  Propagation of coherent light through a disordered network is accompanied by
randomization and possible conversion into thermal light. Here, we show that
network topology plays a decisive role in determining the statistics of the
emerging field if the underlying lattice satisfies chiral symmetry. By
examining one-dimensional arrays of randomly coupled waveguides arranged on
linear and ring topologies, we are led to a remarkable prediction: the field
circularity and the photon statistics in ring lattices are dictated by its
parity -- whether the number of sites is even or odd, while the same quantities
are insensitive to the parity of a linear lattice. Adding or subtracting a
single lattice site can switch the photon statistics from super-thermal to
sub-thermal, or vice versa. This behavior is understood by examining the real
and imaginary fields on a chiral-symmetric lattice, which form two strands that
interleave along the lattice sites. These strands can be fully braided around
an even-sited ring lattice thereby producing super-thermal photon statistics,
while an odd-sited lattice is incommensurate with such an arrangement and the
statistics become sub-thermal. Such effects are suppressed in the limit of high
lattice disorder, whereupon Anderson localization arrests the spread of light
around the lattice and eliminates topology-dependent phenomena.
