  We investigate the asymptotic number of induced subgraphs in power-law
uniform random graphs. We show that these induced subgraphs appear typically on
vertices with specific degrees, which are found by solving an optimization
problem. Furthermore, we show that this optimization problem allows to design a
linear-time, randomized algorithm that distinguishes between uniform random
graphs and random graph models that create graphs with approximately a desired
degree sequence: the power-law rank-1 inhomogeneous random graph. This
algorithm uses the fact that some specific induced subgraphs appear
significantly more often in uniform random graphs than in rank-1 inhomogeneous
random graphs.
