Universal properties of anyon braiding on one-dimensional wire networks

We demonstrate that anyons on wire networks have fundamentally different braiding properties than anyons in two dimensions (2D). Our analysis reveals an unexpectedly wide variety of possible non-Abelian braiding behaviors on networks. The character of braiding depends on the topological invariant called the connectedness of the network. As one of our most striking consequences, particles on modular networks can change their statistical properties when moving between different modules. However, sufficiently highly connected networks already reproduce the braiding properties of 2D systems. Our analysis is fully topological and independent on the physical model of anyons.