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Graph Surgery and the Do-Operator: A Precise Correspondence for Acyclic Structural Causal Models

This paper establishes a precise mathematical correspondence between the functional replacement of mechanisms and the graphical deletion of arrows (graph surgery) in acyclic structural causal models, proving that the dependencies removed by the do-operator exactly match those removed by graph surgery when the model's graph accurately reflects the underlying mechanism dependencies.

Original authors: Satpreet Makhija

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Satpreet Makhija

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of cause and effect, scientists often build maps to understand how one thing leads to another. Imagine a network of variables, like a series of switches and levers, where the position of one determines the behavior of the next. In these models, a "mechanism" is simply the rule that explains how a specific variable gets its value based on the inputs it receives. Sometimes, researchers want to see what happens if they force a variable to take a specific value, ignoring its usual rule. This act of forcing a value is called an intervention. To visualize this, scientists have long used two different methods. One method involves looking at the map and cutting away the arrows that point toward the variable being forced, effectively saying, "This part of the network no longer listens to its neighbors." The other method involves going into the code of the system and replacing the rule for that variable with a fixed number, saying, "This variable is now stuck at this value." For years, it was assumed these two methods were just different ways of describing the same reality, but no one had proven they were mathematically identical in every detail.

A researcher named Satpreet Makhija has now provided that proof, showing exactly how these two approaches align for a specific class of systems. The work focuses on deterministic models, which are systems where the outcome is completely fixed by the inputs, with no element of chance involved. The study confirms that if you take a set of rules and replace the rules for certain variables with fixed numbers, the resulting pattern of dependencies is exactly the same as if you had started with the map of those dependencies and simply erased the arrows pointing to the variables you changed. This might sound like a technicality, but it is a crucial step in making the language of cause and effect precise. It bridges the gap between the visual map and the functional code, ensuring that when we say we are "cutting" a connection, we are truly removing the influence of that connection in the system's behavior.

The paper also clarifies a subtle but important distinction that often gets overlooked. In many models, the map provided by a researcher might include arrows that represent potential connections, even if the actual rules of the system do not use them. Think of a blueprint that shows a pipe running to a wall, even though the faucet on that pipe is never turned on. If a researcher performs an intervention on such a system, the visual map might still show that unused pipe, while the functional rules would show that the connection is effectively gone. Makhija's work proves that the two methods only produce the exact same map if the original map was perfectly accurate to begin with, containing no unused arrows. If the map had extra, unused lines, the visual surgery would leave them there, while the functional replacement would not. This finding tells us that for the two views to match perfectly, the map must be a true reflection of the active rules, not just a list of possibilities.

Beyond this core correspondence, the study explores what happens when multiple interventions are applied one after another. It establishes a clear rule for how these actions combine: if you change a variable, and then later change it again, the second change is the one that matters. The system does not remember the first change; it simply adopts the new value. This holds true whether you are looking at the map or the rules. The research also investigates how far the effects of an intervention travel. It turns out that the final value of a specific variable depends only on the interventions applied to the variables that directly or indirectly feed into it. If you change a variable that has no path leading to your variable of interest, your change will have no effect on the outcome. This allows researchers to ignore large parts of a complex system when they are only interested in a specific result, focusing only on the relevant history of that result.

The significance of this work lies in its precision. By proving that the visual act of cutting arrows and the functional act of replacing rules lead to the same dependency structure, the study removes ambiguity from the way we think about interventions. It confirms that the standard way of thinking about these problems is not just a convenient shortcut, but a mathematically sound description of reality. The results apply to systems where the variables are finite and the relationships do not loop back on themselves, which covers a vast array of practical scenarios in science and engineering. The paper does not claim to solve every problem in causal reasoning, but it does settle a fundamental question about how we represent changes in these systems. It ensures that when we speak of "doing" something to a system, whether we are drawing on a map or writing a rule, we are describing the same physical change in the world.

Technical Summary: Graph Surgery and the Do-Operator

Problem Statement
In Structural Causal Models (SCMs), the do-operator represents an intervention where specific variables are forced to take constant values, decoupling them from their usual causal mechanisms. Two distinct perspectives describe this operation:

Graphical View (Graph Surgery): Deleting all incoming arrows to the targeted variables in the causal graph.
Functional View (Mechanism Replacement): Replacing the structural equations (mechanisms) of the targeted variables with constant functions.

