An occupancy problem arising in power law fitting

The power law arises commonly in modeling the number of vertices of a given degree in large graphs. In estimating the degree of the power law, the typical approach is to truncate by eye the log-log plot, then fit a linear equation to the remaining log-transformed data. Here we formulate a hard-coded truncation rule to replace the visual truncation, justify it by showing that the truncation point goes to infinity and misses a vanishing fraction of the data with probability tending to one, and refine the subsequent regression with a weighting and a way to use the covariation between slope and intercept to optimize the slope estimate.