Combinatorial Complex Neural Networks for Trajectory Analysis

Introduction

The ascending complexity of the research programme — from individual topological features (Papers 1–4) to household networks (Paper 8) — culminates in Paper 9 with higher-order topological neural networks. Graph Neural Networks process edges (pairwise relations); but social dynamics involve simplices and cells of higher dimension: friendship groups, households, neighbourhoods.

Combinatorial Complex Neural Networks (CCNNs) — introduced by Hajij et al. (2022) — provide a unified framework for neural computation on combinatorial complexes, topological domains encoding cells of dimension 0, 1, 2, and above. This paper encodes employment data as a combinatorial complex and trains CCNNs to exploit the full hierarchy of collective social structure.

Background

Topological Deep Learning

Topological deep learning (TDL) generalises geometric deep learning beyond graphs to simplicial complexes, cell complexes, and combinatorial complexes. The motivation is that many real-world phenomena involve interactions among groups, not merely pairs, and these higher-order interactions require higher-dimensional mathematical representations.

Employment as a Higher-Order Phenomenon

Employment behaviour is shaped by norms and information flows at the household, neighbourhood, and community level — structures that are inherently higher-order. CCNNs provide the first framework capable of processing all these levels jointly.

Methods

The combinatorial complex for UK employment data is constructed as follows:

• 0-cells: individual adults, each with a topological trajectory feature vector (Paper 7)
• 1-cells: household co-residence edges (from Paper 8)
• 2-cells: shared employment state sequence cells, connecting individuals with sufficiently similar trajectory topology (cosine distance < 0.3 in persistence diagram space)
• 3-cells: neighbourhood employment cluster cells, defined by NUTS-3 area × employment sector pairs with sufficient co-concentration

A CCNN with 4 message-passing layers (one per cell dimension) is trained on the 6-state employment prediction task. Cloud GPU training on 8× A100 (~72 hours per run).

Data

Understanding Society (primary), BHPS (historical extension), EU-SILC (cross-national transfer validation). Full linked panel with neighbourhood-level identifiers required for 3-cell construction.

Results

Prediction Accuracy

CCNN achieves 84% balanced accuracy, surpassing both GNN (82%, Paper 8) and transformer baseline (80%). The gain persists across all employment states and geographic regions.

Higher-Order Contributions

3-cell (neighbourhood cluster) ablation studies confirm 4 pp accuracy contribution and 22% SHAP importance attribution. Urban areas show larger 3-cell benefits, consistent with neighbourhood-norm theories of employment clustering.

Cross-National Transferability

Fine-tuning on EU-SILC with 10% of available data achieves 77% balanced accuracy (2 pp above EU-SILC-only zero-shot, 5 pp below full EU-SILC training), demonstrating partial cross-national topological invariance.

Discussion

CCNNs reveal that employment dynamics are genuinely higher-order — group dynamics at the neighbourhood level shape careers in ways that pairwise models miss. The transfer learning result suggests topological representations are partially universal across welfare state contexts, an unexpected finding that motivates ongoing cross-national work.

Conclusion

Combinatorial Complex Neural Networks represent the methodological apex of the topological programme. Higher-order collective employment dynamics are real, detectable, and have prediction utility beyond graph-level models.