  We study an extension of Duncan Watts' 2002 model of information cascades in
social networks where edge weights are taken to be random, an innovation
motivated by recent applications of cascade analysis to systemic risk in
financial networks. The main result is a probabilistic analysis that
characterizes the cascade in an infinite network as the fixed point of a
vector-valued mapping, explicit in terms of convolution integrals that can be
efficiently evaluated numerically using the fast Fourier transform algorithm. A
second result gives an approximate probabilistic analysis of cascades on "real
world networks", finite networks based on a fixed deterministic graph.
Extensive cross testing with Monte Carlo estimates shows that this approximate
analysis performs surprisingly well, and provides a flexible microscope that
can be used to investigate properties of information cascades in real world
networks over a wide range of model parameters.
