Understanding causal relations in dynamic systems is essential

in epidemiology.
While causal inference methods have been extensively studied, they often rely on fully specified causal graphs, which may not always be available

in complex dynamic systems. Partially specified causal graphs, such as summary causal graphs (SCGs), provide a simplified representation of causal relations, omitting temporal information and focusing on high-level causal structures.
This simplification introduces new challenges concerning the types of queries of interest: macro queries, which involve relationships between clusters represented as vertices in the graph, and micro queries, which pertain to relationships between variables that are not directly visible through the vertices of the graph.
In this paper, we first clearly distinguish between macro conditional independencies and micro conditional independencies and between macro total effects and micro total effects. Then, we demonstrate the soundness and completeness of the d-separation to identify macro conditional independencies in SCGs. Furthermore, we establish that the do-calculus is sound and complete for identifying macro total effects in SCGs.
Finally, we give a graphical characterization for the non-identifiability of macro total effects in SCGs.

§ INTRODUCTION

In observational studies, causal graphs are vital for decision-making regarding interventions and can be useful for detecting independencies between variables. Several standard tools have been developed for fully specified causal graphs, typically represented as directed acyclic graphs (DAGs) or acyclic directed mixed graphs (ADMGs). For example, d-separation [Pearl, 1988] enables the detection of conditional independencies directly from the graph. Similarly, the do-calculus [Pearl, 1995] allows for the derivation of total effect estimands from observational data. These results were also extended to directed mixed graphs (DMGs) [Richardson, 1997, Forré and Mooij, 2017, Forré and Mooij, 2018, Forré and Mooij, 2020].
However, constructing a fully specified causal graph is challenging because it requires knowledge of the causal relations among all pairs of observed variables. This knowledge is often unavailable, particularly in complex, high-dimensional settings, thus limiting the applicability of causal inference theory and tools.
Therefore, recently there has been more interest in partially specified causal graphs [Perkovic, 2020, Anand et al., 2023, Ferreira and Assaad, 2024, Assaad et al., 2024].

An important type of partially specified graphs is the cluster-DMG which provides a coarser representation of causal relations through vertices that represent a cluster of variables (and can contain cycles).
The partial specification of cluster-DMG arises from the fact that each vertex in an cluster-DMG represents a cluster of variables.
A well-known special case of cluster-DMGs is the cluster-ADMG, which imposes an acyclicity assumption on the graph [Anand et al., 2023]. Another notable special case of cluster-DMGs in dynamic systems is the summary causal graph (SCG), where each cluster represents a time series or repeated measurements through time of the same variable in a longitudinal study, and all temporal information is omitted from the graph [Assaad et al., 2022].

In SCGs (or any cluster-DMG), there are two primary types of queries of interest. The first, called a micro query, involves considering one or a few specific variables (but not all) represented by a vertex in an SCG. The second, called a macro query, involves considering the whole cluster of variables represented by a vertex in an SCG.
This paper focuses on two types of macro queries: macro conditional independencies and macro total effects. Specifically, we show that the d-separation criterion and the do-calculus to respectively identify macro conditional independencies and macro total effects in SCGs. By addressing these macro queries, we aim to provide practical solutions for inferring causal effects in real-world applications where causal relations are only partially understood.
In particular, the concept of macro queries is crucial for public health strategies and epidemiological modeling. For example, the macro total effect can be used to understand the impact of the COVID-19 pandemic on the flu epidemic and vice versa [Boëlle et al., 2020, Ferguson et al., 2003], shedding light on the broader impacts of interventions and natural immunity dynamics over extended periods.

The remainder of the paper is organized as follows: Firstly, we formally presents SCGs as well as micro and macro queries.

Secondly, we show that d-separation is sound and complete for macro conditional independencies in SCGs.
Thirdly, we show that the do-calculus is sound and complete for macro total effects in SCGs and present a graphical characterization for the non-identifiability of these effects.

After that, we discuss related works.
Finally, we conclude the paper while showing its limitations.

§ SUMMARY CAUSAL GRAPHS AND MACRO QUERIES

We suppose that in a dynamic system, observations are generated from a discrete-time dynamic structural causal model
(DTDSCM), an extension of structural causal models [Pearl, 2009] to dynamic systems. In the following we use the convention that uppercase letters represent variables and lowercase letters represent their values, moreover letters in blackboard bold represent sets.

