The equilibrium properties of block copolymer star networks (BCS) are studied via computer simulations. We employ both molecular dynamics and multiparticle collisional dynamics simulations to investigate the self-organization of BCS with f = 9 functionalized arms close to their overlap concentrations under conditions of different fractions of functionalization and varying attraction strength. We find three distinct macroscopic self-organized states depending on fraction of attractive end-monomers and the strength of the attraction. At weak attractions, ergodic, diffusive liquids result, with short-lived bonds between the stars. As the attraction strength grows, the whole system forms a percolating cluster, while at the same time the individual molecules are diffusive. Finally, arrested gels emerge when the attractions become strong. The conformation of the BCS in these solutions is found to be strongly affected by the concentration, with the stars assuming typically spherical, open configurations in seeking to maximize inter-star associations as opposed to the inter-star collapse that results at infinite dilution, giving rise to strongly aspherical shapes and reduced sizes.