Hidden Symmetries in Real and Theoretical Networks

Abstract

Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called \emph{latent symmetry}, which is an extension of the standard notion of symmetry. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a \emph{size}, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erd{\H o}s-Rényi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that \emph{latent symmetries} are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.

18 pages

References in corpus (14)

• Emergence of scaling in random networks
• Modularity and community structure in networks
• Topology of evolving networks: local events and universality
• Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
• Degree Distributions of Growing Networks
• Scaling Behaviour of Developing and Decaying Networks
• Spectra and eigenvectors of scale-free networks
• On Automorphism Groups of Networks
• Dynamical and spectral properties of complex networks
• Spectral Characteristics of Network Redundancy
• Emergence of Symmetry in Complex Networks
• Complex Networks and Symmetry I: A Review
• Detecting degree symmetries in networks
• Network Quotients: Structural Skeletons of Complex Systems

Cited by in corpus (22)

• Flat bands by latent symmetry
• Does the brain behave like a (complex) network? I. Dynamics
• Latent symmetry induced degeneracies
• Designing pretty good state transfer via isospectral reductions
• Hidden symmetries in acoustic wave systems
• Emergent non-Hermitian models
• Temporal network modeling with online and hidden vertices based on the birth and death process
• Duality of Topological Edge States in a Mechanical Kitaev Chain
• Strain topological metamaterials
• Cospectrality preserving graph modifications and eigenvector properties via walk equivalence of vertices
• Higher-order topology protected by latent crystalline symmetries
• Quasi-symmetries in complex networks: a dynamical model approach
• Topological states constructed by two different trivial quantum wires
• Eigenmodes of latent-symmetric quantum photonic networks
• State Transfer in Latent-Symmetric Networks
• Topological Anderson insulators by latent symmetry
• Topological band insulators without translational symmetry
• High-root topological edge-state bands
• Stability of Stochastically Switched and Stochastically Time-Delayed Systems
• Constructing Cospectral Vertices Through Orbits of Subgraphs
• On symmetries of a matrix and its isospectral reduction
• Isospectral Reductions of Non-negative Matrices