  Conditional graph generation tasks involve training a model to generate a
graph given a set of input conditions. Many previous studies employ
autoregressive models to incrementally generate graph components such as nodes
and edges. However, as graphs typically lack a natural ordering among their
components, converting a graph into a sequence of tokens is not
straightforward. While prior works mostly rely on conventional heuristics or
graph traversal methods like breadth-first search (BFS) or depth-first search
(DFS) to convert graphs to sequences, the impact of ordering on graph
generation has largely been unexplored. This paper contributes to this problem
by: (1) highlighting the crucial role of ordering in autoregressive graph
generation models, (2) proposing a novel theoretical framework that perceives
ordering as a dimensionality reduction problem, thereby facilitating a deeper
understanding of the relationship between orderings and generated graph
accuracy, and (3) introducing "latent sort," a learning-based ordering scheme
to perform dimensionality reduction of graph tokens. Our experimental results
showcase the effectiveness of latent sort across a wide range of graph
generation tasks, encouraging future works to further explore and develop
learning-based ordering schemes for autoregressive graph generation.
