To cover the basic principle behind the Gaussian graphic model, let G = (V, E) be an undirected graph with a set V of nodes and a set E of edges and X = (X¹, X², … , X^p) = (X₁, X₂, …, X_n)^T be an n × p dimensional design matrix, where X^j denotes a j-th variable and X_i denotes an 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, Xⁱ and X^j are conditionally independent if (Σ^-1)_ij = 0 [30]. With this property, the Gaussian graphical model represents a conditional dependency between two variables into an edge (i.e., (i, j) ∈ E if (Σ^-1)_ij ≠ 0). Thus, we can obtain a network structure by estimating the inverse of covariance matrix (Σ^-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.

To resolve this problem and obtain the sparse solution, l₁-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 l₁ 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 l₁ 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 l₁ 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 l₁-minimization for inverse matrix estimation (CLIME), which directly minimizes the l₁-norm of the precision matrix with a constraint for relaxation of the precision matrix condition.

A Bayesian network is a probabilistic framework for representing a directed acyclic graph (DAG) structure, while Gaussian graphical models consider undirected graph structures. With the structure of DAG, the joint probabilities can be explicitly calculated from a simple conditional distribution. To obtain the network structure, Bayesian network methods generally select the graph structure by following three steps. Step 1-a candidate DAG structure is chosen among all possible DAG structures. Step 2-the posterior probability of the candidate DAG structure, given the gene expression data, is calculated. Step 3-repeat steps 1 and 2 until the highest posterior probability is obtained. Thus, searching candidate DAG structures becomes a critical part of the Bayesian network methods when the number of nodes increases. To avoid this problem, many heuristics are applied to improve efficiency [41-43]. For instance, B-Course [41], which is a web-based application, uses a combination of stochastic and greedy search, and BNT [42] and Werhli's Bayesian network [43] adopt Markov Chain Monte Carlo (MCMC) methods to find the best network structure for given gene expression data.

Correlation networks represent pairwise correlations between two nodes into edges. Although the construction of correlation networks can be straightforward to obtain from Pearson's correlations, it leads to a problem-that all the edges in the network are connected, since these correlations are generally non-zero. Thus, it is required that meaningless relationships in the correlation network be removed to focus on important edges that highly correlate nodes. To remedy this problem, hard thresholds [44] or soft thresholds [45] are applied to Pearson's correlations. By using threshold cutoffs, correlation network methods can capture biologically meaningful relationships. Moreover, this representation, based on correlations, is useful not only to identify modules of genes but also to interpret regulation interactions. "WGCNA" [46], an R implemented package, is an example that makes use of correlation networks. WGCNA provides the construction of unweighted (or weighted) networks with hard and soft threshold schemes [44, 45]. In addition, WGCNA offers various functions to analyze the network, including module detection, calculation of topological properties, and visualization.

Information theory-based methods construct the GRN, based on information theoretic scores, such as the mutual information (MI), to measure dependencies between variables [47-50]. Unlike correlations, the MI does not assume the linearity and the continuity of variables. Thus, information theory-based methods generally outperform other methods that are based on correlation coefficients when the true network structure contains non-linearity dependencies. We first review two popular methods that reconstruct GRNs based on MI values and then introduce recently developed methods.

The relevance network method [47] uses the MI values to determine the edges in the GRN. To select significant edges, the relevance network proposes a threshold rule based on the distribution of the permuted MI values. The maximum value of the average of the permuted MI values is considered the threshold value. On the other hand, ARACNE [48, 49] additionally takes into account data processing inequality (DPI), in which the MI value of an indirect interaction is less than or equal to each MI value of the direct interactions for all triplets of nodes that only have two direct interactions. This DPI enforces that ARACNE [48, 49] discards indirect interactions and some direct interactions with small MI values.

To improve the accuracy of the information theory-based methods, the context likelihood of relatedness (CLR) algorithm [50] and the gene expression and transcription factor activity-based relevance network (GTRNetwork) algorithm [51] have been developed. These methods focus on the transcriptional regulatory interactions between known TFs and their target genes. Instead of using the MI values directly, the CLR algorithm proposes a likelihood value based on z-scores derived from the empirical distribution of the MI values in order to adjust random noises in the MI values. In addition to the CLR algorithm, the GTRNetwork algorithm additionally considers transcription factor activities (TFAs) between TFs and target genes as a hidden layer. The GTRNetwork algorithm first estimates changes of TFAs with known TF-gene networks and then identifies transcriptional regulatory interactions between the estimated TFAs and genes with the CLR algorithm. More recently, Zhang et al. [52] described the transcription procedure with TFAs by using the ordinary differential equation. Unlike the GTRNetwork algorithm, the NARROMI algorithm considers the gene expression data for candidate TFs as the TFAs and solves the minimization of the absolute errors between inferred and observed expressions with l₁-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.

