Structural Causal Models (SCMs)

Updated 16 October 2025

Structural Causal Models are formal frameworks that use structural equations, directed graphs, and interventions to precisely represent causal relationships.
They extend classical structural equation models by enabling equilibrium analysis of dynamic systems, including cyclic and latent structures.
SCMs facilitate the prediction of intervention effects in fields such as biology, engineering, and social sciences by formalizing cause-effect mechanisms.

A Structural Causal Model (SCM) is a formal framework for encoding, analyzing, and manipulating the causal structure of a system, combining structural equations, a directed graph, and an explicit specification of interventions and counterfactuals. SCMs generalize classical structural equation models by treating variables as generated by (possibly stochastic or deterministic) functions of their parents and exogenous noise. The resulting framework underlies the modern theory of causal inference, providing both mathematical rigor and clear intervention semantics. SCMs are foundational for modeling equilibrium outcomes of dynamical systems, handling systems with cycles or latent confounders, and enabling identification of interventional and counterfactual effects.

1. Fundamental Structure and Mathematical Definition

An SCM is represented as a tuple $\mathcal{M} = \langle \mathcal{X}, \mathcal{U}, \mathcal{F}, P(\mathcal{U})\rangle$ :

$\mathcal{X}$ : a set of endogenous (observed) variables, each with a finite domain.
$\mathcal{U}$ : a set of exogenous (latent, unobserved, or background) variables.
$\mathcal{F}$ : a set of measurable structural functions $\{f_i\}$ , where each $X_i \in \mathcal{X}$ is assigned via $X_i = f_i(\mathrm{pa}(X_i), U_i)$ , with $\mathrm{pa}(X_i) \subseteq \mathcal{X} \setminus \{X_i\}$ the parent variables.
$P(\mathcal{U})$ : a joint probability distribution over the exogenous variables.

The functional relationships $\mathcal{F}$ define a (possibly cyclic) directed graph, with edges from each parent $X_j$ to $X_i$ if $X_j$ appears as an input to $f_i$ . In acyclic settings, the model can be solved sequentially; with cycles, simultaneous solutions or additional solvability constraints are required (Bongers et al., 2016).

Interventions are defined via the do-operator: an intervention $do(X_i = \xi_i)$ replaces the structural function $f_i$ with a constant assignment $X_i = \xi_i$ . This operation truncates the graphical and functional structure, yielding a modified model with new interventional semantics.

2. Equilibrium, Dynamics, and Cyclic SCMs

SCMs can be naturally derived from the equilibrium states of dynamical systems described by ordinary differential equations (ODEs). Consider a system of ODEs: $\dot{X}_i(t) = f_i(\mathbf{X}_{\operatorname{pa}_\mathcal{D}(i)}), \quad X_i(0) = (X_0)_i, \quad \forall i \in \mathcal{I}$ where $\operatorname{pa}_\mathcal{D}(i)$ indexes the variables that $\dot{X}_i$ depends on. At equilibrium ( $t \to \infty$ ), the derivatives vanish: $0 = f_i(\mathbf{X}_{\operatorname{pa}_\mathcal{D}(i)}), \quad \forall i$ Under strong stability and structural solvability conditions, one can solve explicitly to obtain assignments: $X_i = h_i(\mathbf{X}_{\operatorname{pa}_\mathcal{M}(i)}), \text{ where } \operatorname{pa}_\mathcal{M}(i) = \operatorname{pa}_\mathcal{E}(i) \setminus \{i\}$ forming the canonical deterministic SCM equations.

Interventions are reflected by replacing the relevant equilibrium equation with $0 = X_i - \xi_i$ (for perfect interventions) (Mooij et al., 2013, Mooij et al., 2014).

For cyclic models (i.e., systems where the causal graph contains feedback loops), this procedure enables a causal interpretation at equilibrium without requiring time ordering, provided unique, stable equilibrium points exist for every possible intervention. The construction extends SCM applicability beyond acyclic systems, allowing for the rigorous causal analysis of many physical, engineering, and biological systems modeled via cyclic ODEs (Mooij et al., 2013, Mooij et al., 2014).

3. Assumptions and Solvability Conditions

The validity of an SCM representation, particularly one induced from equilibrium ODEs or for cyclic/latent systems, rests on several crucial conditions:

Global Stability: For any initial condition, the dynamical system converges to a unique equilibrium $X^*$ . Formally,

$\lim_{t\to\infty} X(t) = X^*$

for all $X(0)$ (Mooij et al., 2014).

