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.

翻译：复杂网络的对称性是一种全局属性，自MacArthur等人2008年发现许多真实网络包含大量对称性以来，这一特性近期引起了广泛关注。这些作者基于网络的自同构定义了一种严格的对称性概念。该方法潜在的问题在于，即便图结构发生微小变化，也可能导致某些对称性的消除或产生。近期，Liu在2020年提出使用近似自同构替代严格自同构，该方法能够在接受结构细微不完美性的前提下发现网络中的对称性。然而，所提出的数值方法存在性能问题，且因假设不存在不动点而具有局限性。本研究利用近年来为解决图匹配问题而开发的替代方法，提出了一种名为二次对称近似器（QSA）的方法来克服上述缺陷。为测试该方法，我们提出了一套适用于评估广泛近似对称算法的随机图模型，并在脑网络上验证了该方法的性能。