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 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 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 (Erdős-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 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.

Characterizing cospectral vertices via isospectral reduction

“…In [20], in connection with hidden symmetries and isospectral reduction, the notion of measure of latency is introduced. This concept gives a measure of how "hidden" the symmetry is when there is a latent automorphism.…”

“…Isospectral graph reductions have also been used to study "hidden symmetries" in real and theoretical networks [20]. A set of vertices in a simple graph G is referred to as being symmetric or automorphic if there is a nontrivial automorphism φ : V → V of the graph's vertices that fixes this set.…”

“…As this notion of symmetry can be extended to weighted graphs, a hidden or latent symmetry is an automorphism of some set of vertices in an isospectral reduction of a graph. Although these vertices may not be automorphic in the original unreduced graph, they share a number of properties that are known to hold for standard symmetries (for details see [20]).…”

Link prediction in networks using effective transitions

“…The following proof of Proposition 4.3 relies on the following result of [26]. Property (a) in Theorem 3 states that the spectral radius of a matrix is unaffected by an isoradial reduction.…”

Designing pretty good state transfer via isospectral reductions

“…For this reason, graphs (or, just as well, matrices) that lack direct symmetries, but whose eigenstates fulfill Eqs. (7) and (8) were recently termed latently symmetric [18,19]. However, although these symmetries may indeed seem hidden, we would like to mention here that all the graphs shown in Fig.…”