  The symmetry of complex networks is a global property that has recently
gained attention since MacArthur et al. 2008 showed that many real-world
networks contain a considerable number of symmetries. These authors work with a
very strict symmetry definition based on the network's automorphism. The
potential problem with this approach is that even a slight change in the
graph's structure can remove or create some symmetry. Recently, Liu 2020
proposed to use an approximate automorphism instead of strict automorphism.
This method can discover symmetries in the network while accepting some minor
imperfections in their structure. The proposed numerical method, however,
exhibits some performance problems and has some limitations while it assumes
the absence of fixed points. In this work, we exploit alternative approaches
recently developed for treating the Graph Matching Problem and propose a
method, which we will refer to as Quadratic Symmetry Approximator (QSA), to
address the aforementioned shortcomings. To test our method, we propose a set
of random graph models suitable for assessing a wide family of approximate
symmetry algorithms. The performance of our method is also demonstrated on
brain networks.
