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Two nodes are linked if the similarity between their abundance profiles across all samples is equal to 1. Once MENs are determined, various network topology indices can be calculated based on the adjacency matrix Table 1. The overall topological indices describe the overall network topology in different views and thus are useful in characterizing various MENs identified under different situations.

The indices for describing individual nodes are useful in assessing their roles in the network. Scale-free, small world, modularity and hierarchy are most common network characteristics of interest [ 8 , 53 , 93 ]. A scale-free network is a network whose connectivity follows a power law, at least asymptotically [ 94 ], that is, only a few nodes in the network have many connections with other nodes while most of nodes have only a few connections with other nodes.

A small-world network is the network in which most nodes are not neighbors of one another, but most nodes can be reached by a few paths typically, less than 6.

Small world network has a small average shortest path GD typically as the logarithm of the number of nodes [ 43 ]. In addition, there is no formal definition for hierarchical topology [ 95 ]. Modularity is a fundamental characteristics of biological networks as well as many engineering systems [ 53 ]. In MENs, a module in the network is a group of OTUs that are highly connected within the group, but very few connections outside the group.

The maximum modularity score is used to separate the graph into multiple dense sub-graphs or modules. The modularity of each network M is estimated using the equation [ 66 ]:. M measures the extension whose nodes have more links within their own modules than expected if linkage is random. Several different algorithms can be used to separate modules, including short random walks, leading eigenvector of the community matrix, simulated annealing approach, and fast greedy modularity optimization [ 56 , 57 ].

The algorithm of short random walks is based on the idea that all random walks tend to stay in the densely connected parts of a graph that was corresponding to the modules [ 56 ]. After calculating a distance between two nodes or between sets of nodes by random walk algorithm, it uses a hierarchical clustering approach to present the structural similarities between all nodes. Thereafter this approach will choose the best partition automatically.

The advantage of this algorithm is efficient and fast computation. Once the network modularity value M was explicitly defined, theoretically the module structure can be determined by maximizing M values over all possible divisions of network.

However, exhaustive maximization over all divisions is computational intractable [ 57 ]. The algorithm of leading eigenvector is one of several approximate optimization methods have been proven effectively obtained higher M values with high speed. It simplified the maximization process in terms of a modularity matrix B nxn that can be obtained by the adjacent matrix A nxn subtracting an expected edges matrix P nxn from a null model.

Then the network can be split into two groups by finding the leading eigenvector that was corresponding to the largest positive eigenvalue of modularity matrix. This splitting process can be looped until any further divisions will not increase the M value [ 57 ].

This method shows more accurate separations than other algorithms in several well-studied social networks [ 57 ]. The algorithm of simulated annealing approach usually produces the best separation of the modules by direct maximization of M [ 58 ]. It carries out the exhaustive search on network structures to merge and split priori -modules and move individual nodes from one module to another.

Although this is a time-consuming process, it is expected to obtain clear module separations with a higher M. The algorithm of fast greedy modularity optimization is to isolate modules via directly optimizing the M score [ 66 , 97 ]. It starts with treating each node as the unique member of one module, and then repeatedly combines two modules if they generate the largest increase in modularity M. This algorithm has advantages with fast speed, accurate separations and ability to handle huge networks [ 66 , 97 ].

After all modules are separated, each node can be assigned a role based on its topological properties [ 59 ], and the role of node i is characterized by its within-module connectivity z i and among-module connectivity P i as follows. The within-module connectivity, z i , describes how well node i is connected to other nodes in the same module, and the participation coefficient, P i , reflects what degree that node i connects to different modules.

P i is also referred as the among-module connectivity [ 98 ]. The topological roles of individual nodes can be assigned by their position in the z -parameter space. Originally, Guimera et al. Olesen et al. From ecological perspective, peripheral nodes represent specialists whereas the other three are generalists.

One of the grand challenges in dealing with high throughput metagenomics data is the high dimensionality. Various statistical approaches are used to reduce dimensions and extract major features, including principal component analysis PCA , detrended correspondence analysis DCA , and singular value decomposition SVD.

SVD is an orthogonal linear transformation of data e. Based on SVD analysis, the Eigengene is a linear combination of genes and eigenvalues.

In the diagonalized data, each eigengene is just expressed in the corresponding eigen arrays. Langfelder and Horvath [ 61 ] proposed eigengene network analysis to summarize the gene expression data from each module as a centroid.

Eigengene network analysis is powerful to reveal higher order organization among gene co-expression modules [ 33 , 61 , 62 ]. Here, we have adopt this method to analyze modules in MENs. Suppose there are n b OTUs in the b -th module. In SVD analysis, X b can be decomposed as follows:. Assuming that the singular values are arranged in decreasing order, the first column of V b is referred as the Module Eigen-gene, E b , for the b -th module.

The relative abundance profile of the OTUs within a module is represented by the eigen-gene. Module eigen-gene provides the best summary of variation in relative abundance of OTUs within a module, but it is a centroid of a module rather than a real OTU. In practice, it is always important to understand how close it is between a given actual OTU and its eigen-gene. The correlation of the eigen-gene in module b to the i -th actual OTU across all experimental samples is defined as.

Since only a single data point is available for each network parameter, we are not able to perform standard statistical analyses to assess statistical significances. Similar to the concept of hypothesis testing, the null model is generated to assess the performance of the alternative model. Thus, the random networks are generated to compare different complex networks using the Maslov-Sneppen procedure [ 68 ].

The Maslov-Sneppen method keeps the numbers of nodes and links unchanged but rewires the positions of all links in the MENs so that the sizes of networks are the same and the random rewired networks are comparable with original ones. This method has been typically used for ecological network analyses [ 4 ]. For each network identified, a total of randomly rewired networks are usually generated by the Maslov-Sneppen procedure [ 68 ] and all network indices are calculated individually for each randomized network.

Then the average and standard deviation for each index of all random networks are obtained. The statistical Z -test is able to test the differences of the indices between the MEN and random networks. Meanwhile, for the comparisons between the network indices under different conditions, the Student t -test can be employed by the standard deviations derived from corresponding random networks. In gene expression network analyses, the gene significance GS i,h is the correlation between the expression profile of the i -th gene and the h -th sample trait, T h [ 33 ].

The higher GS i,h , the more biologically significant gene i is related to the sample trait h. Similarly, in this study, the trait-based OTU significance is defined as:.

Since the measurement units for different traits vary, all trait data should be standardized prior to statistical analysis. To discern the relationships between molecular ecological networks and soil properties, Mantel tests can be performed [ ]. The relationships between the MENs and environmental variables were determined as follows: First, the significances of variables are calculated with the above equation Eq 13 and the OTU significance matrix is generated.

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