Individual variation in structural brain network topology has been associated with heritable behavioral phenotypes such as intelligence and schizophrenia, making it a candidate endophenotype. However, little is known about the genetic influences on individual variation in structural brain network topology. Moreover, the extent to which structural brain network topology overlaps with heritability for integrity and volume of white matter remains unknown. In this study, structural network topology was examined using diffusion tensor imaging at 3T. Binary connections between 82 structurally defined brain regions per subject were traced, allowing for estimation of individual topological network properties. Heritability of normalized characteristic path length (λ), normalized clustering coefficient (γ), microstructural integrity (FA), and volume of the white matter were estimated using a twin design, including 156 adult twins from the newly acquired U-TWIN cohort. Both γ and λ were estimated to be under substantial genetic influence. The heritability of γ was estimated to be 68%, the heritability estimate for λ was estimated to be 57%. Genetic influences on network measures were found to be partly overlapping with volumetric and microstructural properties of white matter, but the largest component of genetic variance was unique to both network traits. Normalized clustering coefficient and normalized characteristic path length are substantially heritable, and influenced by independent genetic factors that are largely unique to network measures, but partly also implicated in white matter directionality and volume. Thus, network measures provide information about genetic influence on brain structure, independent of global white matter characteristics such as volume and microstructural directionality.