With the results of sequencing genomes, the efforts to predict protein functions have been focused on the functionalities of genomic annotations. In the initial stage, the prediction of protein function begins with the sequence homologies to annotated proteins [54-57]. These methods have been successful, but 70% of proteins still remain unannotated [58]. Accordingly, various types of methods have been developed to characterize unannotated proteins. In this review, we introduce four approaches in protein function prediction, based on direct interactions in the network: neighborhood-based, graph-based, Bayesian, and Kernel-based approaches. The methods are summarized in Table 3 [10, 11, 13-16, 59-83].

In neighborhood-based approach, the proposed methods [1, 62, 63, 65] commonly consider the number of edges connected to annotated proteins in a neighborhood. The neighborhood counting method [1] only takes into account the frequencies of annotated proteins in the neighborhood and chooses the top three functions, with the calculated frequencies ranked in descending order for each protein. Other methods also have derived their own scores, such as χ² 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.

The graph-based approach is similar to the neighborhood-based approach, but the graph-based approach focuses more on how to label the unannotated proteins with graph theoretical properties, such as the distance and the adjacency matrix. The label assignment models have been proposed with the adjacency matrix [66] and the weighted adjacency matrix [67], respectively. Since these assignment problems are computationally intractable, the heuristic methods, such as the simple threshold rule or simulated annealing [84], have been applied. To avoid these computational problems, several propagation-flavored methods have been developed. First, the label propagation methods [69, 70] obtain the optimal assignments and the optimal combination of the weight matrices that reflects different types of networks. Second, the functional flow method [68] iteratively spreads the flow from the annotated protein to the unannotated proteins by connected edges. The functional flow score, defined as an amount of the flow, is the criterion of the prediction. Third, the sequential linear neighborhood propagation method [71] sequentially updates unlabeled proteins according to their shortest path distance to the set of labeled proteins. Finally, the neighbor relativity coefficient (NRC) method [72] derives the NRC score by integrating various graph topological properties, such as the shortest path distance, path connectivity, and common neighbors.

The Bayesian approach takes into account the posterior probabilities of binary label random variables to obtain the prediction from the observed network and annotated proteins. Markov random field (MRF)-based [85] methods [74, 75] have been proposed and modified to Bayesian MRF [77] recently. To predict protein functions, these methods commonly derive the marginal posterior probability of the binary label variable given other variables and then estimate the posterior probability by Gibbs sampling. In addition to MRF-based models, other probabilistic models [78-80] have been developed under hierarchical structures from gene ontology [64]-which provides gene product annotation data that are characterized into three categories: biological processes, cellular components, and molecular functions-with some models, such as the hierarchical binomial neighborhood model [79]. In particular, Jiang et al. [80] considered the auto-probit model with a weighted network information from STRING [16], and their auto-probit model also reflects the uncertainty of the annotation [64].

The Kernel-based approach considers the protein function prediction problem as a classification problem. To reflect network information into the classification state, the network information is converted into a kernel matrix. Lanckriet et al. [81] proposed the kernel-based support vector machine (SVM) method, which incorporates heterogeneous types of data, such as amino acid sequence, gene expression data, and PPI network data, by converting these data into kernel matrices. The SVM method can be reformulated as semi-definite programming (SDP) [86] with kernels. Although the SVM method performs well, this method becomes slow when the dimension increases, caused by the computational complexity of the SDP. To remedy this problem, Lee et al. [82] proposed the kernel-based logistic regression (KLR) method by combing MRF-based methods [74, 75] with the diffusion kernel [87]. The KLR can contain multiple functions and various types of datasets at once. It has been shown that the KLR method is faster than the SVM and is comparable to the SVM in prediction accuracy [82]. Recently, Wang et al. [83] proposed the function-function correlated multi-label method, which treats all function categories in the prediction at once, while other methods only consider one function at a time, except for KLR.

