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Higher-Order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes
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Proposes HiGCN, a spectral graph network built on flower-petals (FP) Laplacians derived from two-step random walks between a node core and simplex petals, with learnable per-order polynomial filters whose weights quantify higher-order interaction strength.
Problem
Pairwise GNNs cannot capture higher-order group interactions and are bounded by the WL test, while existing simplicial GNNs are complex, rigid, and need full simplex enumeration (Hasse diagrams), and interaction strength is hard to quantify.
Approach
Simplifies the Hasse diagram into a flower-petals model where each order-p petal connects only to the 0-simplex core via a bipartite incidence graph; a two-step random walk yields FP adjacency/Laplacians, and per-order learnable polynomial filters on these Laplacians are concatenated, with filter weights reported as interaction strength S_p.
Key results
- On node classification over 10 graphs, HiGCN (best of 1-4 orders) attains the top result on 9 of 10 datasets; e.g., 2-HiGCN reaches 89.23% on Cora and 68.47% on Chameleon.
- On TUD graph classification, HiGCN is best on MUTAG (91.3), IMDB-B (76.2) and IMDB-M (52.7) among the compared methods.
- On simplicial data imputation at 10% known signals, HiGCN reaches Kendall's tau 0.463 on Geology versus 0.265 for SNN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Cora | accuracy (%): 88.57 ± 0.69 -> 89.23 ± 0.23 | Helped |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Citeseer | accuracy (%): 80.52 ± 0.71 -> 81.12 ± 0.28 | Helped |
| Use 4-HiGCN instead of the strongest listed baseline | HiGCN node classification · PubMed | accuracy (%): 89.63 ± 0.46 -> 89.95 ± 0.13 | Helped |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Computers | accuracy (%): 87.54 ± 0.43 -> 90.76 ± 0.27 | Helped |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Photo | accuracy (%): 93.85 ± 0.28 -> 95.33 ± 0.37 | Helped |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Chameleon | accuracy (%): 68.29 ± 1.58 -> 68.47 ± 0.45 | No clear effect |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Actor | accuracy (%): 41.79 ± 1.91 -> 41.81 ± 0.52 | No clear effect |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Squirrel | accuracy (%): 51.35 ± 0.73 -> 51.86 ± 0.42 | Helped |
| Use 2-HiGCN instead of the strongest listed baseline | HiGCN node classification · Texas | accuracy (%): 93.12 ± 0.65 -> 92.15 ± 0.73 | Hurt |
| Use 4-HiGCN instead of the strongest listed baseline | HiGCN node classification · Wisconsin | accuracy (%): 91.82 ± 0.38 -> 94.89 ± 0.65 | Helped |
| Use HiGCN instead of the strongest listed baseline | HiGCN graph classification · MUTAG | accuracy: 90.6 ± 8.7 -> 91.3 ± 6.4 | Helped |
| Use HiGCN instead of the strongest listed baseline | HiGCN graph classification · PTC | accuracy: 66.8 ± 1.7 -> 66.2 ± 6.9 | Hurt |
Surprising findings
- Performance does not consistently increase with more petals (orders); 2- and 3-HiGCN beat 4-HiGCN on several datasets, which the authors attribute to training complexity.
- HiGCN underperforms on Texas (90.36 for 1-HiGCN, 92.15 for 2-HiGCN vs. BernNet 93.12), which the authors link to few triangles and weak higher-order interactions.
Limitations
- Performance does not monotonically improve with more higher-order petals, suggesting training difficulty when more orders are added.
- Underperforms on heterogeneous Texas where higher-order structures are sparse.
- Non-SC graphs require a one-time clique complex lifting preprocessing step.
Experimental setup
Datasets: Cora CiteSeer PubMed Computers Photo Chameleon Squirrel Actor Texas Wisconsin PROTEINS MUTAG
Benchmarks: Cora CiteSeer PubMed Computers Photo Chameleon Actor Squirrel Texas Wisconsin PROTEINS MUTAG
Metrics: accuracy Kendall's tau
Baselines: MLP GAT ChebNet BernNet GGCN APPNP GPRGNN SNN SGAT SGATEF MPSN GIN
Builds on: GPRGNN GCN Hodge Laplacian simplicial GNNs (SNN, SCNN) MPSN Weisfeiler-Lehman test
Closest papers by meaning
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