Equality of opportunity and integration in social networks

We propose the notion of k-integration as a measure of equality of opportunity in social networks. A social network is k-integrated if there is a path of length at most k between any two individuals, thus guaranteeing that everybody has the same network opportunities to find a job, a romantic partner, or valuable information. We compute the minimum number of bridges (i.e. edges between nodes belonging to different components) or central nodes (those which are endpoints to a bridge) required to ensure k-integration. The answer depends only linearly on the size of each component for k = 2, and does not depend on the size of each component for k ≥ 3. Our findings provide a simple and intuitive way to compare the equality of opportunity of real-life social networks.