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We propose a model which explains how power-law crossover behaviour can arisein a system which is capable of experiencing cascading failure. In our modelthe susceptibility of the system to cascades is described by a single number,the propagation power, which measures the ease with which cascades propagate.Physically, such a number could represent the density of unstable material in asystem, its internal connectivity, or the mean susceptibility of its componentparts to failure. We assume that the propagation power follows an upwarddrifting Brownian motion between cascades, and drops discontinuously each timea cascade occurs. Cascades are described by a continuous state branchingprocess with distributional properties determined by the value of thepropagation power when they occur. In common with many cascading models, purepower law behaviour is exhibited at a critical level of propagation power, andthe mean cascade size diverges. This divergence constrains large systems to thesubcritical region. We show that as a result, crossover behaviour appears inthe cascade distribution when an average is performed over the distribution ofpropagation power. We are able to analytically determine the exponents beforeand after the crossover.

Crossover Behaviour In Driven Cascades