  Social networks affect the diffusion of information, and thus have the
potential to reduce or amplify inequality in access to opportunity. We show
empirically that social networks often exhibit a much larger potential for
unequal diffusion across groups along paths of length 2 and 3 than expected by
our random graph models. We argue that homophily alone cannot not fully explain
the extent of unequal diffusion and attribute this mismatch to unequal
distribution of cross-group links among the nodes. Based on this insight, we
develop a variant of the stochastic block model that incorporates the
heterogeneity in cross-group linking. The model provides an unbiased and
consistent estimate of assortativity or homophily on paths of length 2 and
provide a more accurate estimate along paths of length 3 than existing models.
We characterize the null distribution of its log-likelihood ratio test and
argue that the goodness of fit test is valid only when the network is dense.
Based on our empirical observations and modeling results, we conclude that the
impact of any departure from equal distribution of links to source nodes in the
diffusion process is not limited to its first order effects as some nodes will
have fewer direct links to the sources. More importantly, this unequal
distribution will also lead to second order effects as the whole group will
have fewer diffusion paths to the sources.
