Proportional Selection in Networks

Consider the problem of selecting a fixed number of k nodes from a network. Our goal is twofold: to identify the most influential nodes, and to ensure that the selection proportionally represents the diversity within the network. For instance, consider a network composed of three groups of densely connected nodes. Assume the groups contain 50%, 30%, and 20% of all nodes, respectively, and connections between groups are relatively sparse. If the objective is to select k = 10 nodes, a proportional approach would involve selecting five most influential nodes from the first group, three from the second, and two from the third group.