Topological Data Analysis and Geometric Graph Theory in Complex Networks

Topological Data Analysis and Geometric Graph Theory in Complex Networks: From Social Dynamics to Spatial Omics.

Abstract

Complex networks have gone through a radical change from just a few graph-theoretic representations of networks that consider only pairs of nodes to advanced geometric and higher-order topological representations. This paper introduces a detailed methodological approach to build a bridge between the classical structural network analysis and contemporary spatial biological systems. We investigate the development of network modeling by the use of dynamic influence matrices, spectral geometry, and persistent homology, introducing a mathematical hierarchy. In conclusion, we show how a mathematical framework such as topological data analysis (TDA) and geometric graph theory are needed to unlock the architectural complexity of biological tissues, especially in the fast evolving field of spatial omics.

Introduction

Traditionally, complex networks have been thought of by using graph-theoretical approach based on adjacency, which has been successful in quantifying the classical structural topologies and social dynamics. With the shift into biologically multi-plexed systems, traditional models were unable to capture the multi-scale and continuous nature of cell micro-environments. In the past few years, new spatially resolved omics technologies have allowed the high dimensional molecular profiling of cells in the native context of their tissue structure (Isik et al., 2026). Hence, a fundamental shift from pairwise interaction to higher-order geometric and topological models is needed for the modelling of these networks.

In this paper, we tackle the important issue of mathematical modeling of complex biological networks from a unified, hierarchical perspective. We study the transition from static graphs to dynamic networks and then from dynamic networks to geometric and topological representations based on the tools of spectral geometry and Ricci curvature as well as those of persistent homology. A central goal of this research is to create a comparative mathematical pipeline to make the use of higher-order topology in spatial omics uniform. We bring these mathematical languages together and offer a solid basis for the detection of spatially structured patterns of gene expression and cellular organization (Xu & Sankaran, 2021).

The classical network approaches are very limited in application to bio data with high dimensionality for a number of basic reasons. Firstly, higher order, multi-way interactions are common in dense multicellular environments in biological systems and are not captured by classical adjacency-based graphs (Noorbakhsh et al., 2025). Second, common pairwise measures are sensitive to spatial noise and cannot maintain the continuous shape characteristics of the manifolds of underlying data (Anai et al., 2018). Last, simple graph metrics are not able to integrate morphological features across multiple scales with complex layers of transcriptomics data seamlessly (Chelebian et al., 2024).

Acknowledging these deep methodological deficiencies, the present paper contributes the following major elements:

We construct a mathematically hierarchical sequence of spaces of classical graph metrics (adjacency and Laplacian matrices) and advanced geometric and topological spaces that are mapped in a systematic way.

A rigorous, comparative methodological approach for using topological data analysis (TDA) to unsupervised feature selection and spatial enrichment in large-scale spatial omics networks is proposed.

Related Work

Spatial Statistics and Neighborhood Models

The first class of related work is on spatial statistics and analytical neighborhood enrichment models. Permutation-based Monte Carlo tests are commonly used to measure the level of spatial enrichment or depletion of categorical cellular labels in traditional spatial omics workflows (Andersson & Nyström, 2025). Spatial statistics have also been shown to be versatile tools for modeling local gene expression as point patterns and lattice data, with advanced computational toolkits emerging (Emons et al., 2024). These approaches offer powerful basic analytics and statistical acceleration, but typically are not capable of representing multi-scale topological shapes and complex geometrical deformations found in tissue structures.

Topological Data Analysis and Filtrations

The second class of methods are based on Algebraic Topology, namely Persistent Homology and Topological Data Analysis (TDA). High-performance TDA has been widely implemented in machine learning through well-designed and robust preprocessing and C++ implementations (Tauzin et al., 2020). Classical Čech or Vietoris-Rips filtrations are very sensitive to outliers, so researchers have developed Distance-to-Measure (DTM) filtrations that offer greater noise-stability in Euclidean point clouds (Anai et al., 2018). Moreover, there have been several new tools developed to visualize multivariate data structures without loss of information, such as the TDA Ball Mapper (Rudkin, 2025). A remaining limitation in these methods is the very high computational burden it takes to produce higher dimensional simplicial complexes in large-scale databases of biology.

Ion-Channel Diseases and the Integration of AI and Multimodal approaches into Biological Networks

The third category is a combined overview of the field of artificial intelligence and multimodal data integration in biological networks. To fully explore the potential of integrating spatial transcriptomics with imaging AI, cutting-edge frameworks are actively developed to extract morphological features that correlate with the spatial pattern of expression of the genes (Chelebian et al., 2024). The creation of AI models that can be understood and interpreted in space requires advanced data integration algorithms and a new way of thinking about AI (Noorbakhsh et al., 2025). TDA is underpinned by mathematical models offering interpretability, which black-box AI models are currently lacking, and represents a powerful link between spatial coordinates and biological function (Boyle et al., 2025).

