Quasi-symmetries in complex networks: a dynamical model approach

The existence of symmetries in complex networks has a significant effect on network dynamic behaviour. Nevertheless, beyond topological symmetry, one should consider the fact that real-world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. Therefore, the resulting approximate symmetries remain hidden to standard symmetry analysis - fully accomplished by discrete algebra software. There have been a number of attempts to deal with approximate symmetries. In the present work we provide an alternative notion of these weaker symmetries, which we call `quasi-symmetries'. Differently from other definitions, quasi-symmetries remain free to impose any invariance of a particular network property and they are obtained from the phase differences at the steady-state configuration of an oscillatory dynamical model: the Kuramoto-Sakaguchi model. The analysis of quasi-symmetries unveils otherwise hidden real-world networks attributes. On the one hand, we provide a benchmark to determine whether a network has a more complex pattern than that of a random network with regard to quasi-symmetries, namely, if it is structured into separate quasi-symmetric groups of nodes. On the other hand, we define the `dual-network', a weighted network (and its corresponding binnarized counterpart) that effectively encodes all the information of quasi-symmetries in the original network. The latter is a powerful instrument for obtaining worthwhile insights about node centrality (obtaining the nodes that are unique from that act as imitators with respect to the others) and community detection ( quasi-symmetric groups of nodes).