  Gaussian Boson Sampling is a promising method for experimental demonstrations
of quantum advantage because it is easier to implement than other comparable
schemes. While most of the properties of Gaussian Boson Sampling are understood
to the same degree as for these other schemes, we understand relatively little
about the statistical properties of its output distribution. The most relevant
statistical property, from the perspective of demonstrating quantum advantage,
is the anticoncentration of the output distribution as measured by its second
moment. The degree of anticoncentration features in arguments for the
complexity-theoretic hardness of Gaussian Boson Sampling, and it is also
important to know when using cross-entropy benchmarking to verify experimental
performance. In this work, we develop a graph-theoretic framework for analyzing
the moments of the Gaussian Boson Sampling distribution. Using this framework,
we show that Gaussian Boson Sampling undergoes a transition in
anticoncentration as a function of the number of modes that are initially
squeezed compared to the number of photons measured at the end of the circuit.
When the number of initially squeezed modes scales sufficiently slowly with the
number of photons, there is a lack of anticoncentration. However, if the number
of initially squeezed modes scales quickly enough, the output probabilities
anticoncentrate weakly.
