## 1 Introduction  
In recent years, technological developments in data visualization, especially the sub-field of topological data analysis (TDA), has illuminated the structure of biological data with features such as clusters, holes, and skeletons across a range of scales [1]. The TDA approach has proven to be especially useful with recent advancements in experimental techniques at the single-cell resolution in both genomics and neuroscience, such as radiomics [2] and brain imaging [3, 4]. The utility of the topology comes from the idea of persistence, which extracts the underlying structures within data while discarding noisy elements in the single-cell data collection. Unlike graph-based data such as human connectomes, most of the time, the high-dimensional data collected from single-cell techniques are similiarity-based. Under the assumption that these data were sampled from an underlying space \(\mathcal{X}\), the goal is to first approximate \(\mathcal{X}\) with a combinatorial representation and then compute some sort of invariant features to recover the topology of \(\mathcal{X}\). For interested readers, Refs. [5, 6, 7] are a few recent reviews of the applications of TDA in various field of biology, Ref. [8] is a practical introduction and guide on how to apply TDA to data science and understand its results, and [9] is a gentle introduction and tutorial to the computation of a persistent homology. Single-cell topological data analysis (scTDA) is one of the first attempts to apply topology-based computational analyses to study temporal, unbiased transcriptional regulation given the single-cell RNA sequencing data [10]. In order to visualize the most invariant features of the entire gene expression data, scTDA clusters low-dispersion genes with significant gene connectivity according to their centroid in the topological representation and visualizes the data points in a low-dimension space with the Mapper algorithm [11]. Computing the library complexity as the number of genes whose expression is detected in a cell, scTDA observes a mild dependence by the library complexity on the timescale of the single-cell data of 1529 cells collected at 5 time points. This is expected because the number of genes expressed by cells in the early stages of a developmental process is larger than in the adult case, as pointed out in [12]. As a result, in scTDA, the library complexity is not used for any purpose in the topological data analysis and not related to any topological properties. Intuitively thinking, if we were to introduce a definition for \"cell complexity\" which characterizes the behaviors of cell-cell co-expression or interactions, the quantities of cell complexity should be agnostic to the number of genes expressed by the cells and should be different across differentiated cells and across the developmental process. Can we introduce a better summary statistic for the cell complexity that can capture the developmental trajectory with more distinctions between time points? To clarify, unlike the previous definition of \"library complexity\", which simply quantifies the number of genes expressed in a cell, we wish to define a cell complexity measure to better model higher-order networks and dynamic interactions in single-cell data. Understanding the cell-cell interactions can help identify intercellular signaling pathways, and previous analytical studies have focused on computing a communication score between the ligand-receptor pair of interacting proteins [13]. For instance, the authors of [14; 15] inferred the intercellular signaling pathways of cell-cell communications by computing the co-expression of all genes or other cell markers. The alternative would be to compute the similarity between the gene expression profiles, as in [16]. In this work, we aim to focus directly on the cell level and use the similarity between each cell's gene expression profiles as a graph to compute a topological descriptor of the complexity.