Symmetry in complex networks

We consider the size and structure of the automorphism groups of a variety of empirical ‘realworld’ networks and find that, in contrast to classical random graph models, many real-world networks – including a variety of biological networks and technological networks such as the internet – are richly symmetric. We then discuss how knowledge of the structure of the automorphism group of a network can be used to further understand network properties. In particular, since matrix representations of network automorphisms commute with matrix representations of the network (such as the adjacency and Laplacian matrices) symmetry can give rise to highly degenerate eigenvalues which manifest as spikes in the network’s spectral density. We discuss how a network’s automorphism group naturally provides a ‘geometric’ factorization of its characteristic polynomial, enabling association of specific eigenvalues and eigenvectors with specific network motifs. PACS: 89.75.-k, 89.75.Fb, 02.20.-a keywords: Complex Network – Automorphism Group – Eigenvalue Spectra