Figure 3 |. Branching process null models.

a, Subset of the mathematician genealogy network. Mentors/parents (black circles) are connected to each of their protégés/children (white circles). The horizontal positions of mathematicians represent their graduation/birth dates, t. The bottom two parents were born in 1924, the top two parents were born in 1937, and all four parents have a child born in 1958. From a parent’s perspective, three essential features of the empirical network must be preserved in random networks generated from the two branching process null models: the birth date, t_p, the fecundity, k_p, and the chronology of child births, {t_c}. b, Random networks from ensemble I preserve these three essential features. Solid red lines highlight the links in the empirical network whose end points can be randomized. Dashed red lines illustrate one of the possible randomization moves after switching the corresponding pair of links. We note that the age difference between parent and child is not preserved. c, Random networks from ensemble II preserve the three essential features as well as the age difference between parent and child. Solid blue lines of the same colour highlight the links in the empirical network whose end points can be randomized. Dashed blue lines illustrate one of the possible randomization moves after switching the corresponding pair of links. Random networks for each ensemble are generated by attempting 100 switches per link (Methods).