Communities represent important functional modules in networked systems. A key goal in preserving such communities is understanding their robustness under perturbations. Previous research has studied the impact of node removals and edge removals on the community structure. However, the impact of edge additions on the robustness of the community structure is relatively unknown. Edge additions or false positive edges may simulate measurement errors or external exceptional events that threaten the functionality of networked systems. Here, we study the impact of edge additions on the community structure using Lancichinetti-Fortunato-Radicchi (LFR) benchmark networks. We show that, for a fixed network size, the impact of edge additions is greater on networks with initially weak community structure than on networks with strongly clustered structures. In addition, we find that the perception of the impact is also dependent on the community detection algorithm used to uncover communities. In particular, we found that modularity-based methods such as Leiden and Louvain are less affected than information-theoretic and message passing-based methods such as Infomap and Label Propagation. Our results demonstrate that edge addition can (a) significantly impact the community structure of networks based on their initial conditions, and (b) the perception of the impact is dependent on the community detection algorithm used. We describe limitations, open challenges, and how this methodology can inform the design of resilient networked systems under edge additions.