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Robustness Analysis of the Mouse Gene Co-expression Networks

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Text S2: Robustness Analysis of the MouseGene Co-expression NetworksJun Dong and Steve Horvath∗Dept. of Human Genetics, David Geffen School of Medicine, UCLADept. of Biostatistics, School of Public Health, UCLA∗Correspondence: [email protected] is a supplement of the article “Geometric Interpretation of Gene Co-Expression Network Anal-ysis”. Here we illustrate our theoretical results using gene expression data from the mouse application.In particular, we study the robustness of our theoretical findings with regard to alternative methods ofconstructing a network. We describe the results for weighted co-expression networks constructed usingdifferent soft-thresholds β ≥ 1 inaij= |cor(xi, xj)|β.Further, we report the analogous findings for unweighted networks constructed on the basis ofaij= Ind(|cor(xi, xj)| ≥ τ),where τ is the ‘hard’ threshold parameter, and Ind(·) is the indicator function taking value of 1 if thecondition is true, and 0 otherwise. We provide empirical evidence that co-expression modules tend tohave high eigengene factorizability and that the maximum conformity assumption is satisfied for lowpowers of β. Our robustness analysis shows that many of our theoretical results apply even if ourunderlying assumptions are not satisfied.1 Mouse Gene Co-expression Network ApplicationIn this supplement, we illustrate our theoretical derivations using an F2 intercross between two mousestrains C3H/HeJ and C57BL/6J. Liver gene expression data from 135 female mice were used to construct aweighted network. The biological significance of the network and its 12 modules is described in (Ghazalpouret al., 2006). In this supplement and in Figure 9 of the main article, we focus on the relationships betweenthe network concepts and a gene significance measure based on body weight. We find that many of ourtheoretical results hold approximately even if the expression factorizability is low and when an unweightednetwork is used.We have constructed weighted networks with β = 1, 2, 3, 4, 5 and 6, and unweighted networks withτ = 0.65 and 0.5. For the unweighted networks, we use the eigengene-based network concepts of weightednetworks with β = 1 for demonstration purposes.12 Robustness of Module DefinitionOur module definition was based on the topological overlap measure (TOM) in conjunction with averagelinkage hierarchical clustering (refer to the Methods Supplement). In this application, we used TOM of anweighted network with power β = 6. To facilitate a comparison, we used this network module assignmentfor the other weighted and unweighted network analysis as well. In each of the figure below, the upperpanel is the dendrogram of the average linkage hierarchical clustering method using the specific networkconstruction parameter, and the lower panel shows genes colored by their module membership. As thefigures show, our module definition is quite robust with regard to the choice of network constructionmethods.A B C0.50 0.60 0.70 0.80Network Dendrogram, β = 1Colored by module membership0.5 0.6 0.7 0.8 0.9Network Dendrogram, β = 2Colored by module membership0.4 0.6 0.8 1.0Network Dendrogram, β = 3Colored by module membershipD E F0.4 0.6 0.8 1.0Network Dendrogram, β = 4Colored by module membership0.3 0.5 0.7 0.9Network Dendrogram, β = 5Colored by module membership0.3 0.5 0.7 0.9Network Dendrogram, β = 6Colored by module membershipG H0.0 0.4 0.8Network Dendrogram, τ = 0.65Colored by module membership0.0 0.4 0.8Network Dendrogram, τ = 0.5Colored by module membershipFigure 1: Robustness of module definition. In each figure, genes are colored by their module membership.23 Summary of Robustness AnalysisTo make this supplement self-contained, we repeat the following summary table from our main article. Inthe rest of this supplement, we provide the details on how we arrived at the R2values of this table.Table 1: Robustness Analysis of the Mouse Co-expression Network. The table reports how the relationbetween network concepts changes as function of different soft threshold parameters β or hard thresholdsused in the network construction. For each relationship and each network construction method, the tableentry reports the squared correlation R2across the proper modules. For within module comparisons thetable reports median R2values.Squared Correlation R2Weighted Networks Unweighted NetAcross Modules. Soft Threshold β Hard Threshold τRelation 1 2 3 4 5 6 0.65 0.5Centralization ≈ CentralizationE0.69 0.74 0.90 0.95 0.94 0.92 0.007 0.66Heterogeneity ≈ HeterogeneityE0.54 0.59 0.71 0.82 0.88 0.86 0.30 0.33ClusterCoefi≈ ClusterCoefE0.94 0.84 0.70 0.59 0.50 0.44 0.09 0.33Modul eSignif ≈ ModuleSignifE0.96 0.96 0.96 0.97 0.98 0.99 0.96 0.96HubGeneSignif ≈ HubGeneSignifE0.98 0.98 0.98 0.99 1.0 1.0 0.88 0.91EigengeneSignif ≈ HubGeneSignif 0.98 0.98 0.98 0.99 1.0 1.0 0.89 0.92ClusterCoefi≈¡1 + (Heterogeneity)2¢2× Density 0.89 0.78 0.70 0.62 0.54 0.48 0.08 0.31Modul eSignif ≈√Density × HubGeneSignif 0.99 0.99 0.99 0.99 0.99 0.99 0.90 0.96Centralization ≈√Density(1 −√Density) 0.52 0.21 0.43 0.73 0.82 0.84 0.60 0.82kmaxn−1≈√Density 0.95 0.97 0.97 0.98 0.98 0.98 0.93 0.80Ki≈ ae,i(median R2) 1.0 0.99 0.98 0.96 0.95 0.94 0.74 0.86Overall, we find that our theoretical results are highly robust in weighted networks. The relationshipbetween the clustering coefficient and its eigengene-based analog is diminished (down to 0.44) for β > 3.The relationship between heterogeneity and its eigengene-based analog is diminished (down to 0.54 whenβ is low (β < 3)). The relation Centralization(q)≈pDensity(q)(1 −pDensity(q)) has a relatively lowR2value (down to 0.21) for low values of β ≤ 3 but the other relationships among network concepts arehighly robust with respect to β. For unweighted networks, the R2values tend to be lower and severalrelationships show a marked dependency on the hard threshold τ (Table 1).Our robustness analysis shows that many of our theoretical results apply even if our underlying assump-3tions are not satisfied. We find that the correspondence between network concepts and their eigengene-based analogs is often better in weighted networks than in unweighted networks. Further, we find thatresults in weighted networks tend to be more robust than those in unweighted networks with regard tochanging the network construction thresholds β and τ , respectively. Thus, weighted co-expression networksare preferable


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