A range of work has been done analyzing the effect of specific pruning rules on the dynamics of recurrent neural networks (RNNs) with a given connectivity structure. However, a more comprehensive study that explains the effect of different, biologically plausible pruning rules across a variety of structures has yet to be done. Specifically, previous work has investigated uniform random sparsification on RNNs with low rank connectivity matrices [1], and utilized theoretical results from graph sparsification to propose a pruning rule shown to be optimal for preserving the dynamics of symmetric and diagonally dominant connectivity matrices [3]. Our work seeks to unify some of these ideas, presenting a framework to understand the relationship between the rank of the connectivity matrix and the effects of different probabilistic pruning rules on the network dynamics. We evaluate random pruning under 2 different rules: 1) the strengthen or prune method, and 2) the regular prune method. In the “strengthen or prune” method, connections that are not pruned are strengthened, and in the “regular prune” method, unpruned connections are left unaltered. The main results of our analysis include: i) strengthen or prune rules preserve full low dimensional dynamics not just the dimensionality for low rank connectivity matrices and ii) for high rank connectivity matrices, strengthen or prune rules perform worse than regular pruning due to the stronger pruning-induced, high-dimensional dynamics. Our work thus explains the relationship between connectivity structure and dynamics induced by pruning in RNN’s, which in turn provides insights into dynamics of sparse biological neuronal networks.