Degree Variance in Bernoulli Graphs

Jan Hagberg

**Abstract**

Exact and asymptotic distributions
of the degree variance are investigated for Bernoulli graphs and uniform
random graphs. In particular the range of values of the degree variance
and its maximum value are considered. We show that the degree variance
is approximately gamma distributed with parameters obtained from the first
two moments of the degree variance. Since centrality of a graph can be
interpreted as a measure of its heterogeneity in terms of vertex degrees,
we can perform a centrality test with a critical value obtained from the
gamma distribution.

Key words: Centrality Testing, Bernoulli Graphs, Degree Variance, Gamma Approximation, Uniform Random Graphs