Graph Surgery and Do-Operator Equivalence in Causal Models

The paper addresses a fundamental concept in causal inference: the relationship between graphical interventions (graph surgery) and functional interventions (the do-operator). While often treated as equivalent, the authors highlight that this equivalence hasn't been mathematically formalized in a precise way, as one operation yields a graph and the other modifies mechanisms and values. For deterministic acyclic structural causal models with a finite number of endogenous variables, the research provides a rigorous mathematical statement. It demonstrates that replacing target mechanisms functionally removes exactly the same dependencies that are removed by performing graph surgery on the model's dependency graph. This establishes a precise correspondence between these two representations of intervention. The work further defines the intervened model, characterizes its behavior, and explains how sequential interventions combine. It also proves that an outcome's dependency is solely on interventions affecting its actual dependency ancestors, providing a clearer understanding of causal effects within these models.