A very interesting matter of Network Science is assessing how complex a given network is. In other words, by how much does such a network departs from any general patterns which could be evoked for its description. Among other choices, these patterns can be defined in terms of node or edge properties, as one of the various modes of centrality. Although there are centrality indexes defined for weighted graphs, the discussions on this subject are far from over, especially because the influence of edge weights in this regard can vary not only in form but also in intensity, a still incipient approach so far. For the aforementioned complexity evaluation, the situation is even more acute: to the best of our knowledge, this was never addressed in the literature, from the standpoint of centrality in weighted graphs. This paper details in a colorful fashion a sound methodology covering both topics, as well as experiments confirming its practical applicability and future trends in this context.