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latent/.append style=white, fill=black

[fill=red!30] (X) at (0,0) $X$ ;
[fill=blue!30] (Y) at (2.4,0) $Y$;
(Z) at (1.2,0) $W$;

[transform canvas=yshift=-.25em]
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[->,>=latex] (Z) – (Y);

[transform canvas=yshift=.25em]
[<-,>=latex] (Z) – (Y);

[<->,>=latex, dashed] (X) to [out=90,in=90, looseness=1] (Y);

[->,>=latex] (X) to [out=165,in=120, looseness=2] (X);
[->,>=latex] (Y) to [out=15,in=60, looseness=2] (Y);

[->,>=latex] (Z) to [out=15,in=60, looseness=2] (Z);

[<->,>=latex, dashed] (X) to [out=-155,in=-110, looseness=2] (X);
[<->,>=latex, dashed] (Y) to [out=-25,in=-70, looseness=2] (Y);
[<->,>=latex, dashed] (Z) to [out=-25,in=-70, looseness=2] (Z);

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latent/.append style=white, fill=black

(Z) at (1.2,1.2) $W_t$;
(Z-1) at (0,1.2) $W_{t-1}$;
(Z-2) at (-1.2,1.2) $W_{t-2}$;
[fill=blue!30] (Y) at (1.2,0) $Y_t$;
[fill=blue!30] (Y-1) at (0,0) $Y_{t-1}$;
[fill=blue!30]  (Y-2) at (-1.2,0) $Y_{t-2}$;
[fill=red!30] (X) at (1.2,2.4) $X_t$;
[fill=red!30] (X-1) at (0,2.4) $X_{t-1}$;
[fill=red!30]  (X-2) at (-1.2,2.4) $X_{t-2}$;

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[->,>=latex] (X-1) to (Z-1);
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[->,>=latex] (Z-1) – (Y-1);
[->,>=latex] (Z) – (Y);

[->,>=latex] (X-2) to (Z-1);
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[->,>=latex] (Z-2) – (Y-1);
[->,>=latex] (Z-1) – (Y);
[->,>=latex] (Y-2) to (Z-1);
[->,>=latex] (Y-1) to (Z);

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[->,>=latex] (Y-2) – (Y-1);
[->,>=latex] (Y-1) – (Y);
[->,>=latex] (Z-2) – (Z-1);
[->,>=latex] (Z-1) – (Z);

[<->,>=latex, dashed] (X-2) to [out=180,in=180, looseness=1] (Y-2);
[<->,>=latex, dashed] (X-1) to [out=30,in=-30, looseness=0.5] (Y-1);
[<->,>=latex, dashed] (X) to [out=0,in=0, looseness=1] (Y);
[<->,>=latex, dashed] (X-2) to (Y-1);
[<->,>=latex, dashed] (X-1) to (Y);
[<->,>=latex, dashed] (X-2) to [out=80,in=100, looseness=1] (X-1);
[<->,>=latex, dashed] (X-1) to [out=80,in=100, looseness=1] (X);
[<->,>=latex, dashed] (Y-2) to [out=-80,in=-100, looseness=1] (Y-1);
[<->,>=latex, dashed] (Y-1) to [out=-80,in=-100, looseness=1] (Y);

[dashed,>=latex] (X-2) to[left] (-2,2.4);
[dashed,>=latex] (Z-2) to[left] (-2,1.2);
[dashed,>=latex] (Y-2) to[left] (-2,0);
[dashed,>=latex] (X) to[right] (2,2.4);
[dashed,>=latex] (Z) to[right] (2,1.2);
[dashed,>=latex] (Y) to[right] (2,0);
Compatible FT-ADMG 1.

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(Z) at (1.2,1.2) $W_t$;
(Z-1) at (0,1.2) $W_{t-1}$;
(Z-2) at (-1.2,1.2) $W_{t-2}$;
[fill=blue!30] (Y) at (1.2,0) $Y_t$;
[fill=blue!30] (Y-1) at (0,0) $Y_{t-1}$;
[fill=blue!30]  (Y-2) at (-1.2,0) $Y_{t-2}$;
[fill=red!30] (X) at (1.2,2.4) $X_t$;
[fill=red!30] (X-1) at (0,2.4) $X_{t-1}$;
[fill=red!30]  (X-2) at (-1.2,2.4) $X_{t-2}$;

[->,>=latex] (Z-2) to (X-2);
[->,>=latex] (Z-1) to (X-1);
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[->,>=latex] (Y-1) – (Z-1);
[->,>=latex] (Y) – (Z);

[->,>=latex] (X-2) to (Z-1);
[->,>=latex] (X-1) to (Z);
[->,>=latex] (Z-2) – (X-1);
[->,>=latex] (Z-1) – (X);
[->,>=latex] (Z-2) – (Y-1);
[->,>=latex] (Z-1) – (Y);
[->,>=latex] (Y-2) to (Z-1);
[->,>=latex] (Y-1) to (Z);

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[->,>=latex] (Z-2) – (Z-1);
[->,>=latex] (Z-1) – (Z);

