A theory on power in networks

The eigenvector centrality equation λx = A x is a successful compromise between simplicity and expressivity. It claims that central actors are those connected with central others. For at least 70 years, this equation has been explored in disparate contexts, including econometrics, sociometry, bibliometrics, Web information retrieval, and network science. We propose an equally elegant counterpart: the power equation x = Ax÷, where x÷ is the vector whose entries are the reciprocal of those of x. It asserts that power is in the hands of those connected with powerless others. It is meaningful, for instance, in bargaining situations, where it is advantageous to be connected to those who have few options. We tell the parallel, mostly unexplored story of this intriguing equation. In particular, we reveal its intimate relationship with the matrix balancing problem.