Computing Assortative Mixing by Degree with the s-Metric in Networks Using Linear Programming

Estimation of assortative mixing by degree in networks is important because it points to the existence of a hub-like core in networks with scaling degree distributions. Such networks show self-similarity and other emergent properties of complex networks. Degree correlation has generally been used to measure assortative mixing of a network. But it has been shown that degree correlation cannot always distinguish properly between different networks. In the commonly used class of simple connected networks, the so-called s-metric is a better choice to estimate assortative mixing. The s-metric is normalized with respect to this specific class of networks, while degree correlation is always normalized with respect to unrestricted networks, where self-loops, multiple edges, and multiple components are allowed. The challenge in computing the normalized s-metric is in obtaining the minimum and maximum value within a specific class of networks. We show that this can be solved by using linear programming. We use Lagrangean relaxation and the subgradient algorithm to obtain a solution to the s-metric problem. Several examples are given to illustrate the principles and some simulations indicate that the solutions are generally accurate.