[<->,>=latex, dashed] (X-2) to [out=180,in=180, looseness=1] (Y-2);
[<->,>=latex, dashed] (X-1) to [out=30,in=-30, looseness=0.5] (Y-1);
[<->,>=latex, dashed] (X) to [out=0,in=0, looseness=1] (Y);
[<->,>=latex, dashed] (X-2) to (Y-1);
[<->,>=latex, dashed] (X-1) to (Y);
[<->,>=latex, dashed] (X-2) to [out=80,in=100, looseness=1] (X-1);
[<->,>=latex, dashed] (X-1) to [out=80,in=100, looseness=1] (X);
[<->,>=latex, dashed] (Y-2) to [out=-80,in=-100, looseness=1] (Y-1);
[<->,>=latex, dashed] (Y-1) to [out=-80,in=-100, looseness=1] (Y);

[dashed,>=latex] (X-2) to[left] (-2,2.4);
[dashed,>=latex] (Z-2) to[left] (-2,1.2);
[dashed,>=latex] (Y-2) to[left] (-2,0);
[dashed,>=latex] (X) to[right] (2,2.4);
[dashed,>=latex] (Z) to[right] (2,1.2);
[dashed,>=latex] (Y) to[right] (2,0);

Compatible FT-ADMG 2.

An SCG with two compatible FT-ADMGs. Each pair of red and blue vertices represents the macro query we are interested in.

A discrete-time dynamic structural causal model is a tuple $\mathcal{M}=(\mathbb{L}, \mathbb{V}, \mathbb{F},  \proba{\mathbb{l}})$,
where $\mathbb{L} = \bigcup \{\mathbb{L}^{v^i_t} \mid i \in [1,d], t \in [t_0,t_{max}]\}$ is a set of exogenous variables, which cannot be observed but affect the rest of the model.
$\mathbb{V} = \bigcup \{\mathbb{V}^i \mid i \in [1,d]\}$ such that $\forall i \in [1,d]$, $\mathbb{V}^{i} = \{V^{i}_{t} \mid t \in [t_0,t_{max}]\}$, is a set of endogenous variables, which are observed and every $V^i_t \in \mathbb{V}$ is functionally dependent on some subset of $\mathbb{L}^{v^i_t} \cup \mathbb{V}_{\leq t} \backslash \{V^i_t\}$ where $\mathbb{V}_{\leq t} = \{V^j_{t'} \mid j \in [1,d], t'\leq t\}$.
$\mathbb{F}$ is a set of functions such that for all $V^i_t \in \mathbb{V}$, $f^{v^i_t}$ is a mapping from $\mathbb{L}^{v^i_t}$ and a subset of $\mathbb{V}_{\leq t} \backslash \{V^i_t\}$ to $V^i_t$.
$\proba{\mathbb{l}}$ is a joint probability distribution over $\mathbb{L}$.

We suppose that the direct causal relations $\mathbb{F}$ in a DTDSCM can be qualitatively represented by a full-time acyclic directed mixed graph (FT-ADMG), a special case of ADMGs [Richardson, 2003], where bidirected dashed arrows represent hidden confounding.

Consider a DTDSCM $\mathcal{M}$. The full-time acyclic directed mixed graph (FT-ADMG) $\mathcal{G} = (\mathbb{V}, \mathbb{E})$ induced by $\mathcal{M}$ is defined in the following way:

\begin{equation*}
\begin{aligned}
\mathbb{E}^{1} :=& \{X_{t-\gamma}\rightarrow Y_{t} \mid \forall Y_{t} \in \mathbb{V},~X_{t-\gamma} \in \mathbb{X} \subseteq \mathbb{V}_{\leq t} \backslash \{Y_t\} \\
&\st Y_t := f^{y_t}(\mathbb{X}, \mathbb{L}^{y_t}) \text{ in } \mathcal{M}\},\\
\mathbb{E}^{2} :=& \{X_{t-\gamma}\longdashleftrightarrow Y_{t} \mid \forall X_{t-\gamma}, Y_{t} \in \mathbb{V}\\
&\st \notindep{\mathbb{L}^{x_{t-\gamma}}}{\mathbb{L}^{y_t}\}}{\probasymbol},
\end{aligned}
\end{equation*}

where $\mathbb{E}=\mathbb{E}^{1}\cup \mathbb{E}^{2}$.

In the remainder, we will use the notations of parents, children, ancestors and descendants for FT-ADMGs. Consider an FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$ for every vertex $Y_t \in \mathbb{V}$ we note:
$\parents{Y_t}{\mathcal{G}} = \{X_{t'} \in \mathbb{V} \mid X_{t'} \rightarrow Y_t\}$, $\children{Y_t}{\mathcal{G}} = \{X_{t'} \in \mathbb{V} \mid Y_t \rightarrow X_{t'}\}$, $\ancestors{Y_t}{\mathcal{G}} = \bigcup_{n\in \mathbb{N}}P_n$ where $P_0 = \{Y_t\}$ and $P_{k+1} = \bigcup_{X_{t'} \in P_k} \parents{X_{t'}}{\mathcal{G}}$, $\descendants{Y_t}{\mathcal{G}} = \bigcup_{n\in \mathbb{N}}C_n$ where $C_0 = \{Y_t\}$ and $C_{k+1} = \bigcup_{X_{t'} \in C_k} \children{X_{t'}}{\mathcal{G}}$.

