The complex topological characteristics of network-like structural systems, such as lattice structures, cellular metamaterials, and mass transport networks, pose a great challenge for uncertainty qualification (UQ). Various UQ approaches have been developed to quantify parametric uncertainties or high dimensional random quantities distributed in a simply connected space (e.g., line section, rectangular area, etc.), but it is still challenging to consider the topological characteristics of the spatial domain for uncertainty representation and quantification. To resolve this issue, a network distance-based Gaussian random process uncertainty representation approach is proposed. By representing the topological input space as a node-edge network, the network distance is employed to replace the Euclidean distance in characterizing the spatial correlations. Furthermore, a conditional simulation-based sampling approach is proposed for generating realizations from the uncertainty representation model. Network node values are modeled by a multivariate Gaussian distribution, and the network edge values are simulated conditionally on the node values and the known network edge values. The effectiveness of the proposed approach is demonstrated on two engineering case studies: thermal conduction analysis of 3D lattice structures with stochastic properties and characterization of the distortion patterns of additively manufactured cellular structures.