Expression quantitative trait locus (eQTL) analysis associates genotype with gene expression, but most eQTL studies only include cis-acting variants and generally examine a single tissue. We used data from 13 tissues obtained by the Genotype-Tissue Expression (GTEx) project v6.0 and, in each tissue, identified both cis- and trans-eQTLs. For each tissue, we represented significant associations between single nucleotide polymorphisms (SNPs) and genes as edges in a bipartite network. These networks are organized into dense, highly modular communities often representing coherent biological processes. Global network hubs are enriched in distal gene regulatory regions such as enhancers, but are devoid of disease-associated SNPs from genome wide association studies. In contrast, local, community-specific network hubs (core SNPs) are preferentially located in regulatory regions such as promoters and enhancers and highly enriched for trait and disease associations. These results provide help explain how many weak-effect SNPs might together influence cellular function and phenotype.