Interventional Stability: Modified dynamics after interventions (even those fixing arbitrary subsets of variables) must also converge to unique equilibria.
Structural Solvability: For each variable $X_i$ , there exists a subset $I_i \supseteq \operatorname{pa}_\mathcal{D}(i) \setminus \{i\}$ such that, under intervention $do(\mathbf{X}_{I_i} = \boldsymbol{\xi}_{I_i})$ , the equilibrium equation for $X_i$ admits a unique solution.
Jacobian Condition: In standard examples (e.g., damped oscillators), local stability is ensured if the Jacobian of the equilibrium dynamics has eigenvalues with strictly negative real parts, precluding oscillatory or divergent behaviors.

These requirements guarantee the well-posedness of both the equilibrium and interventional semantics, and allow the labeled equilibrium equations (LEEs) to be translated faithfully into an SCM. The equilibrium SCM is, in general, not unique to the underlying ODE system: multiple dynamical systems may induce the same equilibrium SCM, which has implications for identifiability (Mooij et al., 2013, Mooij et al., 2014).

4. Causality, Intervention Semantics, and Cycles

SCMs interpret causality via their directed structure and intervention semantics. Each equation $X_i = h_i(\text{parents})$ encodes a local, modular causal mechanism, which can be "mutated" independently by interventions, a property critical for counterfactual and interventional analysis. For interventions on $X_i$ , the structural equation is replaced, modifying the joint distribution in accordance with the truncated factorization: $P(\mathbf{X}) = \prod_{j} P(X_j \mid \mathrm{pa}_S(X_j)); \quad \text{after } do(X_i = \xi_i), f_i \text{ replaced by } X_i := \xi_i$ where $\mathrm{pa}_S(\cdot)$ are parents in the current structure.

In cyclic models, despite the presence of feedback, interventions remain well-defined by the replacement of equilibrium equations, as long as system solvability is retained (Mooij et al., 2013). The equilibrium approach thereby obviates the need for time ordering to assign causal directionality, situating SCMs as a natural framework for static causal analysis of dynamical systems.

5. Applications and Implications

SCMs provide a unified formalism applicable to a range of scientific domains:

Population Biology: The Lotka–Volterra predator–prey system, which is cyclic and admits multiple equilibria, can have its steady-state interventions modeled via an equilibrium SCM (Mooij et al., 2013).
Physical & Engineering Systems: Systems such as damped harmonic oscillators and mass–spring networks, influenced by external control, can be studied causally in steady state even with limited temporal data.
Biochemical Networks: Equilibrium causal inference is applicable when only stationary concentrations or outputs are available.
Data with Partial Time Resolution: When system dynamics are faster than measurement granularity, SCMs encode causal constraints at the measurement time scale.

In all these domains, intervention analysis via the SCM allows one to predict the effect of perturbations, control policies, or environmental changes, relying only on equilibrium data and known structure, rather than requiring full time-series observations.

6. Theoretical Developments and Extensions

The equilibrium SCM framework constitutes a rigorous bridge from time-dependent ODEs to static causal analysis, underlining the role of labeled equilibrium equations as the mediators between dynamics and causal semantics. The labeling is essential for preserving the mapping between structural equations and intervention targets, particularly in cyclic models.

This approach clarifies that the same SCM can arise from disparate underlying dynamics, emphasizing the partial identifiability of system mechanisms from equilibrium observations alone. The method points toward extensions for stochastic systems and hidden confounding, which are active research areas.

Further work generalizes SCMs to account for situations where equilibrium outcomes depend on initial conditions or conservation laws, leading to frameworks such as causal constraints models (CCMs) (Blom et al., 2018), and enhancing the handling of functional laws and deterministic constraints outside the expressive power of standard SCMs.

7. Summary Table: Key Elements of SCMs from ODEs

Concept	Mathematical Formulation	Required Condition
ODE Dynamics	$\dot{X}_i(t) = f_i(\mathbf{X}_{\operatorname{pa}_\mathcal{D}(i)})$	Dynamics specified
Equilibrium Equation	$0 = f_i(\mathbf{X}_{\operatorname{pa}_\mathcal{D}(i)})$	Global stability
Labeled Eqn (LEE)	As above, labeled by $i$	Structural solvability
Intervened Eqn	$0 = X_i - \xi_i$ (for $i \in I$ )	Interventional stability
SCM Structural Eqn	$X_i = h_i(\mathrm{pa}_\mathcal{M}(i))$	Unique solvability/solutions

This formalization supports both rigorous causal analysis at equilibrium and a systematic mapping between dynamic and static causal modeling.

For a thorough treatment of the above topics, including formal definitions, stability criteria, the treatment of cycles, and detailed proofs, see (Mooij et al., 2013) and (Mooij et al., 2014).