Fig. 1

Overview of SINUM (SIngle-cell Network Using Mutual information) method. A Generation of scatter diagrams for every two genes from the gene expression matrix (GEM), where the x- and y-axes are the expression values of every two genes within the n cells. Each point denotes a cell. B The statistical model of SINUM for estimating the association between genes X and Y. First, each scatter diagram containing n cells was split into \(\lfloor\sqrt n + \frac{1}{2}\rfloor\) grids. Next, SINUM produces two boxes \(G_{X}^{\left( c \right)}\) (light blue) and \(G_{Y}^{\left( c \right)}\) (medium blue) close to cell c to represent its neighborhoods of expression values for genes X and Y, respectively; thus, the intersection region can directly produce the third box \(G_{XY}^{\left( c \right)}\) (dark blue). The entropies \(H\left( {G_{X} } \right)^{\left( c \right)}\), \(H\left( {G_{Y} } \right)^{\left( c \right)}\), and \(H\left( {G_{X} ,G_{Y} } \right)^{\left( c \right)}\) of the boxes \(G_{X}^{\left( c \right)}\), \(G_{Y}^{\left( c \right)}\), and \(G_{XY}^{\left( c \right)}\), respectively, were then evaluated for calculating mutual information, \(I\left( {G_{X} ;G_{Y} } \right)^{\left( c \right)}\). The mutual information is used to evaluate whether any given two genes X and Y, are a dependent or independent gene pair in cell c among all cells. If the value of \(I\left( {G_{X} ;G_{Y} } \right)^{\left( c \right)}\) is larger than the threshold, it suggests that genes X and Y are dependent on each other in cell c and will be represented by an edge in a network. C Construction of n single-cell networks (SCNs) for n cells. For m genes, a total of \(m\left( {m - 1} \right)/2\) scatter diagrams were generated to measure all possible associations between two genes. In each SCN, the red solid line represents that there’s an edge between two genes for a specific cell inferred by our SINUM method; otherwise, there’s no edge. D Generation of degree matrix (DM) by counting the number of edges connected to every gene in each SCN. Note that the size of DM is the same as GEM with m rows and n columns