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

Calculation of assortative mixing by degree in networks indicates whether nodes with similar degree are connected to each other. In networks with scale‐free distribution high values of assortative mixing by degree can be an indication of a hub‐like core in 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 with nodes that have the same degrees. The so‐called s‐metric has been shown to be a better choice to calculate assortative mixing. The s‐metric is normalized with respect to the class of networks without self‐loops, multiple edges, and multiple components, 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 Lagrangian 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.