Revising the simple measures of assortativity in complex networks

We find that traditional statistics for measuring degree mixing are strongly affected by superrich nodes. To counteract and measure the effect of superrich nodes, we propose a paradigm to quantify the mixing pattern of a real network in which different mixing patterns may appear among low-degree nodes and among high-degree nodes. This paradigm and the simple revised measure uncover the true complex degree mixing patterns of complex networks with superrich nodes. The alternate method indicates that some networks show a false disassortative mixing induced by superrich nodes and have no tendency to be genuinely disassortative. Our results also show that the previously observed fragility of scale-free networks is actually greatly exacerbated by the presence of even a very small number of superrich nodes.