  Graph transformer has been proven as an effective graph learning method for
its adoption of attention mechanism that is capable of capturing expressive
representations from complex topological and feature information of graphs.
Graph transformer conventionally performs dense attention (or global attention)
for every pair of nodes to learn node representation vectors, resulting in
quadratic computational costs that are unaffordable for large-scale graph data.
Therefore, mini-batch training for graph transformers is a promising direction,
but limited samples in each mini-batch can not support effective dense
attention to encode informative representations. Facing this bottleneck, (1) we
start by assigning each node a token list that is sampled by personalized
PageRank (PPR) and then apply standard multi-head self-attention only on this
list to compute its node representations. This PPR tokenization method
decouples model training from complex graph topological information and makes
heavy feature engineering offline and independent, such that mini-batch
training of graph transformers is possible by loading each node's token list in
batches. We further prove this PPR tokenization is viable as a graph
convolution network with a fixed polynomial filter and jumping knowledge.
However, only using personalized PageRank may limit information carried by a
token list, which could not support different graph inductive biases for model
training. To this end, (2) we rewire graphs by introducing multiple types of
virtual connections through structure- and content-based super nodes that
enable PPR tokenization to encode local and global contexts, long-range
interaction, and heterophilous information into each node's token list, and
then formalize our Virtual Connection Ranking based Graph Transformer
(VCR-Graphormer).
