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.

The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches zeta(local)=0.65+/-0.03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Lévy distribution and the Lévy characteristics contribute to the local roughness exponent.

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. Large transportation networks like the road system, pipelines and the electrical power grid are sensitive to local failures. When failures occur the transport on these networks must be redistributed on the still intact part of the network, occasionally exceeding the local capacity and causing further failures[1, 2]. The resulting avalanches may finally end in major breakdowns: megajams in vehicular traffic or blackouts in the electrical power system. Such systems get increasingly complex with time as they and the transport on them grow. In addition, the liberalisation of the electrical power distribution market adds more complexity to the power grid system, since the market mechanisms give feedback to the production and transport of power over large regions. As the complexity of these systems increases the ability to predict the behaviour of large unwanted events becomes more and more difficult. Studies of vulnerability and avalanche statistics in complex networks have been done using different models. In the Motter and Lai model [3] the load on the network is described by betweenness centrality. Avalanches generated by this model on scale free networks were found to follow power law distribution [1]. Alberts et al. studied a similar model on the North American power grid.[4]

We test methods for measuring and characterizing rough profiles with emphasis on measurements of the self-affine roughness exponent, and describe a simple test to separate between roughness exponents originating from long range correlations in the signs of the profile, and roughness exponents originating from Lévy distributions of jumps. Based on tests on profiles with known roughness exponents we find that the power spectrum density analysis and the averaged wavelet coefficients method give the best estimates for roughness exponents in the range 0.1-0.9. The error bars are found to be less than 0.03 for profile lengths larger than 256, and there is no systematic bias in the estimates. We present quantitative estimates of the error bars and the systematic error and their dependence on the value of the roughness exponent and the profile length. We also quantify how power-law noise can modify the measured roughness exponent for measurement methods different from the power spectrum density analysis and the second order correlation function method.

We analyze the statistical distribution function for the height fluctuations of brittle fracture surfaces using extensive experimental data sampled on widely different materials and geometries. We compare a direct measurement of the distribution to an analysis based on the structure functions. For length scales delta larger than a characteristic scale Lambda that corresponds to a material heterogeneity size, we find that the distribution of the height increments Deltah=h(x+delta)-h(x) is Gaussian and monoaffine, i.e., the scaling of the standard deviation sigma is proportional to delta(zeta) with a unique roughness exponent. Below the scale Lambda we observe a deviation from a Gaussian distribution and a monoaffine behavior. We discuss for the latter, the relevance of a multiaffine analysis and the influences of the discreteness resulting from material microstructures or experimental sampling.

We measure the roughness exponent for fracture profiles in the two-dimensional central force lattice model using different measurement methods. We find that the profiles are self-affine for a system with narrow disorders and that broader disorders introduces overhangs in the fracture surface leading to deviation from self-affinity for small length scales and to nontrivial finite size scaling.

We measure the roughness exponent and the correlation length exponent of a stress-weighted percolation process in the central force model in 2D. The roughness exponent is found to be zeta = 0.75 ±0.03 and the correlation length exponent is found to be nu = 1.7 ±0.3. This result supports a conjecture that the fracture roughness for large scales is controlled by a stress weighted percolation process, and the fracture roughness can by calculated from the correlation length exponent by zeta = 2*nu/(1+2*nu). We also compare global and local measurements of the fracture roughness and do not find sign of anomalous scaling in the central force model.

Fracture morphology has for the last two decades gained much interest seen from a fundamental point of view. Experimental evidence point out a self-affine scaling of such fracture surfaces, which leads to the fact that the roughness of the fracture scales with the size of the system as w ∼L ζ . We have implemented a numerical network of random fuses in three dimensions in order to study breakdown processes of brittle materials. With this model we have been able to measure critical exponents at breakdown, and in that sense gained information of the scaling laws involved in the morphology of fracture surfaces. We have studied both the scaling of roughness and behavior of the correlation length ξ for broad distributions in order to examine the linkage between them. A newly proposed relation by Hansen and Schmittbuhl gives ( = 2v/(1 + 2v). By using the relation ξ ∼ ‖p - p c ‖ - v from percolation theory we measure the critical correlation length exponent v = 0.83 ′ 0.06. From this value we derive a value for the roughness exponent ζ = 0.62 ′ 0.05 in the three-dimensional fuse model. This is consistent with previous studies on the fuse model in three dimensions.