  Testing network effects in weighted directed networks is a foundational
problem in econometrics, sociology, and psychology. Yet, the prevalent edge
dependency poses a significant methodological challenge. Most existing methods
are model-based and come with stringent assumptions, limiting their
applicability. In response, we introduce a novel, fully nonparametric framework
that requires only minimal regularity assumptions. While inspired by recent
developments in $U$-statistic literature (arXiv:1712.00771, arXiv:2004.06615),
our approach notably broadens their scopes. Specifically, we identified and
carefully addressed the challenge of indeterminate degeneracy in the test
statistics $-$ a problem that aforementioned tools do not handle. We
established Berry-Esseen type bound for the accuracy of type-I error rate
control. Using original analysis, we also proved the minimax optimality of our
test's power. Simulations underscore the superiority of our method in
computation speed, accuracy, and numerical robustness compared to competing
methods. We also applied our method to the U.S. faculty hiring network data and
discovered intriguing findings.
