Minimizing the Importance Inequality of Nodes in a Social Network Graph

Network graphs are widely used to model a variety of real-world interactions. In such graphs, nodes do not have the same importance in the graph structure as a result of the graph's topological properties. This may have various implications concerning a network's behaviour as, for example, how different nodes operate (even a node's failure) may not have the same impact for the whole network. The differences in the structural properties of the nodes imply that each node has different importance, which, in turn, gives rise to the notion of importance inequality in a graph. This paper defines and addresses the problem of importance inequality minimization, which may be useful to achieve certain properties in a network. Given a network graph and an integer $k$, the problem aims to identify $k$ edges to connect non-adjacent nodes, in a way that minimizes the importance inequality of the graph. The paper provides a formal definition of the problem and proves its NP-hardness. Then, a naive greedy method is proposed, which is enhanced by heuristics that make its use practical. Experiments using 8 real-world networks are conducted to evaluate the proposed methods in terms of effectiveness and efficiency.