In many domains, such as epidemiology, it is generally challenging for practitioners to validate, analyze, or even provide an FT-ADMG due to the difficulty in determining the temporal lag between a cause and its effect. However, practitioners can usually provide summary causal graphs, which are compact versions of FT-ADMG. These summary causal graphs represent the causal relations between the exposure, the disease and the different factors without specifying the temporal lags of these relations.

Consider an FT-ADMG $\mathcal{G} = (\mathbb{V}, \mathbb{E})$. The summary causal graph (SCG) $\mathcal{G}^{s} = (\mathbb{S}, \mathbb{E}^{{s}})$ compatible with $\mathcal{G}$ is defined in the following way:

\begin{equation*}
\begin{aligned}
% &\mathbb{S} &:=& \{Y &\mid& \forall Y_t \in \mathbb{V}\},&\\
\mathbb{S} :=& \{V^i = (V^i_{t_0},\cdots,V^i_{t_{max}}) \mid \forall i \in [1, d]\},\\
\mathbb{E}^{s1} :=& \{X\rightarrow Y \mid \forall X,Y \in \mathbb{S},~\exists t'\leq t\in [t_0,t_{max}] \\
&\st X_{t'}\rightarrow Y_{t}\in\mathbb{E}\},\\
\mathbb{E}^{s2} :=& \{X\longdashleftrightarrow Y \mid \forall X,Y \in \mathbb{S},~\exists t',t \in [t_0,t_{max}] \\
&\st X_{t'}\longdashleftrightarrow Y_{t}\in\mathbb{E}\}.
\end{aligned}\\
\end{equation*}

where $\mathbb{E}^{s} = \mathbb{E}^{s1}\cup \mathbb{E}^{s2}.$

The transformation from an FT-ADMG to an SCG can be made explicit using the natural transformation from $\mathbb{V}$ to $\mathbb{S}$, $\tau_0$ :

$(v^1_{t_0},\cdots,v^1_{t_{max}},\cdots,v^d_{t_0},\cdots,v^d_{t_{max}}) \mapsto ((v^1_{t_0},\cdots,v^1_{t_{max}}),\cdots,(v^d_{t_0},\cdots,v^d_{t_{max}})).$

The abstraction of SCGs entails that, even though there is exactly one SCG compatible with a given FT-ADMG, there are in general several FT-ADMGs compatible with a given SCG. For example, we give in Figure <ref> an SCG with two of its compatible FT-ADMGs.

While SCGs, like cluster-DMGs, can contain cycles, the cycles in an SCG arise from the partial specificity of the graph.
Notice that, like cluster-DMGs, SCGs may have directed cycles and in particular two directed edges oriented in opposite directions, , if in the FT-ADMG we have $X_{t'}\rightarrow Y_{t}$ and $Y_{t''}\rightarrow X_{t}$ then in the SCG we have $X\rightarrow Y$ and $Y\rightarrow X$ which we often write $X\rightleftarrows Y$.
However, unlike in cluster-DMGs, the cycles in SCGs necessarily arise from the partial specificity of the graph.
Moreover, if $\mathcal{G}^{s} = (\mathbb{S}, \mathbb{E}^{{s}})$ and $\mathcal{G} = (\mathbb{V}, \mathbb{E})$ are compatible, we abuse the notation $\forall V^i \in \mathbb{S}$, $\mathbb{V}^i=\{V^i_{t} \mid t \in [t_0,t_{max}]\}$ given in Definition <ref> by writing $\forall V^i = Y \in \mathbb{S}$, $\mathbb{V}^{Y}=\mathbb{V}^i$ and $\forall \mathbb{Y}\subseteq \mathbb{S}$, $\mathbb{V}^{\mathbb{Y}}=\bigcup\limits_{Y \in \mathbb{Y}}\mathbb{V}^Y$.

In the remainder, we will adopt the terminology of parents, children, ancestors, and descendants for SCGs, which are defined in a manner consistent with those used for FT-ADMGs, , the sets of parents, children, ancestors, and descendants of a vertex $Y$ in an SCG $\mathcal{G}^s$ are respectively given by $\parents{Y}{\mathcal{G}^s}$, $\children{Y}{\mathcal{G}^s}$, $\ancestors{Y}{\mathcal{G}^s}$, and $\descendants{Y}{\mathcal{G}^s}$. Furthermore, a strongly connected component of $Y$ in $\mathcal{G}^s$ denoted as $\scc{V}{\mathcal{G}^s}$ is the set $\ancestors{Y}{\mathcal{G}^s} \cap \descendants{Y}{\mathcal{G}^s}$.

