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. © 2015 lourens j. waldorp and verena d. schmittmann.