Due to recent technology progress based on big-data processing, many applications present irregular or unpredictable communication patterns among compute nodes in high-performance computing (HPC) systems. Traditional communication infrastructures, e.g., torus or fat-tree interconnection networks, may not handle well their matchmaking problems with these newly emerging applications. There are already many communication-efficient application mapping algorithms for these typical non-random network topologies, which use nearby compute nodes to reduce the network distances. However, for the above unpredictable communication patterns, it is difficult to efficiently map their applications onto the non-random network topologies. In this context, we recommend using random network topologies as the communication infrastructures, which have drawn increasing attention for the use of HPC interconnects due to their small diameter and average shortest path length (ASPL). We make a comparative study to analyze the impact of application mapping performance on non-random and random network topologies. We propose using topology embedding metrics, i.e., diameter and ASPL, and list several diameter/ASPL-based application mapping algorithms to compare their job scheduling performances, assuming that the communication pattern of each application is unpredictable to the computing system. Evaluation with a large compound application workload shows that, when compared to non-random topologies, random topologies can reduce the average turnaround time up to 39.3% by a random connected mapping method and up to 72.1% by a diameter/ASPL-based mapping algorithm. Moreover, when compared to the baseline topology mapping method, the proposed diameter/ASPL-based topology mapping strategy can reduce up to 48.0% makespan and up to 78.1% average turnaround time, and improve up to 1.9x system utilization over random topologies.