Cascading activity is commonly found in complex systems with directed
interactions such as metabolic networks, neuronal networks, or disease spreading
in social networks. Substantial insight into a system's organization
can be obtained by reconstructing the underlying functional network architecture
from the observed activity cascades. Here we focus on Bayesian approaches and
reduce their computational demands by introducing the Iterative Bayesian (IB)
and Posterior Weighted Averaging (PWA) methods. We introduce a special case of
PWA, cast in nonparametric form, which we call the normalized count (NC)
algorithm. NC efficiently reconstructs random and small-world functional network
topologies and architectures from subcritical, critical, and supercritical
cascading dynamics and yields significant improvements over commonly used
correlation methods. With experimental data, NC identified a functional and
structural small-world topology and its corresponding traffic in cortical
networks with neuronal avalanche dynamics