The brain's functional network exhibits many features facilitating functional specialization, integration and robustness to attack. Using graph theory to characterize brain networks, studies demonstrate their small-world, modular, and "rich-club" properties, with deviations reported in many common neuropathological conditions. Here we estimate the heritability of five widely used graph theoretical metrics (Mean Clustering Coefficient (γ), Modularity (Q), Rich Club Coefficient (ϕnorm), Global Efficiency (λ), Small Worldness (σ)) over a range of connection densities (k=5-25%) in a large cohort of twins (N=592, 84 MZ and 89 DZ twin pairs, 246 single twins, age 23±2.5). We also considered the effects of global signal regression (GSR). We found the graph metrics were moderately influenced by genetic factors h(2)(γ=47-59%, Q=38-59%, ϕnorm=0-29%, λ =52-64%, σ=51-59%) at lower connection densities (≤15%), and when global signal regression was implemented heritability estimates decreased substantially h(2)(γ=0-26%, Q=0-28%, ϕnorm=0%, λ =23-30%, σ=0-27%). Distinct network features were phenotypically correlated (|r|=0.15-0.81) and γ, Q and λ were found to be influenced by overlapping genetic factors. Our findings suggest that these metrics may be potential endophenotypes for psychiatric disease and suitable for genetic association studies, but that genetic effects must be interpreted with respect to methodological choices.