## 4 Discussion
To demonstrate the sensitivity of persistent homology to the sampling size and reduced dimensions, we perform the following experiment. We use the full dimensions of the standard scaled dataset, varied the sampling size from 50 to 100, 500, and 1000 data points, and computed their persistence diagrams. We then set the sample size to 1000, varied the PCA dimensions to be the first 2, 10, and 103 (full) dimensions, and computed their persistence diagrams. We observed no clear difference. Then, we perform simplicial analysis with witness sampling using sample sizes from 10 to 100, 200, and 300. In this case, we observe a slightly higher number of higher-order simplical complexes, but the overall shape and distinction between the time steps were maintained (Figure 11). Future studies can investigate strategies for increasing the stability of simplicial analysis for the sample size. In the introduction, we posed some open questions we wish to engage the field to discuss and investigate together, instead of answering them directly in this first work. Here, we will briefly share our preliminary takes on some of the specific ones. Why does expression similarity deserve the name of complexity? To clarify, the expression similarity may not be a measure of complexity. However, the temporally connected higher order co-expression structure characterized by similarity can be a useful measure of complexity. If the task requires several agents to work together at the same time or follow a specific sequence of actions by different agents, then it is more complex than a task which only requires a few agents or does not need to follow a specific sequence. The notion of similarity is usually related to clustering and thus the separation of homogeneous groups. To expand on this understanding, the similarity relationships that are further constrained by temporal sequences would relate to functionally separating groups of homogeneous agents, thus potentially being informative of their interactions. Figure 11: Sensitivity analysis of the effectiveness of witness sampling and PCA dimension reduction on the persistent homology and simplicial analysis. **(A)** Persistence diagrams of the dataset when sampling 50, 100, 500, and 1000 data points. **(B)** Persistence diagrams when choosing the first 2, 10, and 103 (all dimensions) principal components. **(C)** The overall distribution of the normalized simplicial complexity did not change much when the sampling size at each time step rose from 10 to 100, 200, and 300. Is there a reason to believe gene expression similarity has something to do with interactions rather than reflecting the number of similar cells that happen to be present in the sample? This temporally constrained gene expression similarity can both reflect the number of similar cells that coexist at the same time and also potentially be related to some level of functional interaction, as discussed above. We wish to leave further investigations on the types of interaction for future works and welcome discussions on and critique of these interpretations. Finally, there are other potentially applicable questions we can explore. Can we determine the developmental stages without physiological features? Can we generate pseudo-time series based on single-cell sequencing data? Finally, and most importantly, does the vast presence of high-dimensional cliques suggest that the interaction between these cells is organized into fundamental building blocks of increasing complexity? Through this inquiry with topological simplicial analysis, we can form a hypothesis that the cells organize themselves into high-dimensional cliques for certain functional or developmental reasons. Further research includes developing mechanistic theories behind the emergence of such high-dimensional cellular cliques and experimentally testing these hypotheses to reveal the missing link between functions and cellular complexity.