Geometric Interpretation of Correlation Networks Using the Singular Value Decomposition
The nodes of a correlation network correspond to the columns of a numeric data matrix datX. Based on the singular value decomposition (SVD) of datX, we are able to characterize approximately factorizable correlation networks, i.e., adjacency matrices that factor into node-specific contributions. The SVD yields singular vectors that have important practical applications. For example, the first left singular vector (referred to as the eigenvector) explains the maximum amount of variation of the columns of datX. The eigenvector is also known as module eigengene in the context of a gene co-expression network module. Right singular vectors can be used for signal balancing, e.g., to remove batch effects and other technical artifacts. Based on the eigenvector (the first left singular vector), we define a new type of network concept, referred to as eigenvector-based network concept. Eigenvector-based concepts are analogous to approximate conformity-based network concepts but have a major advantage: they often allow for a geometric interpretation based on the angular interpretation of correlations. The underlying structure of correlation networks affects network analysis results. For example, there are geometric reasons why intramodular hub nodes in important modules tend to be important, and why hub nodes in one module cannot be hubs in another distinct module. The hub node significance of a module can be interpreted as angle between a sample trait and the module eigengene. Since the intramodular connectivity kIM i is highly related to the module membership measure kME i , it can be interpreted as angle between x i and the module eigenvector ME. A short dictionary for translating between data mining- and network theory language may facilitate the communication between the two fields. Mouse and brain gene co-expression network applications are used to illustrate the results. This work reviews and extends work with Jun Dong (Horvath and Dong PLoS Comput Biol 4(8):e1000117, 2008).
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