In this paper, we focus on two distinct queries in SCGs: one that pertains to finding conditional independencies and another that infers total effects. Each of these queries can be categorized into two types: macro query and micro query[Papers considering micro queries usually assume stationarity (, consistency throughout time), because it permits the division of a single multivariate time-series into multiple instances, thereby facilitating estimation when each time-point represents a single observation. However, when focusing solely on macro queries, this assumption loses its relevance because stationarity offers no distinct advantage. Instead, it is assumed that there are multiple observations available for each variable because estimating the macro total effect from just a single observation per variable is infeasible.
Below, we define micro conditional independency and macro conditional independency.

A micro independence in the FT-ADMG $\mathcal{G} = (\mathbb{V}, \mathbb{E})$ is a conditional independence between two disjoint sets of temporal variables $\mathbb{X}, \mathbb{Y} \subseteq \mathbb{V}$ conditioned on any subset of temporal variables $\mathbb{W} \subseteq \mathbb{V}\backslash (\mathbb{X}, \mathbb{Y})$.

A macro independence in the FT-ADMG $\mathcal{G} = (\mathbb{V}, \mathbb{E})$ compatible with the SCG $\mathcal{G}^{s} = (\mathbb{S}, \mathbb{E}^{{s}})$ is a conditional independence between two sets of temporal variables of the form $\mathbb{V}^X$ and $\mathbb{V}^Y$ for $X, Y \in \mathbb{S}$ conditioned on a subset of temporal variables of the form $\mathbb{V}^\mathbb{W}$ for $\mathbb{W} \subseteq \mathbb{S}\backslash\{X,Y\}$.

For example, in Figure <ref>, we are interested to know if there is a macro conditional independence between $X$ and $Y$ given $W$ in the SCG which means to know if $\{X_{t} \mid t \in [t_0, t_{max}]\}$ is independent of $\{Y_t \mid t \in [t_0, t_{max}]\}$ given $\{W_t \mid t \in [t_0, t_{max}]\}$. In contrast, a micro conditional independence query would be of the form: "is $X_t$ independent of $Y_{t-1}$ given $W_{t-1}$?".

Note that there exists a fundamental tool called d-separation [Pearl, 1988] that serves for deriving conditional independence properties from causal graphs.
In the following the d-separation of two subsets $\mathbb{X}$ and $\mathbb{Y}$ conditionally on a third one $\mathbb{W}$ in a graph $\mathcal{G}$ is denoted as $\dsepc{\mathbb{X}}{\mathbb{Y}}{\mathbb{W}}{\mathcal{G}}$.

Now, we define micro total effect and macro total effect.

A micro total effects in the FT-ADMG $\mathcal{G} = (\mathbb{V}, \mathbb{E})$ is a total effect from a set of temporal variables $\mathbb{X}$ to another set of temporal variables $\mathbb{Y}$ where $\mathbb{X}, \mathbb{Y} \subseteq \mathbb{V}$, denoted as $\probac{\mathbb{Y} = \mathbb{y}}{\interv{\mathbb{X} = \mathbb{x}}}$.

A macro total effect in the FT-ADMG $\mathcal{G} = (\mathbb{V}, \mathbb{E})$ compatible with the SCG $\mathcal{G}^{s} = (\mathbb{S}, \mathbb{E}^{{s}})$ is a total effect from $\mathbb{V}^X$ to $\mathbb{V}^Y$ where $X, Y \in \mathbb{S}$, denoted as $\probac{\mathbb{V}^Y = \mathbb{v}^y}{\interv{\mathbb{V}^X = \mathbb{v}^x}}$.

Similarly, we will also define the micro and macro interventions in the FT-ADMG respectively as $\mathbb{I}^m = \{\interv{V^i_t = v^i_t} \mid i \in [1,d], t \in [t_0,t_{max}]\}$ and $\mathbb{I}^M = \{\interv{\mathbb{V}^i = \mathbb{v}^i} \mid i \in [1,d]\}$.
Consider also the complete set of interventions in the SCG $\mathbb{I}^s = \{\interv{Y = y} \mid Y \in \mathbb{S}\}$ which corresponds to the macro interventions from the point of view of the SCG.

By a slight abuse of notation, in the remainder we will denote macro total effects as $\probac{\mathbb{v}^y}{\interv{\mathbb{v}^x}}$.
Notice that macro queries are a special case of micro queries.

The following two sections show that, given only the SCG, it is possible to answer the considered macro queries[A bidirected dashed edge on a single vertex in an SCG does not affect the results of the following two sections. Therefore, modelers can choose whether to include these edges in this context or not. However, these edges can be more valuable for micro total effects.] using existing methods.

