  Network alignment task, which aims to identify corresponding nodes in
different networks, is of great significance for many subsequent applications.
Without the need for labeled anchor links, unsupervised alignment methods have
been attracting more and more attention. However, the topological consistency
assumptions defined by existing methods are generally low-order and less
accurate because only the edge-indiscriminative topological pattern is
considered, which is especially risky in an unsupervised setting. To reposition
the focus of the alignment process from low-order to higher-order topological
consistency, in this paper, we propose a fully unsupervised network alignment
framework named HTC. The proposed higher-order topological consistency is
formulated based on edge orbits, which is merged into the information
aggregation process of a graph convolutional network so that the alignment
consistencies are transformed into the similarity of node embeddings.
Furthermore, the encoder is trained to be multi-orbit-aware and then be refined
to identify more trusted anchor links. Node correspondence is comprehensively
evaluated by integrating all different orders of consistency. {In addition to
sound theoretical analysis, the superiority of the proposed method is also
empirically demonstrated through extensive experimental evaluation. On three
pairs of real-world datasets and two pairs of synthetic datasets, our HTC
consistently outperforms a wide variety of unsupervised and supervised methods
with the least or comparable time consumption. It also exhibits robustness to
structural noise as a result of our multi-orbit-aware training mechanism.
