Enhanced Cellular Isomorphism Networks (CIN++)

• The model generalizes CIN by expanding topological message pathways in cell complexes, enabling direct interactions between higher-order structures like rings.
• CIN++ employs a multi-way message-passing layer that aggregates boundary, upper, and lower messages to accelerate information flow and improve color refinement.
• Empirical benchmarks show CIN++ achieves state-of-the-art performance on molecular and peptide datasets, with efficient handling of long-range graph dependencies.

Enhanced Cellular Isomorphism Networks (CIN++) are a family of message-passing neural architectures designed to mitigate the limitations of conventional Graph Neural Networks (GNNs) in modeling long-range and higher-order dependencies. CIN++ generalizes the Cellular Isomorphism Network (CIN) framework by expanding topological message pathways within cell complexes, supporting direct interactions between higher-order structures (“rings”) and enabling more expressive higher-dimensional representations. The approach establishes improved routes for information flow, accelerates the refinement of cell colorings, and empirically sets state-of-the-art benchmarks for tasks involving molecular, peptide, and general graph-structured data (Giusti, 2024, Giusti et al., 2023).

1. Cell Complex Construction and Feature Lifting

CIN++ operates on a 2-dimensional cell complex $\mathcal{C}$ derived from a base graph $G=(V,E)$ via a structural and functional lift. The procedure first constructs $\mathcal{C}$ by attaching 2-cells (“rings”) to all induced (chordless) cycles in $G$ of size up to a specified maximal ring size $R$. The resulting structure comprises:

• 0-cells: vertices of $G$
• 1-cells: edges of $G$
• 2-cells: rings (one for each eligible cycle)

For each edge $e=(u,v)\in E$, edge features $x_e$ are derived from node features through a two-layer MLP: $x_e = \text{MLP}_1([x_u \| x_v])$, optionally augmented with application-specific labels. This feature lifting ensures that edge and ring-level semantics are available for subsequent message passing.

Each cell $G=(V,E)$0 is assigned neighborhoods reflecting:

• Boundary $G=(V,E)$1: strict faces of $G=(V,E)$2
• Co-boundary $G=(V,E)$3: cofaces containing $G=(V,E)$4
• Upper adjacency $G=(V,E)$5: same-dimensional cells sharing a coface with $G=(V,E)$6
• Lower adjacency $G=(V,E)$7: same-dimensional cells sharing a face with $G=(V,E)$8

Typical values for $G=(V,E)$9 (maximal ring size) are dictated by domain priors (e.g., aromatic rings in chemistry).

2. CIN++ Message-Passing Layer

At each layer $\mathcal{C}$0, CIN++ updates cell signals $\mathcal{C}$1 by aggregating three message types, followed by a local update:

1. Boundary messages (from faces in $\mathcal{C}$2):

$\mathcal{C}$3

2. Upper messages (from upper neighbors in $\mathcal{C}$4 via shared cofaces $\mathcal{C}$5):

$\mathcal{C}$6

3. Lower messages (from lower neighbors in $\mathcal{C}$7 via shared faces $\mathcal{C}$8):

$\mathcal{C}$9

4. Update:

$G$0 with $G$1 a linear layer plus nonlinearity, distinct per dimension and per layer.

Readout operations pool features across dimensions $G$2 (nodes, edges, rings) via permutation-invariant aggregators to yield global graph embeddings.

A pseudocode formulation is supplied in (Giusti et al., 2023), supporting parallelization over cells.

3. Multi-Way Communication and Expressivity

Compared to CIN and other cell-complex GNNs, CIN++ introduces explicit lower adjacency message paths, enabling direct intra-dimensional communication (e.g., ring–ring via shared edges, edge–edge via shared nodes). This addition addresses the over-squashing bottleneck by increasing the effective communication bandwidth across the complex, facilitating rapid propagation of signals between distant or structurally related regions.

