GLORI: GRAPH-LOCAL ORDERING FROM RESIDUAL INSTABILITY ACROSS ENVIRONMENTS

We study causal ordering from multi-environment data when an undirected graph skeleton is available but intervention targets are unknown. Our starting observation is that a stable causal mechanism can leave similar prediction errors across envi- ronments, whereas reverse prediction can change with the input distribution. We introduce GLORI (Graph-Local Ordering from Residual Instability). For each node, GLORI trains one predictor using all environments together, evaluates the same predictor separately in each environment, and measures changes in its environment- wise residual mean. The resulting node instability scores define a causal order. We establish the residual-instability contrast theoretically in a tractable one-root setting and then evaluate whether the same signal remains effective beyond this setting. Across 1,440 synthetic graph tasks spanning diverse graph structures and causal mechanisms, as well as real-world CausalBench data, GLORI consistently demonstrates strong causal-ordering performance while remaining computationally efficient relative to the compared methods.