains every vertex set whose members are pairwise adjacent, whereas a Dowker complex contains sets observed together and their subsets.The clique complex is determined by the graph alone; the Dowker complex is defined from the observed groups.Source: §2 · p. 3
• The threshold graph includes edges whose co-occurrence count is at least t; decreasing t gives nested clique and Dowker filtrations, with stronger co-occurrences entering first.The paper computes persistent homology over F2.Source: §2 · p. 3

• Network-science work has built simplicial complexes directly from observed groups, while reviews discuss when clique complexes add interactions that were never observed.Examples include papers as author simplices and concepts co-occurring in mathematical texts.Source: §3 · p. 4
• In topology, prior work studies flagification of Dowker complexes as a cheaper approximation for metric data, whereas this paper contrasts that setting with count filtrations of co-occurrence data.Source: §3 · p. 4
• Learning-analytics TDA has mainly used Vietoris–Rips complexes of point clouds and Mapper, while Nestimate uses clique complexes of pairwise matrices.The authors state that clique-construction loss on co-occurrence data had not been quantified and that no learning-analytics tool built complexes from observed groups.Source: §3 · p. 4

4 What the clique complex erases

• The Dowker complex is a subcomplex of the clique complex on the same 1-skeleton, and inclusion induces a surjection on H1.The kernel is spanned by boundaries of phantom triangles: triangles whose edges meet the threshold but whose three vertices do not co-occur often enough.Source: §4 · p. 4
• The H1 Betti-number difference equals the kernel dimension and is bounded above by the number of phantom triangles.The clique construction’s added triangle boundaries account for the loss.Source: §4 · p. 4
• Every H1 class is born at the same filtration value in both complexes and dies no later in the clique complex.Classes may be filled at the same step, earlier by the clique complex, only by the clique complex, or in neither.Source: §4 · p. 4
• Count filtrations have no metric-style approximation guarantee, so the loss from using the clique complex must be measured on each data set.The result follows from the fact that co-occurrence counts lack the metric structure used for the Dowker–Rips guarantee.Source: §4 · pp. 4–5
• Dowker complexes on either side of a filtered relation have isomorphic persistent homology in every dimension, but clique flagifications are guaranteed to agree only in dimensions 0 and 1.In the four-learner, six-pair example, both clique complexes have β1 = 0 although the Dowker complexes have β1 = 3.Source: §4 · p. 5

5 Which construction for which data

• For pairwise data, a clique complex is appropriate; for observed groups, the pairwise network is a lossy projection, so the Dowker complex preserves the primary information.The choice should follow the data type rather than reconstructing higher-order structure from pairwise information.Source: §5 · p. 6
• Nestimate’s simplicial functions apply the clique construction to observed-group data, while version 0.8.5 cannot pass its hypergraph representation into those functions.The authors reimplemented the clique construction and reproduced the package’s simplices for both TNA data sets at all four thresholds.Source: §5 · p. 6
• Elementwise-maximum symmetrization discards TNA transition direction before building a complex; the reported matrix asymmetry is 0.66 for group_regulation and 0.55 for human_long.Direction-preserving constructions are left to future work; this paper treats co-occurrence only.Source: §5 · p. 6

6 Data and methods

• Four Stack Exchange data sets represent either question tags as groups or a question asker together with answerers as a group.The tag sets comprise tags-matheducators, tags-academia, and tags-physics; threads-matheducators uses asker-answerer groups.Source: §6 · pp. 6–7
• Table 2 covers phantom rates, maximum β1, and H1 classes filled only by the clique complex or earlier by it; Table 3 compares phantom rates and maximum-β1 z-scores with the null model.Figure 2 depicts clique phantom rates against null means and maximum β1 for clique and Dowker complexes across Stack Exchange and TNA data.Source: Table 2 · p. 7, Table 3 · p. 8, Figure 2 · p. 8
• The TNA examples are group_regulation and human_long, with groups defined as distinct codes in sliding windows of w consecutive events.At w = 3, the data contain 23,625 and 9,948 windows, respectively; results for w = 4 are in the Appendix.Source: §6 · p. 7
• For each data set, the authors compute H0 and H1 persistence over F2 for both complexes on the same 1-skeleton, recording and classifying each H1 class’s death in each complex.There is no cap on group size.Source: §6 · p. 7
• The null model preserves each group’s size and each entity’s group-membership count, using 200 independently sampled data sets and one-sided empirical p-values.Curves are indexed by the fraction of 1-skeleton edges present; the minimum attainable p-value is approximately 0.005.Source: §6 · p. 7

Stack Exchange data

• In four Stack Exchange data sets, 46–91% of clique triangles are phantom, 522–1,970 Dowker classes remain unfilled by the clique complex, and Dowker β1 reaches 680–10,951 versus at most 49 for clique.The clique also fills a further 1,015–39,456 classes earlier than the data warrant.Source: §7 · pp. 8–9, Figure 3 · p. 9

TNA example data

• On TNA examples, clique β1 is 0 at every threshold, while Dowker opens 10–11 cycles and all 28 H1 classes are filled earlier by clique.The full-complex phantom rates are 0.00 for group_regulation and 0.20 for human_long.Source: §7 · p. 9

Null-model comparison

• Dowker departs strongly from the degree- and size-preserving null in five of six data sets, with z-scores from −11.6 to −270.1; clique misses TNA departures and points the other way on two Stack Exchange sets.The null destroys pairwise structure, so “more closed than random” alone is not evidence of irreducible many-body integration.Source: §7 · pp. 9–10, Figure 5 · p. 10

