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. 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]