§ IDENTIFICATION OF MACRO CONDITIONAL INDEPENDENCIES

In this section, we show that d-separation is sound and complete in SCGs for macro queries.
Firstly, since an FT-ADMG is an ADMG, the standard definition of d-separation [Pearl, 1998] is directly applicable for FT-ADMG.
This definition was introduced for ADMGs and was extended to DMGs in [Forré and Mooij, 2018] and to cluster-ADMGs in [Anand et al., 2023].
But it turned out that it is readily extendable to SCGs, which are a special case of cluster-DMGs.

In an SCG $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$, a path $\pi=\langle V^{p_1},\cdots,V^{p_n}\rangle$ is said to be blocked by a set of vertices $\mathbb{W}\subseteq\mathbb{S}$ if:

* $\exists 1 < i < n \st V^{p_{i-1}}\starbarstar V^{p_i}\rightarrow V^{p_{i+1}}$ or $V^{p_{i-1}}\leftarrow V^{p_i}\starbarstar V^{p_{i+1}}$ and $V^{p_i}\in\mathbb{W}$, or

* $\exists 1 < i < n \st V^{p_{i-1}}\stararrow V^{p_i}\arrowstar V^{p_{i+1}}$ and $\descendants{V^{p_i}}{\mathcal{G}^s}\cap\mathbb{W}=\emptyset$.

where $\stararrow$ represents $\rightarrow$ or $\longdashleftrightarrow$, $\arrowstar$ represents $\leftarrow$ or $\longdashleftrightarrow$, and $\starbarstar$ represents any of the three arrow type $\rightarrow$, $\leftarrow$ or $\longdashleftrightarrow$.
A path which is not blocked is said to be active. A set $\mathbb{W}\subseteq \mathbb{S}$ is said to d-separate two sets of variables $\mathbb{X},~\mathbb{Y}\subseteq \mathbb{S}$ if it blocks every path from a variable of $\mathbb{X}$ to a variable of $\mathbb{Y}$.

Before providing the main results of this section, we define the notion of walks in graphs and show how it is related to the notion of paths. This notion is important in SCGs since they contain cycles and the relation between paths and walks will be useful in the proof of the soundness of d-separation.

A walk is a sequence of consecutively adjacent vertices and contrary to paths, walks may contain repeating vertices.
Furthermore, as in Ferreira and Assaad, 2024 we define the notion of primary path of a walk $\tilde{\pi}=\langle V^1,\cdots,V^n \rangle$ as $\pi = \langle U^1,\cdots,U^m \rangle$ where $U^1 = V^1$ and $U^{k+1} = V^{max\{i\mid V^i = U^k\}+1}$. The definition of blocked path given in Definition <ref> is readily extendable to walks.

[Active Walks]

Let $\mathcal{G}=(\mathbb{V},\mathbb{E})$ be an FT-ADMG, $\mathbb{W}\subseteq\mathbb{V}$ and $\tilde{\pi}=\langle V^1,\cdots,V^n\rangle$ be a walk.
If $\tilde{\pi}$ is $\mathbb{W}$-active then there exist a path $\pi$ from $V^1$ to $V^n$ which is $\mathbb{W}$-active.

Let $\mathcal{G}=(\mathbb{V},\mathbb{E})$ be an FT-ADMG, $\mathbb{W}\subseteq\mathbb{V}$ and $\tilde{\pi}=\langle V^1,\cdots,V^n\rangle$ be an $\mathbb{W}$-active walk.
Suppose that its primary path $\pi = \langle U^1,\cdots,U^m \rangle$ is $\mathbb{W}$-blocked and take $1<i<m$ such that $\langle U^{i-1},U^i,U^{i+1}\rangle$ is $\mathbb{W}$-blocked.

If $U^{i-1}\starbarstar U^{i}\rightarrow U^{i+1}$ (or symmetrically if $U^{i-1} \leftarrow U^{i} \starbarstar U^{i+1}$) and $U^{i}\in\mathbb{W}$ then there exists $1<j<n$ such that $U^i=V^j$ and $U^{j+1}=V^{j+1}$ and thus $V^{j-1}\starbarstar V^{j}\rightarrow V^{j+1}$ is in $\tilde{\pi}$ and $V^j=U^{i}\in\mathbb{W}$.
Therefore $\tilde{\pi}$ is $\mathbb{W}$-blocked.

Otherwise, $U^{i-1}\stararrow U^{i}\arrowstar U^{i+1}$ and $\descendants{U^{i}}{\mathcal{G}}\cap\mathbb{W}=\emptyset$.
Then, there exists $1<j<n$ such that $U^{i-1}=V^{j-1}$ and $U^{i}=V^{j}$ and thus $ V^{j-1} \stararrow V^j \cdots  \arrowstar $ is in $\tilde{\pi}$.
Take $j\leq k < n$ such that $V^{j-1} \stararrow V^j \rightarrow \cdots \rightarrow V^{k} \arrowstar$ is in $\tilde{\pi}$ and notice that $\stararrow V^k \arrowstar$ and $\descendants{V^k}{\mathcal{G}} \subseteq \descendants{V^{j}}{\mathcal{G}} = \descendants{U^{i}}{\mathcal{G}}$ so $\descendants{V^k}{\mathcal{G}} \cap \mathbb{W} = \emptyset$.
Therefore, $\tilde{\pi}$ is $\mathbb{W}$-blocked which contradicts the initial assumption.

