  We consider network models of quantum localisation in which a particle with a
two-component wave function propagates through the nodes and along the edges of
an arbitrary directed graph, subject to a random SU(2) rotation on each edge it
traverses. The propagation through each node is specified by an arbitrary but
fixed S-matrix. Such networks model localisation problems in class C of the
classification of Altland and Zirnbauer, and, on suitable graphs, they model
the spin quantum Hall transition. We extend the analyses of Gruzberg, Ludwig
and Read and of Beamond, Cardy and Chalker to show that, on an arbitrary graph,
the mean density of states and the mean conductance may be calculated in terms
of observables of a classical history-dependent random walk on the same graph.
The transition weights for this process are explicitly related to the elements
of the S-matrices. They are correctly normalised but, on graphs with nodes of
degree greater than 4, not necessarily non-negative (and therefore
interpretable as probabilities) unless a sufficient number of them happen to
vanish. Our methods use a supersymmetric path integral formulation of the
problem which is completely finite and rigorous.
