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.

changes in the spatial patterns and rate of urban development will be one of the main determinants of future coastal flood risk. Existing spatial projections of urban extent are, however, often available at coarse spatial resolutions, local geographical scales or for short time horizons, which limits their suitability for broad-scale coastal flood impact assessments. Here, we present a new set of spatially explicit projections of urban extent for ten countries in the Mediterranean, consistent with the Shared Socioeconomic pathways (SSps). to model plausible future urban development, we develop an Urban change Model, which uses input variables such as elevation, population density or road network and an artificial neural network to project urban development on a regional scale. The developed future projections for the five SSPs indicate that accounting for the spatial patterns of urban development can lead to significant differences in the assessment of future coastal urban exposure. The increase in exposure in the Extended Low Elevation Coastal Zone (E-LECZ = area below 20 m of elevation) until 2100 can vary, by up to 104%, depending on the urban development scenario chosen. This finding highlights that accounting for urban development in long-term adaptation planning, e.g. in the form of land-use planning, can be an effective measure for reducing future coastal flood risk on a regional scale. The urban extent in low lying coastal areas is increasing faster than in other regions 1 , thus leading to increased exposure to sea-level rise and associated hazards. Societies' risk from these hazards will, therefore, not only depend on the physical drivers of change but also on the rate and pattern of urban growth which will be guided, to a large extent, by policies on future urban development 2. One way to investigate how urban development influences future coastal flood risk is by accounting for spatiotemporal urban land cover change with the use of spatially explicit future urban projections in coastal impact assessments. Spatially explicit future urban extent scenarios are, however, currently often not available on a regional scale, which is one of the major shortcomings in coastal impact assessments to date 3. Until recently, most studies have considered physical drivers of future change, partly accounting for population and economic development 4 but have neglected changes in the spatial extent of urban agglomerations, where most impacts occur. Large scale urban change models can help to better understand how a non-climatic driver, namely urban development, influences future risk. Different concepts and methods have been developed for modelling future urban change. Consequently, there now exists a vast diversity in modelling approaches, concepts, and models striving to describe and understand the mechanisms of land cover change. These approaches include, among others, the use of Cellular Automata 5 , Monte Carlo simulations 6 , Artificial Neural Networks 7 , as well as process-based and empirical models (e.g. CLUE)...

We apply the cavity method to a spin glass model on a 'small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by Mézard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and ±J long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.

Gridded population projections constitute an essential input for climate change impacts, adaptation, and vulnerability (IAV) assessments as they allow for exploring how future changes in the spatial distribution of population drive climate change impacts. We develop such spatial population projections, using a gravity-based modeling approach that accounts for rural-urban and inland-coastal migration as well as for spatial development patterns (i.e. urban sprawl). We calibrate the model (called CONCLUDE) to the socioeconomically diverse Mediterranean region, additionally considering differences in socioeconomic development in two geographical regions: the northern Mediterranean and the southern and eastern Mediterranean. We produce high-resolution population projections (approximately 1 km) for 2020–2100 that are consistent with the Shared Socioeconomic Pathways (SSPs), both in terms of qualitative narrative assumptions as well as national-level projections. We find that future spatial population patterns differ considerably under all SSPs, with four to eight times higher urban population densities and three to 16 times higher coastal populations in southern and eastern Mediterranean countries compared to northern Mediterranean countries in 2100. In the South and East, the highest urban density (8000 people km−2) and coastal population (107 million) are projected under SSP3, while in the North, the highest urban density (1500 people km−2) is projected under SSP1 and the highest coastal population (15.2 million) under SSP5. As these projections account for internal migration processes and spatial development patterns, they can provide new insights in a wide range of IAV assessments. Furthermore, CONCLUDE can be extended to other continental or global scales due to its modest data requirements based on freely available global datasets.