  We calculate the number of metastable configurations of Ising small-world
networks which are constructed upon superimposing sparse Poisson random graphs
onto a one-dimensional chain. Our solution is based on replicated
transfer-matrix techniques. We examine the denegeracy of the ground state and
we find a jump in the entropy of metastable configurations exactly at the
crossover between the small-world and the Poisson random graph structures. We
also examine the difference in entropy between metastable and all possible
configurations, for both ferromagnetic and bond-disordered long-range
couplings.
