# On the reorderability of node-filtered order complexes

## TL;DR

This paper investigates how the topological properties of growing graphs are affected by changes in the order of node emergence, revealing a spectrum of reorderability among different models and discussing implications for real-world networks.

## Contribution

It introduces the concepts of global and local reorderability in growing graph models and establishes theoretical links between robustness to node pair swaps and arbitrary orderings.

## Key findings

- Six graph models show varying degrees of reorderability.
- Robustness to node pair swaps relates to robustness to full order permutations.
- Theoretical connections between local and global reorderability are established.

## Abstract

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.