  The braiding of the worldlines of particles restricted to move on a network
(graph) is governed by the graph braid group, which can be strikingly different
from the standard braid group known from two-dimensional physics. It has been
recently shown that imposing the compatibility of graph braiding with anyon
fusion for anyons exchanging at a single wire junction leads to new types of
anyon models with the braiding exchange operators stemming from solutions of
certain generalised hexagon equations. In this work, we establish these
graph-braided anyon fusion models for general wire networks. We show that the
character of braiding strongly depends on the graph-theoretic connectivity of
the given network. In particular, we prove that triconnected networks yield the
same braiding exchange operators as the planar anyon models. In contrast,
modular biconnected networks support independent braiding exchange operators in
different modules. Consequently, such modular networks may lead to more
efficient topological quantum computer circuits. Finally, we conjecture that
the graph-braided anyon fusion models will possess the (generalised) coherence
property where certain polygon equations determine the braiding exchange
operators for an arbitrary number of anyons. We also extensively study
solutions to these polygon equations for chosen low-rank fusion rings,
including the Ising theory, quantum double of groups, and Tambara-Yamagami
models. We find numerous solutions that do not appear in the planar theory of
anyons.
