Slowly evolving random graphs II: Adaptive geometry in finite-connectivity Hopfield models

We present an analytically solvable random graph model in which the connections between the nodes can evolve in time, adiabatically slowly compared to the dynamics of the nodes. We apply the formalism to finite connectivity attractor neural network (Hopfield) models and we show that due to the minimisation of the frustration effects the retrieval region of the phase diagram can be significantly enlarged. Moreover, the fraction of misaligned spins is reduced by this effect, and is smaller than in the infinite connectivity regime. The main cause of this difference is found to be the non-zero fraction of sites with vanishing local field when the connectivity is finite.Comment: 17 pages, 8 figure

Spin models on random graphs with controlled topologies beyond degree constraints

We study Ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution $p(k)$ can be chosen arbitrarily, but which allows for further fine-tuning of the topology via preferential attachment of edges on the basis of an arbitrary function Q(k,k') of the degrees of the vertices involved. We solve these models using finite connectivity equilibrium replica theory, within the replica symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution, but also on the choice made for Q(k,k'). The increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real-world interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys

Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices

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).Comment: 22 pages, LaTeX, IOP macros, eps figure

Replicated transfer matrix analysis of Ising spin Models On 'Small . . .

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 e#ective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. `small world' magnets), and that of random sparse long-range bonds (i.e. `small world' spin-glasses)

Analytic Solution of Attractor Neural Networks on Scale-Free Graphs

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 , 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. PACS numbers: 75.10.Nr, 05.20.-y, 64.60.Cn E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] 1

Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

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