Our societies are heterogeneous in many dimensions such as census, education, religion, ethnic and cultural composition. The links between individuals - e.g. by friendship, marriage or collaboration - are not evenly distributed, but rather tend to be concentrated within the same group. This phenomenon, called imbreeding homophily, has been related to either (social) preference for links with own--type individuals ( choice-based homophily) or to the prevalence of individuals of her same type in the choice set of an individual ( opportunity-based homophily). We propose an indicator to distinguish between these effects for minority groups. This is based on the observation that, in environments with unbiased opportunities, as the relative size of the minority gets small, individuals of the minority rarely meet and have the chance to establish links together. Therefore the effect of choice--based homophily gets weaker and weaker as the size of the minority shrinks. We test this idea across the dimensions of race and education on data on US marriages, and across race on friendships in US schools, and find that: for what concerns education i) opportunity--based homophily is much stronger than choice--based homophily and ii) they are both remarkably stationary in time; concerning race iii) school friendships do not exhibit opportunity-based homophily, while marriages do, iv) choice-based homophily is much stronger for marriages than for friendships and v) these effects vary widely across race.