The central limitation the authors identify is architectural. Traditional graph neural networks, from the original graph neural network model through graph convolutional networks, GraphSAGE, and graph attention networks, operate through local message passing between connected nodes. This design encodes pairwise interactions well but struggles to represent higher-order structures explicitly. Research on higher-order network organization has shown that many real systems, from protein interaction networks to transportation systems, cannot be fully described by pairwise edges alone. Group interactions, hierarchies, and multiscale dependencies require richer mathematical objects than simple graphs can provide.

Compounding this expressive bottleneck are well-documented pathologies of message passing. Over-smoothing causes node representations to become indistinguishable as networks grow deeper, while over-squashing compresses information from distant parts of a graph into a fixed-size vector, degrading performance on tasks requiring long-range dependencies. Studies connecting these failures to graph curvature have sharpened the diagnosis, and work on the Weisfeiler-Lehman hierarchy has clarified exactly what graph neural networks can and cannot distinguish. The review positions topological deep learning as a principled response to these constraints rather than an incremental patch.

Topological deep learning draws on topological data analysis, a field rooted in the computation of persistent homology. The idea is elegant: rather than describing data at a single scale, one examines how topological features such as connected components, loops, and voids appear and disappear as a scale parameter varies. The resulting persistence diagrams, often visualized as barcodes, are stable under small perturbations of the data, a property formalized in classical stability theorems for persistence modules. This stability is what makes topological summaries usable as features in machine learning pipelines, since small changes in input should not produce wildly different representations.

The review traces how these topological signals enter neural architectures through several distinct mechanisms. In one family of approaches, persistence diagrams are converted into fixed-length vectors using persistence images, persistence landscapes, or kernels such as the sliced Wasserstein kernel, and then fed to standard learners. In another, differentiable topology layers allow topological quantities to be optimized directly within a network, as in topological autoencoders that regularize latent spaces to preserve topological structure. A third family builds the topology into the architecture itself: simplicial neural networks, cell complex neural networks, and combinatorial complex networks generalize message passing from graphs to higher-dimensional domains, enabling information to flow along boundaries of simplices and between cells of varying dimension.

Sheaf-based models represent a particularly intriguing branch. Neural sheaf diffusion reframes graph neural networks through the lens of cellular sheaves, attaching vector spaces to nodes and restriction maps to edges. This perspective offers a topological explanation for heterophily and over-smoothing problems, suggesting that the failure modes of standard architectures are not accidental but reflect a mismatch between the underlying topology and the assumed uniformity of message passing. The review treats these developments as part of a coherent progression from classical structural features toward increasingly expressive topological models.

Where do topological features actually earn their keep? The review identifies application domains in which they carry meaningful interpretations. In neuroscience, persistent homology has revealed cliques and cavities in the human connectome, and homological scaffolds of brain functional networks have exposed architecture invisible to pairwise correlation analysis. In structural biology, topology-based networks such as TopologyNet have predicted biomolecular properties, while persistent homology has been applied to protein folding, flexibility, and binding analysis. Sensor network coverage has been verified through persistent homology, and neural codes have been decoded through the topology of stimulus spaces. In software engineering, code property graphs and vulnerability detection systems illustrate how structured program representations benefit from richer relational modeling.

The authors are careful to avoid overclaiming. Their review proposes criteria for sound model comparison, noting that a graph baseline alone does not constitute a topology-specific ablation, that asymptotic complexity claims do not substitute for measured preprocessing costs, and that visualizing a persistence diagram counts as localization only when it can be traced back to structure in the original sample. An appendix auditing a subset of method papers found that many empirical studies fall short on these reporting standards, a finding that echoes broader concerns about benchmarking rigor in the field. Frameworks such as TopoBench and the TopoX suite of Python packages are highlighted as infrastructure that could make comparisons more systematic.

The expressivity question remains open. Recent work on topological blindspots has shown that even topological neural networks have limits in what they can distinguish, and the relationship between the Weisfeiler-Lehman hierarchy and simplicial or cellular expressivity is still being mapped. Differentiable lifting, the process of converting graph data into higher-order complexes, introduces its own design choices that can dominate downstream performance. The review emphasizes that these choices are not neutral: how one lifts data into a simplicial or cellular complex determines which topological features become visible to the model at all.

The practical takeaway for practitioners is a decision framework rather than a slogan. If a task is dominated by local pairwise structure, standard graph neural networks remain efficient, well-understood, and supported by mature libraries such as PyTorch Geometric and the Deep Graph Library. But when the signal lives in cycles, cavities, flows, or group interactions, when the global shape of the data matters, topological deep learning offers tools that graph models cannot replicate. Shymkovych and Hryshyn’s synthesis makes a compelling case that the field has matured enough to move from mathematical promise to disciplined application, provided that future studies meet the reporting standards the review lays out. As complex systems generate ever richer structured data, the shape of that data, quite literally, may become the most informative feature of all.

Subject of Research: Topological deep learning methods for analyzing structured data in complex systems

Article Title: Topological deep learning for the analysis of structured data in complex systems: a structured critical review

Article References: Shymkovych, V., & Hryshyn, O. (2026). Topological deep learning for the analysis of structured data in complex systems: a structured critical review. International Journal of Data Science and Analytics, 22(1), Article 314. https://doi.org/10.1007/s41060-026-01304-5

Image Credits: AI Generated

DOI: 10.1007/s41060-026-01304-5

Keywords: topological deep learning, graph neural networks, persistent homology, simplicial complexes, cellular complexes, higher-order networks, topological data analysis, sheaf neural networks, complex systems, structured data, message passing, benchmarking