Algorithm 1 f-divergence-based dynamic gene regulatory network inference algorithm.

Input: Time-series scRNA-seq data matrix $\mathbf{X}$; percentage of randomly sampled single cells $p=80\%$; number of samples $n=100$; f-divergence measures; regularization methods $\left\{\mathrm{LASSO},\mathrm{MCP},\mathrm{SCAD}\right\}$.

Output: Time-varying gene regulatory networks.

Step 1: Random sampling and temporal variation calculation of genes using f-divergence Randomly sample a percentage of single cells from the data at different time points. Apply f-divergence to compute the temporal variation, $D_f^j\left(t_l\right)$, for each gene j’s expression levels across all single cells between two consecutive time points, $t_l$ and $t_{l+1}$, $l=1,2,...,n$. Normalize the temporal variation, $D_f^j\left(t_l\right)$ with respect to time interval between consecutive time points ($\Delta t_l$): ${\widehat{D}}_f^j\left(t_l\right)={\displaystyle \frac{D_f^j\left(t_l\right)}{\Delta t_l}}$. Repeat the above process n times to generate multiple temporal variation vectors. Step 2: Network structure learning using VAR(1) model with regularization Construct sliding windows that contain $D_f^j\left(t_l\right)$ of two consecutive time points. In each window, - build a first-order VAR(1) model for each gene j: $${\widehat{D}}_f^j\left(t_{l+1}\right)={\widehat{\mathbf{D}}}_f\left(t_l\right){\mathit{\alpha }}_j+{ϵ}_t,$$ - Solve the optimization problem using different regularization methods $$\underset{{\mathit{\alpha }}_j}{min}{\displaystyle \frac{1}{2}}{\parallel {\widehat{D}}_f^j\left(t_{l+1}\right)-{\widehat{\mathbf{D}}}_f^j\left(t_l\right){\mathit{\alpha }}_j\parallel }^2+\lambda p\left({\mathit{\alpha }}_j\right),$$ Output a sparse matrix $\mathit{\alpha }$ that represents the inferred connectivity structure at different stages. Step 3: Calculate signed adjacency matrix using partial correlation to identify regulatory relationship In each window, calculate the Spearman rank partial correlation $P_{ij}$ between two genes, and obtain the sign matrix $S_{ij}=\mathrm{sign}\left(P_{ij}\right)$. Construct a signed adjacency matrix $\mathit{A}$ by performing element-wise multiplication of $\mathit{\alpha }$ and $\mathbf{S}$: $A_{ij}={\alpha }_{ij}·S_{ij}$. Reconstruct time-varying regulatory networks based on the matrix $\mathit{A}$.