  Higher-order proximity (HOP) is fundamental for most network embedding
methods due to its significant effects on the quality of node embedding and
performance on downstream network analysis tasks. Most existing HOP definitions
are based on either homophily to place close and highly interconnected nodes
tightly in embedding space or heterophily to place distant but structurally
similar nodes together after embedding. In real-world networks, both can
co-exist, and thus considering only one could limit the prediction performance
and interpretability. However, there is no general and universal solution that
takes both into consideration. In this paper, we propose such a simple yet
powerful framework called homophily and heterophliy preserving network
transformation (H2NT) to capture HOP that flexibly unifies homophily and
heterophily. Specifically, H2NT utilises motif representations to transform a
network into a new network with a hybrid assumption via micro-level and
macro-level walk paths. H2NT can be used as an enhancer to be integrated with
any existing network embedding methods without requiring any changes to latter
methods. Because H2NT can sparsify networks with motif structures, it can also
improve the computational efficiency of existing network embedding methods when
integrated. We conduct experiments on node classification, structural role
classification and motif prediction to show the superior prediction performance
and computational efficiency over state-of-the-art methods. In particular,
DeepWalk-based H2 NT achieves 24% improvement in terms of precision on motif
prediction, while reducing 46% computational time compared to the original
DeepWalk.
