### Introduction

*Saccharomyces cerevisiae*. By using the network representations, it was found that 2,358 among 2,709 total interactions compose a single large network. Also, biological pathway databases, such as Kyoto Encyclopedia of Genes and Genomes (KEGG) [2] and Reactome [3], provide numerous pathways that are represented by networks with nodes of molecules and directed edges of their actions. In addition, various mathematical properties and models of a network have been developed in graph theory. Several reviews [4-6] have illustrated the mathematical properties and topological characteristics of a network with biological systems.

### Network Resources

*S. cerevisiae*to human. On the other hand, MIPS not only provides mammalian PPIs but also offers the functional catalogs that contain descriptions of protein functions. Unlike BioGRID and MIPS, STRING contains identified and predicted functional interactions of proteins with functional similarity scores (i.e., STRING offers weighted networks). For transcriptional regulatory interactions, transcriptional regulatory element database (TRED) [19] offers GRNs and transcriptional factors for three species; human, mouse, and rat. On the other hand, RegulonDB [20] contains both experimental datasets and computational prediction results of transcriptional regulatory interactions for the

*Escherichia coli*K-12 organism.

### Statistical Reconstruction of the Gene Regulatory Network from Gene Expression Data

### Gaussian graphical model

*G*= (

*V*,

*E*) be an undirected graph with a set V of nodes and a set E of edges and X = (

*X*

^{1},

*X*

^{2}, … ,

*X*

^{p}) = (

*X*

_{1},

*X*

_{2}, …,

*X*

_{n})

^{T}be an

*n*×

*p*dimensional design matrix, where

*X*

^{j}denotes a

*j*-th variable and

*X*denotes an

_{i}*i*-th sample. In the Gaussian graphical model, it is assumed that an observation is from a

*p*-dimensional multivariate normal distribution with mean zero and covariance matrix Σ (i.e.,

*X*~

_{i}*N*(0, Σ) for

*i*=1, 2,…,

*n*). From this normality assumption,

*X*and

^{i}*X*are conditionally independent if (Σ

^{j}^{-1})

*= 0 [30]. With this property, the Gaussian graphical model represents a conditional dependency between two variables into an edge (i.e., (*

_{ij}*i*,

*j*) ∈

*E*if (Σ

^{-1})

*≠ 0). Thus, we can obtain a network structure by estimating the inverse of covariance matrix (Σ*

_{ij}^{-1}), which is called a precision matrix. It is known that the maximum likelihood estimator (MLE) of the precision matrix is an inverse matrix of the sample covariance matrix. However, for high-dimensional data (p >

*n*), the MLE of the precision matrix can not be obtained from the sample covariance matrix, since the sample covariance matrix is singular.

*l*

_{1}-regularized methods have been developed. These methods can be categorized into four types: regression-based [31, 32], penalized likelihood [33-38], empirical Bayes [39], and constrained

*l*

_{1}minimization [40]. We briefly introduce a recently developed example for each type of method. First, the sparse partial correlation estimation (SPACE) method [32] jointly solves p regression problems with

*l*

_{1}norm penalty on partial correlations. SPACE performs well in the detection of hub nodes that have many connections with other nodes. Unlike regression based-methods, the penalized likelihood-based methods directly maximize a likelihood function with positive definite constraints and

*l*

_{1}norm penalty on elements of the precision matrix. This maximization problem, proposed by [33], is more difficult to solve than the regression problem. To efficiently solve this problem, Friedman et al. [38] developed the graphical lasso algorithm that is motivated from [34, 35] and faster than other existing methods. In the empirical Bayes-based methods, Schafer and Strimmer [39] proposed the GeneNet method, based on multiple testing procedure on the partial correlations estimated by the Moore-Penrose pseudo inverse and the bootstrap method. Finally, Cai et al. [40] recently proposed constrained

*l*

_{1}-minimization for inverse matrix estimation (CLIME), which directly minimizes the

*l*

_{1}-norm of the precision matrix with a constraint for relaxation of the precision matrix condition.

### Bayesian network

### Correlation network

### Information theory

*l*

_{1}-norm penalty on the strength coefficients, which is referred to as recursive optimization in [52]. The NARROMI algorithm proposes the linear combination of the MI values and the absolute values of estimated strength coefficients as the final scores to construct the GRN.

### Network-Based Applications

*neighborhood*of a node is a set of nodes connected to the node. Second, the

*distance*between two nodes is defined as a length of the shortest path between them if the path exists; otherwise, it is set to infinity. Finally, an

*adjacency matrix*is a binary and symmetric matrix such that its

*ij*-th element is equal to 1 if there is an edge from a node

*i*to a node

*j*; otherwise, it is 0. In some cases, a weighted adjacency matrix can be used to represent the strengths of edges that usually fall between 0 and 1. With these basic concepts, we introduce three network-based applications: protein function prediction, disease gene prioritization, and genome-wide association study.

### Protein function prediction

^{2}statistics [62] and functional similarities [63, 65], based on the annotated protein information in the neighborhood. These methods choose a function with the highest score as a predicted function for each protein.