  We demonstrate that anyons on wire networks have fundamentally different
braiding properties than anyons in 2D. Our analysis reveals an unexpectedly
wide variety of possible non-abelian braiding behaviours 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 braiding properties of 2D systems. Our analysis is
fully topological and independent on the physical model of anyons.
