Weighted gene co-expression network analysis (WGCNA) is used to detect clusters with highly correlated genes. Measurements of correlation most typically rely on linear relationships. However, a linear relationship does not always model pairwise functional-related dependence between genes. In this paper, we first compared 6 different correlation methods in their ability to capture complex dependence between genes in three different tissues. Next, we compared their gene-pairwise coefficient results and corresponding WGCNA results. Finally, we applied a recently proposed correlation method, Hellinger correlation, as a more sensitive correlation measurement in WGCNA. To test this method, we constructed gene networks containing co-expression gene modules from RNA-seq data of human frontal cortex from Alzheimer's disease patients. To test the generality, we also used a microarray data set from human frontal cortex, single cell RNA-seq data from human prefrontal cortex, RNA-seq data from human temporal cortex, and GTEx data from heart. The Hellinger correlation method captures essentially similar results as other linear correlations in WGCNA, but provides additional new functional relationships as exemplified by uncovering a link between inflammation and mitochondria function. We validated the network constructed with the microarray and single cell sequencing data sets and a RNA-seq dataset of temporal cortex. We observed that this new correlation method enables the detection of non-linear biologically meaningful relationships among genes robustly and provides a complementary new approach to WGCNA. Thus, the application of Hellinger correlation to WGCNA provides a more flexible correlation approach to modelling networks in gene expression analysis that uncovers novel network relationships.