Networks of excitatory and inhibitory (EI) neurons form a canonical circuit in the brain. Classical theoretical analyses of dynamics in EI networks have revealed key principles such as EI balance or paradoxical responses to external inputs. These seminal results assume that synaptic strengths depend on the type of neurons they connect but are otherwise statistically independent. However, recent synaptic physiology datasets have uncovered connectivity patterns that deviate significantly from independent connection models. Simultaneously, studies of task-trained recurrent networks have emphasized the role of connectivity structure in implementing neural computations. Despite these findings, integrating detailed connectivity structures into mean-field theories of EI networks remains a substantial challenge. In this talk, I will outline a theoretical approach to understanding dynamics in structured EI networks by employing a low-rank approximation based on an analytical computation of the dominant eigenvalues of the full connectivity matrix. I will illustrate this approach by investigating the effects of pair-wise connectivity motifs on linear dynamics in EI networks. Specifically, I will present recent results demonstrating that an over-representation of chain motifs induces a strong positive eigenvalue in inhibition-dominated networks, generating a potential instability that challenges classical EI balance criteria. Furthermore, by examining the effects of external input, we found that chain motifs can, on their own, induce paradoxical responses, wherein an increased input to inhibitory neurons leads to a counterintuitive decrease in their activity through recurrent feedback mechanisms. Altogether, our theoretical approach opens new avenues for relating recorded connectivity structures with dynamics and computations in biological networks.