Morphological inversion of complex diffusion.

Epidemics, neural cascades, power failures, and many other phenomena can be described by a diffusion process on a network. To identify the causal origins of a spread, it is often necessary to identify the triggering initial node. Here, we define a new morphological operator and use it to detect the origin of a diffusive front, given the final state of a complex network. Our method performs better than algorithms based on distance (closeness) and Jordan centrality. More importantly, our method is applicable regardless of the specifics of the forward model, and therefore can be applied to a wide range of systems such as identifying the patient zero in an epidemic, pinpointing the neuron that triggers a cascade, identifying the original malfunction that causes a catastrophic infrastructure failure, and inferring the ancestral species from which a heterogeneous population evolves.