  We study the thermodynamic properties of spin systems on small-world
hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin
interactions onto a one-dimensional Ising chain with nearest-neighbor
interactions. We use replica-symmetric transfer-matrix techniques to derive a
set of fixed-point equations describing the relevant order parameters and free
energy, and solve them employing population dynamics. In the special case where
the number of connections per site is of the order of the system size we are
able to solve the model analytically. In the more general case where the number
of connections is finite we determine the static and dynamic
ferromagnetic-paramagnetic transitions using population dynamics. The results
are tested against Monte-Carlo simulations.
