  We examine how three different communication processes operating through
social networks are affected by homophily -- the tendency of individuals to
associate with others similar to themselves. Homophily has no effect if
messages are broadcast or sent via shortest paths; only connection density
matters. In contrast, homophily substantially slows learning based on repeated
averaging of neighbors' information and Markovian diffusion processes such as
the Google random surfer model. Indeed, the latter processes are strongly
affected by homophily but completely independent of connection density,
provided this density exceeds a low threshold. We obtain these results by
establishing new results on the spectra of large random graphs and relating the
spectra to homophily. We conclude by checking the theoretical predictions using
observed high school friendship networks from the Adolescent Health dataset.