Disease-gene association studies have focused on identifying relationships between disease phenotypes and genes to reveal and understand human disease mechanisms. Although traditional approaches, including linkage analysis and association studies, have been successful, the specified genomic intervals often contain tens or even hundreds of genes. It may not be possible by experiments to identify the correct causative genes of all the genes that lie on the specified intervals. To reduce experimental costs and efforts, the prioritization of candidate genes becomes one of the major tasks in disease-gene association studies.

Taking into consideration that genes corresponding to similar disease phenotypes tend to be neighbors in either a PPI network [88, 89] or a pathway [90], several network-based disease gene prioritization methods have been proposed recently. The network-based disease gene prioritization methods [91-94] define different similarity scores between the disease and genes, based on either functional annotations or PPI networks, to rank candidate genes. The random walk method [91] considers the random walk on graphs and uses a diffusion kernel matrix [87] derived from a PPI network as the steady-state transition probability matrix. The random walk method then defines the similarity score as the sum of elements of the diffusion kernel corresponding to known disease genes for each candidate gene. CIPHER [92] adopts a regression model for similarities between phenotypes and considers the Gaussian kernel for closeness between genes. The Pearson correlation coefficient between the similarity of phenotypes and the closeness between genes is used as a similarity score. On the other hand, PRINCE [93] and MINProp [94] are based on the iterative label propagation algorithm [95]. PRINCE defines the prioritization function, which combines functional similarities from the network information with prior information for disease phenotypes as the similarity score. The prioritization function is obtained by iteratively applying a label propagation algorithm. MINProp first defines two heterogeneous networks, such as a gene network and a disease network, and then additionally defines bipartite networks between two heterogeneous networks from known disease-gene associations. After initializing the labels of the disease genes, MINProp iteratively propagates label information through three networks until convergence occurs. Finally, the converged label scores are used as the similarity scores. From the comparison study in [93, 94], PRINCE and MINProp outperform the random walk and CIPHER methods, respectively. There is no comparison study between PRINCE and MINProp, but MINProp is more general than PRINCE, since MINProp can be applied to three or more heterogeneous networks. We report these disease gene prioritization methods with their data resources in Table 4 [12, 13, 15-18, 91-94, 96-99].

The GWAS measures DNA sequence variations in the human genome to identify associations between genetic variants and diseases (or phenotypes). To measure genetic variations, the single- nucleotide polymorphism (SNP), which represents a single base-pair change in the DNA sequence, is generally used as a marker of a genomic region in the GWAS. Generally, the GWAS conducts a comparison of the SNPs between case and control groups (i.e., disease and non-disease groups) by statistical methods, such as ANOVA and logistic regression, to identify significant SNPs related to the disease. Genetic risk factors revealed by a GWAS can be used as preventive markers or for therapeutic targets in curing the disease. There have been more than 11,000 SNPs discovered as candidate bio-markers in common diseases [100]. The large number of SNPs detectable in the human genome can, however, lead to multiple testing problems. To control the false positive errors (i.e., type I errors in the context of statistical testing procedures), the Bonferroni correction and false discovery rate [101] methods are commonly adopted. Although these multiple testing rules have been successful in the identification of significant single SNPs, these test procedures often fail to detect genomic regions that are weakly associated with the disease and still ignore the combined effects caused by the interactions between genes.

The network-based GWAS methods [102-104], summarized in Table 5 [12, 13, 17, 102-114], take into account both interactions between genes or proteins with association information available from a GWAS. The HumanNet method [115] combines functional gene networks derived from multiple network sources, such as the PPI network, and mRNA co-expression with the log odds ratio from GWAS data to prioritize disease genes. By combining functional gene networks and the information from GWAS data, HumanNet has higher power to detect disease-related genes. Unlike HumanNet, NIMMI [116] and dmGWAS [117] focus on identifying groups of genes associated with diseases, based on GWAS data and PPI network data. NIMMI constructs a PPI network with weights of genes by using a modified Google PageRank algorithm [118]. The weights of genes are then combined with the gene-based association p-values calculated from GWAS data. The subnetworks of genes are identified by the DAVID method [119, 120] with the network structure and weights of genes. On the other hand, dmGWAS only considers gene-based association p-values as the node weights and proposes a modified version of the dense module searching method [121] to prioritize candidate groups of genes.