  The $\beta$-model has been extensively utilized to model degree heterogeneity
in networks, wherein each node is assigned a unique parameter. In this article,
we consider the hypothesis testing problem that two nodes $i$ and $j$ of a
$\beta$-model have the same node parameter. We prove that the null distribution
of the proposed statistic converges in distribution to the standard normal
distribution. Further, we investigate the homogeneous test for $\beta$-model by
combining individual $p$-values to aggregate small effects of multiple tests.
Both simulation studies and real-world data examples indicate that the proposed
method works well.
