## 4 Discussion  
What is cellular complexity and what does higher-order complexity mean? As an inquiry to this question, we explored the possibility of introducing the mathematical notion of higher-order simplicial complexes into analyzing distance-based single-cell data. Benchmarked on single-cell gene expression data with multiple developmental stages,  
Figure 10: Cell lineage tracing with the simplicial statistics. In this analysis, hierarchical clustering was performed on the summary statistics of transcriptomic data of different cell types. (**A**) The heat map and clustering results when using the Betti numbers as the clustering features. (**B**) The heat map and clustering results when using the normalized simplicial complexity as the features for hierarchical clustering. Figure 9: Temporal filtration identifies the critical stage of cellular complexity change. The colors indicate the time points, and each node corresponds to a small cluster of cells collected at the same time points. The conventional TDA mapping (the left panel) identifies a bifurcation structure, but there are spatial locations that have a mixture of clusters that belong to non-consecutive time points. This makes the identification of a developmental pathway challenging. When applying the temporal filtration (the right panel), the mapping identifies a clean separation of two tracks, or two sub-populations of cells that evolve in the gastrulation stage, matching the observation in our summary statistics from the algebraic topology. we proposed single-cell topological simplicial analysis and demonstrated that simplicial complexity can be a well-defined summary statistic for celluar complexity. This investigation provides a scalable, parameter-free, expressive, and unambiguous mathematical framework to represent the cellular complexity with its underlying structure. By \"parameter-free\", we mean that it does not have arbitrary hyperparameters that the users have to set in order to perform the analysis. The parameter \(\tau\) is instead a user-specified parameter that is relevant to the specific application and problem of interest. An analogy for a prediction model would be that the learning rate is an arbitrary hyperparameter, and the prediction window would be a user-specified parameter relevant to the application. Locally, these structures are characterized in terms of the simplicial complexes. Globally, these structures are characterized in terms of the cavities formed by these simplices. Topological cavities are usually formed and then later filled with the addition of new edges (and potentially nodes). When computing the persistent homology, we performed a filtration process which innately tracked the formation and later filling of topological cavities of different dimensions. The temporal persistent homology characterized the information of cavities with the lifespan of these topological objects. This framework revealed an intricate topology of cellular similarity which included a vast number of cliques of cells and of the cavities that bound these cliques together. These topological summary statistics, which captured the relationships among the high-dimensional cliques, uncovered the transcriptional differences in the connectivity of cells of different types during the graph reconstruction process. From the scTSA visualization, we discovered for the first time in any single-cell data an abundant number and variety of higher-order cliques and cavities. When compared with the control models, the framework measured a much higher number of high-dimensional cliques and cavities in the graph construction filtration process. The critical stage identified by the framework matched the current understanding in developmental biology. Compared with the statistics of the Betti numbers, the normalized simplicial complexity demonstrated better distinctions between time points and cell types. Topological data analysis, as in many other machine learning methods, has many empirical considerations related to sample size and dimensionality selection.