constructed with n nodes (ROIs), with a network degree of E equal to number of edges (links), and a network density (cost) of D = E/ [(N6(N21) )/2] representing the fraction of present connections to all possible connections. Thresholding. Thresholding the association matrices of different groups at an absolute threshold results in networks with a different number of nodes (and degrees) that might influence the network measures and reduce interpretation of between group results [29]. Two approaches are implemented for thresholding the constructed association matrices based on previous studies [12,14,20]: 1) Thresholding the constructed association matrices at a minimum network density (Dmin) in which all nodes become fully connected in the brain networks of both groups (none of the networks are fragmented); 2) Thresholding the constructed association matrices at a range of network densities for comparing the network topologies across that range. The GAT allows the user to specify the range and the interval between densities. The specified range may include the density points prior to the network fragmentation point but the results should be interpreted carefully. Network analysis. The small-worldness of a complex network, as described in the introduction, has two key metrics: the clustering coefficient C and the characteristic path length L of the network. The clustering coefficient of a node is a measure of the number of edges that exist between its nearest neighbors. The clustering coefficient of a network is thus the average of clustering coefficients across nodes and is a measure of network segregation. The characteristic path length of a network is the average shortest path length between all pairs of nodes in the network and is the most commonly used measure of network integration [27]. To evaluate the topology of the brain network, these parameters must be compared to the corresponding mean values of a benchmark random graph [51,52]. Thus, the small-worldness index of a network is obtained as [C/Crand]/[L/Lrand] where Crand and Lrand are the mean clustering coefficient and the characteristic path length of the m random networks [11]. m is the number of networks generated for normalization of clustering and path length and is specified by the user (default value of 20). In a small-world network, the clustering coefficient is significantly higher than that of random networks (C/Crand ratio greater than 1) while the characteristic path length is comparable to random networks (L/ Lrand ratio close to 1). The benchmark random networks are usually constructed using rewiring algorithms that preserves the topology of the graphs; i.e. random graphs with the same number of nodes, total edges and degree distribution as the network of interest [51,52]. However, recent evidence suggests that correla- tion networks are inherently more clustered and partial correlation networks are inherently less clustered than random networks of the same size and degree [53]. Thus, alternative algorithms were proposed for constructing benchmark random networks that only annihilates intrinsic structure in the empirical data. For structural networks, two types of algorithms are implemented in the GAT for construction of benchmark networks: 1) topology randomiza- tion that generates random networks with the same number of nodes, degree, and degree distribution as the network of interest and 2) correlation matrix randomization that generates covariance matrices that are matched to the distributional properties of the observed covariance matrix [53]. It should be noted that the choice of network largely affects the small- world metrics for networks thresholded at lower densities (density ,0.2) [53]. GAT extracts network measures using the codes developed in the Brain Connectivity Toolbox (BCT) [27]. Several network metrics including measures of network segregation (e.g. clustering, transitivity), integration (e.g. path length, efficiency), centrality (e.g. GAT: Graph Analysis Toolbox for Brain Networks nodal betweenness, edge betweenness), and resilience (e.g. assortativity) are quantified (see [27] for a list of network measures). These metrics are quantified at both the network and regional level. GAT generates the plots of changes in global network measures as a function of network density. It also creates the association and adjacency matrices (thresholded at Dmin) for each group network. Comparing network measures between groups. In order to test the statistical significance of the between-group differences in network topology and regional network measures, a non- parametric permutation test with 1000 (user defined) repetitions is used [12,14,20]. In each repetition, the regional data (or residuals) of each participant are randomly reassigned to one of the two groups so that each randomized group had the same number of subjects as the original groups. Then, an association matrix is obtained for each randomized group. The binary adjacency matrices are then estimated by applying the same thresholding procedure as described above. The network measures are then calculated for all the networks at each density. The differences in network measures between randomized groups (at each network density) are then calculated resulting in a permutation distribution of difference under the hypothesis. The actual between-group difference in network measures is then placed in the corresponding permutation distribution and a one/two-tailed p-value (user defined) is calculated based on its percentile position [20]. For global network measures, GAT generates the plots of between- differences in network measures along with the quantified confidence intervals as a function of network density. GAT also compares the areas under a curve (AUC) for each network measure [20,54]. For this purpose, the curves extracted from thresholding across a range of densities are used. Each of these curves depicts the changes in a specific network measure (for each group) as a function of network density. To test the significance of the between-group differences in AUC of each network measure, the actual between-group difference in AUC for each network measure is placed in the corresponding permutation distribution and p-value is calculated based on its percentile position. By performing AUC analysis, the comparison between network measures is less sensitive to the thresholding process. However, the result still depends on the selected minimum and maximum network densities. While the GAT uses user-defined range of network densities for AUC analysis, we strongly recommend using the calculated Dmin as the minimum density; specifically because the networks become fragmented below Dmin and the comparisons might be affected by group differences in the number of nodes and connections [29]. A maximum network density of below 50% is suggested since structural networks with more than 50% connections are likely non-biological [55]. Although AUC analysis alleviates the sensitivity of the between group comparison to the thresholding process, it is too sensitive to the random structure present at higher network density values and is further insensitive to differences in the shape of the curves rather their mean. Thus, in addition to AUC analysis, GAT utilizes functional data analysis (FDA) which is a statistical method for comparing curves and overcomes these limitations [30,31]. In FDA, each network measure curve is treated as a function (y = f(x)) and the summation of differences in y-values (a graph metric) between groups are calculated at a range of density. A nonparametric permutation test is then applied (as for AUC analysis) to test the significance of the observed summation. The same permutation procedure is used to test the significance of the between-group differences in regional network measures. However, for regional measures, GAT performs three separate comparisons: 1) Comparing regional networks measures for the PLoS ONE | www.plosone.org 4 July 2012 | Volume 7 | Issue 7 | e40709