Optimal testing methods measure differences between network groups accurately
Two-Sample Testing for Inhomogeneous Random Graphs in Non-Integral $L_r$ Norms
Machine Learning
Summary
Comparing two groups of networks to see if they have the same connection patterns is a common problem. The difficulty depends on how you measure their difference. The authors studied this problem for certain tricky measurements called non-integer L_r norms and found a new testing method that meets the best expected performance. Their solution smartly combines two known tests to handle different types of changes in connections. Simulations showed the method works well and matches theoretical predictions.
What this means in practice
- •For network analysts: Detect differences in large network datasets with fewer samples by using combined statistical tests optimized for subtle or widespread changes.
- •For cybersecurity teams: Improve detection of changes in communication or infrastructure networks by applying precise tests that adapt to different patterns of anomalies.
Tested on simulated data.
Authors
Soham Dan
Abstract
Testing whether two populations of networks share the same edge probabilities is a basic problem in network inference. How hard it is depends on the norm used to measure the difference. For the inhomogeneous Erdős--Rényi (IER) model, the optimal sample complexity is known for every integer $L_r$ norm and for $1\le r<2$. For non-integral $r>2$, however, the known upper and lower bounds do not match, and the lower bound was conjectured to be tight. We study this gap for two-sample testing on aligned vertices. We propose a test that runs two published statistics, of orders $2$ and $\lceil r\rceil$, on the same data and rejects if either one rejects. Its thresholds come from Hölder interpolation, so that both statistics have the same sample cost. We prove that this test attains the conjectured rate. Combined with earlier results, this shows that for every fixed $r\ge1$ the minimax sample complexity is of order $n^{\max\{4/r-1,\,2/r\}}/ε^2$, even when the separation changes with $n$. In simulations with $n$ between 32 and 256, the number of graphs needed for 80\% power at level $0.05$ grows with $n$ at a rate consistent with the theory. For $r=2.5$, for example, the fitted exponent is $0.78$, against the theoretical value $0.8$. Interestingly, the two statistics split the work as the interpolation argument suggests: the higher-order statistic is more powerful when only a few edges change, and the $L_2$ statistic when many edges change.