The following theorem shows that the d-separation as presented in Definition <ref> is sound in SCGs.

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG and $\mathbb{X},\mathbb{Y},\mathbb{W}\subseteq \mathbb{S}$. If $\mathbb{X}$ and $\mathbb{Y}$ are d-separated by $\mathbb{W}$ in $\mathcal{G}^s$ then, in any compatible FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$, $\mathbb{V}^{\mathbb{X}}$ and $\mathbb{V}^{\mathbb{Y}}$ are d-separated by $\mathbb{V}^{\mathbb{W}}$.

Suppose $\mathbb{X}$ and $\mathbb{Y}$ are d-separated by $\mathbb{W}$ in $\mathcal{G}^s$ and there exists a compatible FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$ and a path $\pi=\langle V^{p_1}_{t^{p_1}},\cdots,V^{p_n}_{t^{p_n}}\rangle$ from $V^{p_1}_{t^{p_1}} \in \mathbb{V}^{\mathbb{X}}$ to $V^{p_n}_{t^{p_n}} \in \mathbb{V}^{\mathbb{Y}}$ which is not blocked by $\mathbb{V}^{\mathbb{W}}$.
The walk $\tilde{\pi}^s=\langle V^{p_1},\cdots,V^{p_n}\rangle$ is in $\mathcal{G}^s$ with $V^{p_1}\in\mathbb{X}$ and $V^{p_n}\in\mathbb{Y}$.
Since $\mathbb{X}$ and $\mathbb{Y}$ are d-separated by $\mathbb{W}$ and using the contraposition of Property <ref>, $\mathbb{W}$ blocks $\tilde{\pi}^s$ and thus there exists $1<i<n$ such that $\langle V^{p_{i-1}},V^{p_{i}},V^{p_{i+1}} \rangle$ is $\mathbb{W}$-blocked.

If $V^{p_{i-1}}\starbarstar V^{p_i}\rightarrow V^{p_{i+1}}$ (or symmetrically $V^{p_{i-1}}\leftarrow V^{p_i}\starbarstar V^{p_{i+1}}$) and $V^{p_i}\in\mathbb{W}$ then $V^{p_{i-1}}_{t^{p_{i-1}}} \starbarstar V^{p_i}_{t^{p_{i}}} \rightarrow V^{p_{i+1}}_{t^{p_{i+1}}}$ and $\forall t\in[t_0,t_{max}],~V^{p_i}_t \in \mathbb{V}^{\mathbb{W}}$ and in particular $V^{p_i}_{t^{p_i}} \in \mathbb{V}^{\mathbb{W}}$.
Therefore, $\pi$ is blocked by $\mathbb{V}^{\mathbb{W}}$.

Otherwise, $V^{p_{i-1}} \stararrow V^{p_i} \arrowstar V^{p_{i+1}}$ and $\descendants{V^{p_i}}{\mathcal{G}^s} \cap \mathbb{W} = \emptyset$ so $V^{p_{i-1}}_{t^{p_{i-1}}} \stararrow V^{p_i}_{t^{p_{i}}} \arrowstar V^{p_{i+1}}_{t^{p_{i+1}}}$ and $\descendants{V^{p_i}_{t^{p_i}}}{\mathcal{G}} \cap \mathbb{V}^{\mathbb{W}} \subseteq \mathbb{V}^{Desc(V^{p_i})} \cap \mathbb{V}^{\mathbb{W}} = \{\mathbb{V}^j \mid V^j \in \descendants{V^{p_i}}{\mathcal{G}^s} \cap \mathbb{W}\} = \{\mathbb{V}^j \mid V^j \in \emptyset\} = \emptyset$.
Therefore, $\pi$ is blocked by $\mathbb{V}^{\mathbb{W}}$ which contradicts the initial assumption.
In conclusion, the d-separation criterion in SCGs is sound.

The following theorem shows that the d-separation as presented in Definition <ref> is complete in SCGs.

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG and $\mathbb{X},\mathbb{Y},\mathbb{W}\subseteq \mathbb{S}$. If $\mathbb{X}$ and $\mathbb{Y}$ are not d-separated by $\mathbb{W}$ in $\mathcal{G}^s$ then, there exists a compatible FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$ such that $\mathbb{V}^{\mathbb{X}}$ and $\mathbb{V}^{\mathbb{Y}}$ are not d-separated by $\mathbb{V}^{\mathbb{W}}$.

