## 1 Introduction  
Systems biology is not only an emerging field, but more importantly, it represents a new approach to biological research [1, 2]. In the past, the structure of gene regulatory networks (GRNs) was inferred from experimental interventions, but such experiments required considerable time and cost. With the rapid development of high-throughput technologies, a large number of research studies have generated a large amount of gene expression data [3, 4], which has made it possible to infer gene regulatory networks from these expression data based on computational methods. In recent years, network inference based on computational methods has become one of the most important goals in the post-genomic era. To this end, the \"Dialogue on Reverse Engineering Evaluation and Methods\" challenge aims to stimulate researchers to develop new and efficient arithmetics [5]. Much progress has been made in inferring GRNs' structure from gene expression data. In the early days, Boolean networks [6, 7] were popular in GRN inferencing, where the states of genes were represented by Boolean variables, and interactions between genes were represented by Boolean functions, which determined the states of genes on top of some other regulatory gene states. At present, information-theoretic approaches are increasingly being used for reconstructing GRNs. Several mutual information (MI)-based methods have been successfully applied to infer GRNs, such as the relevance network (REL), context likelihood of relatedness (CLR), and the ARACNE and minimum redundancy network (MRNET). The REL algorithm [8] calculates MI values between genes and then infers interactions based on the threshold values. The CLR algorithm [9] extends the REL algorithm, which infers interactions based on scores derived from the background distribution of mutual information. For both the REL and CLR algorithms, it is easy to introduce indirect interactions, which can lead to more false edges. In order to eliminate indirect interactions, [PERSON] et al. proposed the ARACNE algorithm [10] based on data-processing inequality, which takes into account indirect interactions in interaction triangles. The MRNET algorithm [11] by [PERSON] is a network inference algorithm using a feature selection strategy, in which an iterative search process is applied to select direct interactions. PCA-CMI [12] measures the dependence between genes through conditional mutual information(CMI), successfully differentiated direct interaction, and indirect association. CMI2 NI [13] calculated the mutual information between two genes when given the third gene by calculating the Kullback-Leibler divergence between the hypothetical distributions at the boundary between the two genes. Although MI can characterize nonlinear dependence, MI's calculation usually requires probability or density estimation, which often requires assumptions. However, it is not accurate because the distribution of gene expression data is uncertain. Recently, a novel and simple statistic of dependency relationships, distance correlation [14], has emerged, which is sensitive to any deviation from independent behaviors, such as nonlinear or non-monotone dependent structures [15]. Better than MI, the distance correlation statistics were calculated very merely without any distribution assumptions. Recently, an approach has been proposed to infer GRNs based on distance correlation. It successfully combined distance correlation with existing CLR algorithms and MRNET algorithms to raise the accuracy of the GRNs [16]. GRNs, or more generally, biochemical networks are sparse, meaning that a gene is regulated by a small number of genes relative to the total number of genes in the network. A range of sparsification-based features have now been proposed to infer GRNs from gene expression data [17; 18]. Currently, we can sparse networks by using graph properties in the network [19; 20]. In the existing studies, most algorithms only consider the connections in the gene expression data.