Dietrich Stauffer

Efficient Hopfield pattern recognition on a scale-free neural network

Abstract:Neural networks are supposed to recognise blurred images (or patterns) of N pixels (bits) each. Application of the network to an initial blurred version of one of P pre-assigned patterns should converge to the correct pattern. In the “standard Hopfield model, the N “neurons” are connected to each other via N2 bonds which contain the information on the stored patterns. Thus computer time and memory in general grow with N2. The Hebb rule assigns synaptic coupling strengths proportional to the overlap of the stored patterns at the two coupled neurons. Here we simulate the Hopfield model on the Barabási-Albert scale-free network, in which each newly added neuron is connected to only m other neurons, and at the end the number of neurons with q neighbours decays as 1/q3. Although the quality of retrieval decreases for small m, we find good associative memory for 1 ≪ m ≪ N. Hence, these networks gain a factor N/m ≫ 1 in the computer memory and time.

Self similarity and correlations in percolation

The infinite cluster above the percolation threshold is shown by a scaling theory and Monte Carlo simulations to be homogeneous on large length scales (compared with the correlation length). On shorter length scales this cluster is self similar, and its measured fractal dimensionality agrees excellently with the scaling law D=d- beta / nu . The exponents beta and nu are also measured, both from the crossover between the two length scale regions and from correlations near the boundaries.

An Infinite Number of Effectively Infinite Clusters in Critical Percolation

An infinite number of effectively infinite clusters are predicted at the percolation threshold, if “effectively infinite” means that a clusters mass increases with a positive power of the lattice size L. All these cluster masses increase as LD with the fractal dimension D = d − β/v, while the mass of the rth largest cluster for fixed L decreases as 1/rλ, with λ = D/d in d dimensions. These predictions are confirmed by computer simulations for the square lattice, where D = 91/48 and λ = 91/96.

Finite-size scaling for percolation above six dimensions

We consider percolation in d dimensions for finite samples of linear size L. Theoretical arguments are presented to show that for d > 6, the shift in the percolation threshold, pc(L)−pc(∞), behaves like A/L2 + B/Ld3; for periodic boundary conditions, A = 0. These predictions are consistent with recent simulations is seven dimensions.

First exit time of termites and random super-normal conductor networks

The resistance of a random super-normal conductor network is calculated through the diffusion of a termite, which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds. The first time, T, that the termite exits at a distance r from the origin (averaged over many configurations and runs) is found to behave as T approximately r2k/ for r > xi (( pc-p)-s describes the divergence of the conductivity as p to pc-, p being the concentration of the superconductor). Scaling arguments are used to show that k=1+s/(2+ theta ), where (2+ theta ) is the fractal dimension of random walks on single clusters at pc. Numerical simulations at two dimensions (d=2) yields s=1.34+or-0.10 and k=1.34+or-0.03, in agreement with scaling. The authors also show that the probability of return to the origin at time t behaves as t-dk2/. Preliminary results at d=1 and other calculations methods are also discussed.

Damage Spreading in a 2D Ising Model with Swendsen-Wang Dynamics

Damage spreading for Ising cluster dynamics is investigated numerically by using random numbers in a way that conforms with the notion of submitting the two evolving replicas to the same thermal noise. Two damage spreading transitions are found; damage does not spread either at low or high temperatures. We determine some critical exponents at the high-temperature transition point, which seem consistent with directed percolation.

Self-similarity and covered neighborhoods of fractals: A random walk test

Abstract A strong version of the property of self-similarity is described, and it is shown that this property is satisfied by random walks on a simple cubic lattice. When each site visited by the walk is surrounded by a small cube, the total volume of these covering cubes depends on the cube size, the size of the region investigated, and the length of the walk. We find that for long walks at a fixed ratio of cube to region size the filling ratio is roughly constant.

The granular fracture model for rock fragmentation

The authors introduce and study a model of granular fracture to mimic the dynamics of rock fragmentation. The model describes a rock as an assembly of interacting grains that evolve in time according to Newtonian dynamics. The main ingredient describing macroscopic rather than microscopic dynamics is a history-dependent attractive potential between pairs of grains, which is set to zero after the pair first moves beyond some threshold distance apart. They study the characteristics of the distribution of cluster sizes and compare them with the corresponding characteristics of the percolation problem. Their results show a decrease of the cluster numbers with sample size and an apparent breakdown of hyperscaling. They also find a dependence of critical exponents on the average initial kinetic energy of the system.

On the width of the critical region in dilute magnets with long range percolation

The Ginzburg criterion is used to estimate the width of the critical region for a dilute magnet with a large range of interactionsR. This width decreases asR−2d/(4−d) at high temperature and asR−2d/(6−d) at very low temperatures, i.e. in the percolation limit. The crossover between the two regimes is discussed.