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 random sparse long-range bonds (i.e. 'small world' spin-glasses).

Abstract. We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs our macroscopic laws close already in terms of the singlespin path probabilities at zero external field. For the general case of arbitrary graph symmetry we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.

We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Finitely connected spin models, although still of a mean-field nature, can be regarded as a convenient level of description in between fully connected and finite-dimensional ones. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d = 2 (finitely connected XY spins with random chiral interactions) and for d = 3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

Abstract. We study the influence of network topology on retrieval properties of recurrent neural networks, using replica techniques for diluted systems. The theory is presented for a network with an arbitrary degree distribution p(k) and applied to power law distributions p(k) ∼ k −γ , i.e. to neural networks on scale-free graphs. A bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin glass phase. Using a population dynamics algorithm, the retrieval overlap and spin glass order parameters may be calculated throughout the phase diagram. It is shown that there is an enhancement of the retrieval properties compared with a Poissonian random graph. We compare our findings with simulations.

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbour bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a 2 n × 2 n matrix (where n → 0) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a onedimensional ring and via more complex long-range connectivity structures, e.g., (1 + ∞)-dimensional neural networks and 'small-world' magnets. Numerical simulations confirm our predictions satisfactorily.