Characterizing branching processes from sampled data

Branching processes model the evolution of populations of agents that randomly generate offspring (children). These processes, more patently Galton-Watson processes, are widely used to model biological, social, cognitive, and technological phenomena, such as the diffusion of ideas, knowledge, chain letters, viruses, and the evolution of humans through their Y-chromosome DNA or mitochondrial RNA. A practical challenge of modeling real phenomena using a Galton-Watson process is the choice of the offspring distribution, which must be measured from the population. In most cases, however, directly measuring the offspring distribution is unrealistic due to lack of resources or the death of agents. So far, researchers have relied on informed guesses to guide their choice of offspring distribution. In this work we propose two methods to estimate the offspring distribution from real sampled data. Using a small sampled fraction of the agents and instrumented with the identity of the ancestors of the sampled agents, we show that accurate offspring distribution estimates can be obtained by sampling as little as 14% of the population.