  Growing graphs describe a multitude of developing processes from maturing
brains to expanding vocabularies to burgeoning public transit systems. Each of
these growing processes likely adheres to proliferation rules that establish an
effective order of node and connection emergence. When followed, such
proliferation rules allow the system to properly develop along a predetermined
trajectory. But rules are rarely followed. Here we ask what topological changes
in the growing graph trajectories might occur after the specific but basic
perturbation of permuting the node emergence order. Specifically we harness
applied topological methods to determine which of six growing graph models
exhibit topology that is robust to randomizing node order, termed global
reorderability, and robust to temporally-local node swaps, termed local
reorderability. We find that the six graph models fall upon a spectrum of both
local and global reorderability, and furthermore we provide theoretical
connections between robustness to node pair ordering and robustness to
arbitrary node orderings. Finally we discuss real-world applications of
reorderability analyses and suggest possibilities for designing reorderable
networks.
