Due to the high directionality and short wavelength of laser transmission in space, satellite laser communication can achieve high speed, wide bandwidth, high precision, and high security with no electromagnetic spectrum constraints. It has become a promising direction to construct the laser space network. Recently, relative products, on-orbit demonstrations, and researches have been conducted on laser space networks. Considering this network as a graph, this paper presents a comprehensive survey from nodes, edges, to architectures of this graph, corresponding to laser communication terminals, laser space links, and laser space network architectures. For each content, current achievements are introduced and future trends are analyzed accordingly. Through this survey paper, we aim to present the prospect to develop laser space networks in the next space generation and provide potential research directions to interested researchers or engineers.