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 $k\ge 3$. Our findings provide a simple and intuitive way to compare the equality of opportunity of real-life social networks.