Fixed-threshold analysis

• At practiced fixed thresholds 0.02–0.2, β1 is identical for both complexes in both TNA data sets, even though human_long contains 17 phantom triangles at t ≤ 0.05.Those triangles fill cycles that exist only partway through the filtration, so fixed-threshold analyses cannot reveal the difference.Source: §7 · pp. 10–11, Table 4 · p. 11

Tooling behaviours

• Nestimate 0.8.5 returned identical co-occurrence matrices for window sizes 2, 3, and 4, and its top-dimension Betti numbers can be truncation artefacts.The example is β2 = 56 for the complete graph on nine codes, whose clique complex is contractible.Source: §7 · p. 11

8.1 Implications for learning-analytics practice

• For observed groups, keep the groups as input and build a Dowker complex before choosing a threshold; pairwise reduction loses which triples actually co-occurred.The discussion recommends comparing whole filtrations and both constructions against a null model, since agreement at individual thresholds can conceal differences.Source: §8.1 · p. 11
• Report the phantom rate, kernel dimension, and classes filled earlier by the clique complex; a zero phantom rate alone does not imply agreement.The diagnostics require no additional data; group_regulation has a phantom rate of 0.00 yet all 28 first-homology classes are filled earlier by the clique complex.Source: §8.1 · p. 12

8.2 Limitations

• The null model preserves group sizes and entity degrees but destroys pairwise co-occurrence, so its comparison cannot establish irreducible many-body integration.Testing irreducibility requires conditioning on all pairwise co-occurrences.Source: §8.2 · p. 12
• Because neighboring TNA windows share events, null-model z-scores and p-values are descriptive rather than exact tests assuming independent observations.Resampling whole sequences to account for dependence is left to future work.Source: §8.2 · p. 12
• The synthetic group_regulation result depends on window length: the Dowker complex is indistinguishable from the null at w = 3 (z = 0.9), but not at w = 4 (z = 3.5).The synthetic data need not reflect real collaborative learning; guidance on window choice and validation on real collaboration data remain future work.Source: §8.2 · p. 12
• The Stack Exchange communities are not classroom records, and the study does not connect topology to learning outcomes; it examines many-body co-occurrence, not higher-order dependencies.Classroom and collaborative-learning data, links to learning processes and outcomes, and a unified treatment of both higher-order meanings are outside this study’s scope.Source: §8.2 · p. 12

8.3 Outlook

• The outlook proposes adding a Dowker construction to Nestimate’s existing pipeline, connecting observed groups to persistent homology and enabling construction-specific diagnostics.It also proposes mapping how data properties such as density, group-size distribution, and closure relate to divergence between constructions.Source: §8.3 · p. 13
• Further work would test irreducibility conditional on pairwise co-occurrences and extend Dowker analyses to time-stamped groups, heterogeneous interactions, and directed transitions.The outlook also points to directed flag complexes and path homology for studying transition structure while retaining direction.Source: §8.3 · p. 13

9 Conclusion

• For observed groups, the Dowker complex is the appropriate construction because it captures group topology that the clique complex can erase.The conclusion says the construction should follow the data type and that the Dowker complex fills a gap in current tooling.Source: §9 · p. 13
• On public co-occurrence data and TNA toolchain examples, the topology loss is large and changes conclusions against a null model.The conclusion notes that the difference is invisible at fixed thresholds used in practice.Source: §9 · p. 13
• For observed-group data, clique complexes built from pairwise co-occurrence can only erase topology, never add to it.The conclusion characterizes the clique complex as a function of the pairwise network.Source: §9 · p. 13

Code and data availability

• Code and scripts for regenerating every figure and table are available in the project repository.The repository URL is https://github.com/igu-lab/beyond-the-clique.Source: Code and data availability · p. 13
• The data are not redistributed; Stack Exchange data derive from public Posts.xml files, and TNA examples come with tna 1.2.3 and Nestimate 0.8.5.The Stack Exchange dump is dated 31 December 2025 and licensed under CC BY-SA.Source: Code and data availability · p. 13
• Preprocessing and reproduction steps are given in Appendix A.4, and Table 2 sizes can be used to check converted data..Source: Code and data availability · p. 13

A.1 Persistence statement of Proposition 1

• In the common 1-skeleton, first-homology classes are born at the same value in both complexes, while clique-complex deaths occur no later than Dowker-complex deaths.Classes that survive in the Dowker complex but die in the clique complex are exactly the kernel of the induced map at the end of the filtration.Source: §A.1 · p. 16

A.2 Proposition 2: sketch and example

• For four learners and six pairwise conversations, the learner-side Dowker complex has β1 = 3 while its clique complex is contractible; the conversation-side Dowker complex also has β1 = 3.The example shows that clique topology can differ across the two sides of the same relation, whereas the Dowker complexes agree up to homotopy.Source: §A.2 · p. 16

A.3 Windows of length four

• At w = 4, the clique complex has β1 = 0 throughout and fills all 28 classes earlier; human_long departs strongly from the null, while group_regulation departs mildly in the opposite direction (z = 3.5).The table caption identifies 21,867 and 9,530 windows for the two TNA examples.Source: §A.3 · p. 16, Table 5 · p. 16

A.4 Preprocessing and reproduction

• Stack Exchange Posts.xml data were converted without a cap on group size; TNA sequences were converted into within-sequence windows, with short sequences forming a single group.The reimplementation of Nestimate::build_simplicial produced identical simplex sets to package output at thresholds 0.02, 0.05, 0.10, and 0.20.Source: §A.4 · p. 16