Method/Approach

We propose a hierarchical continuum and a structured "Comparative Mathematical Framework" to systematically address the complexity of spatial omics. This pipeline takes raw biological coordinates and molecular features and translates them into a series of increasingly sophisticated mathematical spaces. The basic design principle is that biological tissues are not just pairwise graphs but continuous geometric manifolds which need to be represented topologically at multiple scales. The framework moves through different mathematical phases, maintaining the local dynamics as well as the global topological properties.

The framework models classical or dynamic structures during the first two phases based on raw input data that becomes adjacency matrices  and the calculation of the graph Laplacian These matrices represent simple cell-to-cell proximity and local social-like influence relationships between neighbouring cells. The spectrum of the Laplacian gives the first link to geometric representation, and is used to approximate continuous diffusion processes over the discrete tissue network. This allows a basic structural dynamic to be quantified before higher order multi-way interactions are taken into account.

Geometric and topological network representations are the third phase. In this case, we calculate Ricci curvature of the network to measure local neighbourhood density and bottlenecks, thus defining key transition regions between different tumor microenvironments. At the same time, we also create a DTM-filtrations to create robust simplicial complexes from the point cloud data, which reduces the effect of biological noise and outliers (Anai et al., 2018). These complexes are then used in the persistent homology space to monitor how features appear and disappear at different scales in space (Boyle et al., 2025).

The last step addresses more abstract biological networks and spatial omics, using abstract topological summaries. We use methods similar to the TDA Ball Mapper to create abstract 2D representations of the multivariate genomic features (Rudkin, 2025). These topological summaries are then input to machine learning classifiers to perform unsupervised feature selection and discovery of spatially variable genes. High-performance TDA workflows allow for capturing the multi-way cellular interactions and the general tissue morphology (Tauzin et al., 2020).

In order to verify this methodological approach, we suggest a multi-layered spatial transcriptomics benchmarks-based hypothetical evaluation plan. We will compare our TDA based pipeline to the standard neighborhood enrichment scores (Andersson & Nyström, 2025) and with the standard graph neural network. The criteria assessed will be the accuracy of the identification of the genes in space, the computing time needed for increasing spatial resolutions, and the fidelity of the topological features, when coordinate errors are artificially added to the data.

Discussion

Applications of geometric and topological network methods are very promising in real-world complex networks. Topology has been translated into structural biomarkers in the tumor microenvironment that dictates the progression of the disease; these can be discovered through omics analysis (Noorbakhsh et al., 2025). Moreover, these higher order mathematical formalisms can be packaged into usable, interactive computational toolkits, making high-level geometric graph theory a tool for clinical biologists more widely accessible (Xu & Sankaran, 2021).

But these techniques have not yet become widely adopted, due to a number of key limiting factors and failure modes. First, generating higher dimensional simplicial complexes by persistent homology is a process that is exponential in size; large-scale tissue analysis is therefore computationally prohibitive (Tauzin et al., 2020). Second, although DTM-filtrations have been introduced to boost the power, the choice of hyperparameters, such as the filtration radii, is very sensitive to the initial tuning (Anai et al., 2018). Third, higher-order topological features (such as higher dimensional Betti numbers) are difficult to interpret in a strictly biological context and are hard to use clinically (Noorbakhsh et al., 2025).

Ethical issues and risks arise with the use of AI and TDA in biomedical applications. The biggest concern is that the spatial AI models might be biased and create distorted morphological representation, which can affect downstream diagnostic fairness (Chelebian et al., 2024). Second, multimodal spatial omics data are very detailed and include information on the genetic make-up of each cell, making it easy to re-identify patients if the data are not anonymized properly.

In the future, computational and interpretational issues of geometric networks need to be addressed. First, scalable analytical approximations to persistent homology, which are similar to recent speed-ups in neighborhood enrichment tests (Andersson & Nyström, 2025), will be necessary to handle large datasets. Second, future work should take place on embedding the morphological representation learning process of imaging directly into the topological filtration process to create biologically grounded, multimodal geometric models (Chelebian et al., 2024).

Conclusion

Pairwise graph theory has become higher order topological representation, and by using this lens, the understanding of complex systems is indispensable. This paper has illustrated the use of hierarchical models, ranging from dynamic influence to spectral geometry to persistent homology, as means of addressing the shortcomings of the classical adjacency-based models. Most of the domain of the omics of spatial structures can be directly mapped to this mathematical operation, thus providing new tools for deciphering the complex structural and functional architecture of biological tissues.

Finally, the geometric graph theory-spatial biological systems synthesis is a key interdisciplinary research frontier. Topological data analysis will set the foundation for new paradigms of high-dimensional network dynamics as computational limitations are overcome and theory is brought into a more biological context. The ongoing development of these mathematical architectures will no doubt spur further advances in comprehending social interactions and microenvironments of complex diseases.

References

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Noorbakhsh, Javad, pour, Ali Foroughi, & Chuang, Jeffrey (2025). Emerging AI Approaches for Cancer Spatial Omics. https://arxiv.org/pdf/2506.23857v1 https://arxiv.org/pdf/2506.23857v1

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