While these views are widely treated as equivalent, the paper notes that this equivalence is not yet a precise mathematical statement. Graph surgery operates on the graph structure and loses information about the specific values imposed, whereas mechanism replacement operates on functions and retains those values. Furthermore, a causal model often consists of a pair $(G, F)$ where the supplied graph $G$ may contain "unused arrows" (dependencies permitted by the graph but not utilized by the mechanisms $F$ ). It remains unclear when graph surgery performed on the supplied $G$ yields the same result as extracting the dependencies from the modified mechanisms.

Methodology
The paper focuses on deterministic, acyclic structural causal models with a finite set of endogenous variables $V$ . The methodology involves:

Formal Definitions: Defining the dependency graph $Graph(F)$ strictly based on the functional dependencies of the mechanism family $F$ , distinct from any supplied graph $G$ .
Intervention Formalism: Representing interventions as partial maps $\iota$ on $V$ that assign constant values to a target set $T_\iota$ .
Comparative Analysis: Systematically comparing the outcome of two operations:
1. Replacing mechanisms with constants and then extracting the dependency graph ( $Graph(F^\iota)$ ).
2. Extracting the dependency graph first and then performing graph surgery ( $Surg(Graph(F), T_\iota)$ ).
Model Characterization: Analyzing the properties of the resulting "intervened model" $Do(M, \iota)$ , including the behavior of sequential interventions and the dependence of outcomes on specific subsets of variables.

Key Contributions and Results

Exact Correspondence Theorem (Theorem 1):
The paper proves that for any mechanism family $F$ and intervention $\iota$ , the dependency graph of the modified mechanisms is identical to the graph obtained by performing surgery on the original dependency graph:
$Graph(F^\iota) = Surg(Graph(F), T_\iota)$
This establishes that replacing target mechanisms with constants removes exactly the dependencies removed by graph surgery, providing a rigorous mathematical link between the functional and graphical views.
Characterization of Model Exactness (Corollary 1):
The author addresses the relationship between the supplied graph $G$ and the true dependency graph $Graph(F)$. They prove that $Graph(F^\iota) = Surg(G, T_\iota)$ holds for a specific intervention if and only if every arrow in $G$ that is absent from $Graph(F)$ targets a variable in the intervention set. Crucially, the equality holds for every possible intervention if and only if the original model is exact (i.e., $G = Graph(F)$). This clarifies that graph surgery on a supplied graph is only guaranteed to match the functional reality if the supplied graph contains no unused arrows.
Sequential Interventions (Theorem 2):
The paper derives a law for combining sequential interventions. If intervention $\iota$ is followed by $\kappa$ , the resulting model is equivalent to a single intervention $\lambda$ that retains the value of $\kappa$ wherever $\kappa$ is defined, and $\iota$ otherwise. This confirms that later assignments override earlier ones, and the order of non-overlapping interventions is commutative.
Ancestor Theorem (Theorem 3):
The paper proves that the outcome of a set of variables $O$ depends only on the intervention values applied to their actual dependency ancestors in $Graph(F)$. Interventions on variables outside the ancestor set of $O$ (even if they are ancestors in the supplied graph $G$ ) do not affect the outcome of $O$ . This allows for the pruning of irrelevant intervention targets when analyzing specific outcomes.

Significance and Claims
The paper claims to resolve a conjecture by Zhang regarding the equivalence of the syntactic (graph surgery) and semantic (mechanism replacement) views of the do-operator. By formulating this equivalence at the level of dependencies, the work provides a precise mathematical foundation suitable for mechanization (e.g., in proof assistants like Coq).

The author emphasizes that their results distinguish between the actual dependencies of a model and the permitted dependencies of a supplied graph. This distinction is vital for understanding when a supplied causal graph accurately reflects the underlying mechanisms. The paper does not propose new applications or experimental proposals; rather, it offers a foundational clarification of the do-operator's structural and functional components, ensuring that the resulting intervened model is well-defined and that its run (the unique solution to the equations) is correctly characterized. The work concludes that while the two constructions (surgery on $G$ vs. surgery on $Graph(F)$) may yield different graphs if $G$ is not exact, they always yield the same run (outcome), and they yield the same model if and only if the starting model is exact.

Technical Summary: Graph Surgery and the Do-Operator

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