  We calculate equilibrium solutions for Ising spin models on `small world'
lattices, which are constructed by super-imposing random and sparse Poissonian
graphs with finite average connectivity c onto a one-dimensional ring. The
nearest neighbour bonds along the ring are ferromagnetic, whereas those
corresponding to the Poisonnian graph are allowed to be random. Our models thus
generally contain quenched connectivity and bond disorder. Within the replica
formalism, calculating the disorder-averaged free energy requires the
diagonalization of replicated transfer matrices. In addition to developing the
general replica symmetric theory, we derive phase diagrams and calculate
effective field distributions for two specific cases: that of uniform sparse
long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random
sparse long-range bonds (i.e. `small world' spin-glasses).
