Bonacich Network Measures as Minimum Norm Solutions

An undirected connected bimodal network has two Bonacich measures quantifying the centrality of the nodes. We show that the product of Bonacich measures of an undirected bimodal network may be viewed as a product measure that is nearest (w.r.t. Euclidean norm) to the matrix representing the network. A directed bimodal network has four Bonacich measures, two inflow and two outflow measures. Given directed strongly connected bimodal network and its Bonacich inflow (or outflow) measures, there is undirected bimodal network with the same Bonacich measures. So Bonacich measures of directed bimodal network can also be viewed as minimum norm solutions.