One of the key questions in neuroscience is how the connectivity structure of cortical networks determines the collective dynamics of neural activity. Two complementary approaches have been developed to address this question: (i) specifying connectivity in terms of local statistics of excitatory-inhibitory motifs [Trousdale et al. 2012, Aljadeff et al. 2015]; (ii) specifying connectivity through a global low-rank structure that determines the low-dimensional dynamics [Mastrogiuseppe and Ostojic 2018, Beiran et al. 2021, Dubreuil et al. 2020]. It is however currently unclear how local connectivity statistics are related to the global structure and shape the low-dimensional activity. To bridge this gap, here we map local EI statistics onto global statistics of low-rank connectivity and examine the emerging dynamics. We consider a randomly connected, block-like network composed of excitatory and inhibitory subpopulations. Connections in each block are specified by cell-type-dependent statistics and consist of independent and reciprocal parts. We first determine the dominant eigenvalues and eigenvectors of the resulting connectivity matrix, and show that the statistics of their entries universally obey a mixture of Gaussian distribution. We then exploit this result to approximate random EI networks by Gaussian-mixture low-rank networks [Beiran et al. 2021]. Comparing the parameters of local connectivity to the emerging low-rank structure, we show that mean cell-type connectivity determines the dominant low-rank structure, but the reciprocal motifs further modify it by modulating the dominant eigenvalue. Comparing the dynamics in the original EI network and their low-rank approximations, we find that the mean and variance of population activity closely match. In both cases, reciprocal motifs enhance positive feedback and induce bistable dynamics. Altogether, our analytical mapping of the local EI statistics to low-rank description provides an intuitive picture of how local connectivity statistics determine global low-dimensional dynamics and resulting computations.