Jan Øystein Haavig Bakke

Failures and avalanches in complex networks

We study the size distribution of power blackouts for the Norwegian and North American power grids. We find that for both systems the size distribution follows power laws with exponents − 1.65 ± 0.05 and − 2.0 ± 0.1, respectively. We then present a model with global redistribution of the load when a link in the system fails which reproduces the power law from the Norwegian power grid if the simulation are carried out on the Norwegian high-voltage power grid. The model is also applied to regular and irregular networks and give power laws with exponents − 2.0 ± 0.05 for the regular networks and − 1.5 ± 0.05 for the irregular networks. A presented mean-field theory is in good agreement with these numerical results.

Correlation length exponent in the three-dimensional fuse network

We present numerical measurements of the critical correlation length exponent nu in the three-dimensional fuse model. Using sufficiently broad threshold distributions to ensure that the system is the strong-disorder regime, we determine nu to be nu=0.83+/-0.04 based on analyzing the fluctuations of the survival probability. This value is different from that of ordinary percolation, which is 0.88.

Dimensions, maximal growth sites, and optimization in the dielectric breakdown model.

We study the growth of fractal clusters in the dielectric breakdown model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent alpha_{min} ) as a function of the growth exponent eta of the DBM model. We do not find evidence for a phase transition from fractal to nonfractal growth for a finite eta value. Simultaneously, we observe that the limit of nonfractal growth (D-->1) is consistent with alpha_{min}-->12 . Finally, using an optimization principle, we give a recipe on how to estimate the effective value of eta from temporal growth data of fractal aggregates.

Morphology of Laplacian random walks

Roughness of random walks in the presence of a Laplacian field is studied in two dimensions for various strengths of the field parametrized by η. We find an ηc~4.5±0.3 at which a transition occurs from a tortuous fractal structure to a one-dimensional profile of the walk. At ηc, the walks are self-affine with a roughness exponent ζ=0.80±0.05. For increasing η-values, the roughness exponent increases.