Theoretical analysis demonstrates that CIN++ is Cellular Weisfeiler–Lehman (CWL) equivalent:

• For injective aggregation and universal update functions, with sufficient layers, CIN++ can distinguish any two non-isomorphic cell complexes that the CWL test can.
• Although lower messages do not increase expressivity beyond CWL, they empirically accelerate convergence of color refinement, as evidenced for ring colorings in molecular graphs (see, e.g., Figure 1 in (Giusti et al., 2023)).
• Removing boundary or upper messages severely degrades both expressivity and performance.

4. Computational Complexity and Implementation

The computational cost per CIN++ layer is $G$3, where $G$4 and $G$5 is the cell embedding dimension. Each MLP block is $G$6 per cell per neighborhood. Enumeration of chordless cycles for cell complex construction (structural lift) requires $G$7.

CIN++ is implemented in PyTorch and PyTorch-Geometric, using parallelization and graph-tool acceleration for induced cycle enumeration, and supports batch processing. Edge-pooling operations introduce $G$8 overhead when activated.

Memory usage scales with the total number of cells and embedding dimension, fitting 250K molecular graphs of typical size in 32GB RAM.

5. Empirical Performance and Benchmarks

CIN++ sets state-of-the-art results on multiple graph learning benchmarks:

Method	ZINC-Subset MAE↓	ZINC-Full MAE↓	Peptides-func AP↑	Peptides-struct MAE↓
CIN++	0.077 ± 0.004	0.027 ± 0.007	0.6569 ± 0.0117	0.2523 ± 0.0013
Best baseline*	0.081 ± 0.009	0.025 ± 0.004	0.6439 ± 0.0075	0.2529 ± 0.0016

*Best baseline: Graphormer-GD for ZINC, SAN+RWSE for Peptides.

CIN++ achieves the best published MAE on ZINC-Subset and Peptides-struct, and the best AP on Peptides-func, outperforming all listed GNN, transformer, and topological GNN baselines.

On TUDatasets (MUTAG, PTC_MR, PROTEINS, NCI1, NCI109), CIN++ achieves top accuracy on 4 out of 5 benchmarks relative to CIN, CAN, and kernel-based methods. Results were aggregated over 4–10 random seeds with reported standard deviations (Giusti et al., 2023).

Empirical ablations indicate that the lower message path (ring–ring, edge–edge) is critical for gains on long-range tasks: Adding lower messages drops MAE from 0.085 to 0.077 on ZINC-Subset and increases test average precision from 0.643 to 0.657 on Peptides-func.

6. Limitations and Future Prospects

CIN++ is subject to several technical limitations:

• Enumeration of all chordless cycles introduces preprocessing overhead, posing scalability challenges for dense or large graphs.
• Model parameter count and memory footprint increase linearly with the number of cell dimensions; extending to 3-cells (e.g., tetrahedra) further increases these costs.
• The expressive capacity remains upper-bounded by the CWL test; no super-CWL separation is achieved.

Proposed extensions include generalization to $G$9-dimensional cell complexes for richer topological reasoning, learnable structural lifting (attachment of higher-order cells learned end-to-end), optimized batching and sparse kernels for scalability, and the integration of positional or spectral encodings for extremely long-range dependencies (Giusti et al., 2023).

7. Applications

CIN++ is designed for molecular property regression and classification, peptide structure-function prediction, and graph classification in standard benchmarks. Its anisotropic, multi-way message pathways make it particularly effective for domains where higher-order and group interactions are fundamental, including:

• Chemoinformatics (e.g., molecular logP regression, HIV inhibition classification)
• Computational biology (peptide/protein structure and function tasks)
• General graph-structured machine learning tasks requiring expressive and efficient handling of long-range dependencies (Giusti, 2024, Giusti et al., 2023).

The model’s architecture and theoretical foundations also suggest relevance for any setting where the encoding of topological, group-based, or cycle-based relationships is crucial for downstream performance.