Suppose $\mathbb{X}$ and $\mathbb{Y}$ are not d-separated by $\mathbb{W}$ in $\mathcal{G}^s$.

Since $\mathbb{X}$ and $\mathbb{Y}$ are not d-separated by $\mathbb{W}$ in $\mathcal{G}^s$ there exists an $\mathbb{W}$-active path $\pi^s=\langle V^{p_1},\cdots,V^{p_n}\rangle$ with $V^{p_1}\in\mathbb{X}$ and $V^{p_n}\in\mathbb{Y}$.
From the SCG $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ one can build a compatible FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$ in the following way:

\begin{equation*}
\begin{aligned}
\mathbb{V} :=& \mathbb{V}^{\mathbb{S}}&\\
\mathbb{E}^{1}_{\pi^s} :=& \{X_{t} \rightarrow Y_{t} \mid \forall t\in[t_0,t_{max}], \forall X \rightarrow Y \in \mathbb{E}^s &\\
&\st \langle X \rightarrow Y \rangle \subseteq \pi^s \text{ or }\langle Y \leftarrow X \rangle \subseteq \pi^s\}&\\
\mathbb{E}^{1}_{\bar{\pi}^s} :=& \{X_{t} \rightarrow Y_{t+1} \mid \forall t\in[t_0,t_{max}[, \forall X \rightarrow Y \in \mathbb{E}^s &\\
&\st \langle X \rightarrow Y \rangle \not\subseteq \pi^s \text{ and }\langle Y \leftarrow X \rangle \not\subseteq \pi^s\}&\\
\mathbb{E}^{2}_{\pi^s} :=& \{X_{t} \longdashleftrightarrow Y_{t} \mid \forall t\in[t_0,t_{max}], \forall X \longdashleftrightarrow Y \in \mathbb{E}^s &\\
&\st \langle X \longdashleftrightarrow Y \rangle \subseteq \pi^s\text{ or }\langle Y \longdashleftrightarrow X \rangle \subseteq \pi^s\}&\\
\mathbb{E}^{2}_{\bar{\pi}^s} :=& \{X_{t} \longdashleftrightarrow Y_{t+1} \mid \forall t\in[t_0,t_{max}[, \forall X \longdashleftrightarrow Y \in \mathbb{E}^s &\\
&\st \langle X \longdashleftrightarrow Y \rangle \not\subseteq \pi^s\text{ and }\langle Y \longdashleftrightarrow X \rangle \not\subseteq \pi^s\}&\\
\mathbb{E} :=& \mathbb{E}^{1}_{\pi^s} \cup \mathbb{E}^{1}_{\bar{\pi}^s} \cup \mathbb{E}^{2}_{\pi^s} \cup \mathbb{E}^{2}_{\bar{\pi}^s}
\end{aligned}
\end{equation*}

Notice that $\mathcal{G}$ is in indeed acyclic, contains no edges going back in time and is compatible with the SCG $\mathcal{G}^s$.
Moreover, $\mathcal{G}$ contains the path $\pi=\langle V^{p_1}_{t_0},\cdots,V^{p_n}_{t_0}\rangle$ which is necessarily $\mathbb{V}^{\mathbb{W}}$-active since $\pi^s$ is $\mathbb{W}$-active.
In conclusion, the d-separation criterion in SCGs is complete.

The findings of this section establish that identifying a d-separation in SCGs guarantees a macro-level d-separation in all compatible FT-ADMGs. This is crucial because, as shown in Pearl, 2009, a d-separation in an ADMG implies a conditional independence in the underlying probability distribution. By extending the applicability of d-separation to SCGs, this result allows researchers to infer macro-level conditional independencies even when dealing with partially specified graphs that omit specific temporal information. This is particularly valuable in constraint-based causal discovery when the interest is to uncover the structure of the SCG without uncovering an FT-ADMG.

§ IDENTIFICATION OF MACRO TOTAL EFFECTS

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SCGs with identifiable macro total effects. Each pair of red and blue vertices represents the total effect we are interested in.

The do-calculus initially introduced in Pearl, 1995 is an important tool of causal inference that consists of three rules. It allows to express, wheneber it is possible, queries under interventions, , that contains a $\interv{\cdot}$ operator, as queries that can be estimated from observational data, , that does not contain a $\interv{\cdot}$ operator. The do-calculus was initially introduced for acyclic graphs so it is not easily extendable to cyclic graphs.
But it turned out that it is readily extendable to SCGs (containing cycles).

Firstly, we define the notion of mutilated graphs [Pearl, 2009]. Consider a causal graph $\mathcal{G}=(\mathbb{V},\mathbb{E})$ and $\mathbb{A},\mathbb{B}\subseteq\mathbb{V}$, a mutilated graph denoted by $\mathcal{G}_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$ is the graph obtained by removing all edges coming in $\mathbb{A}$ and all edges coming out of $\mathbb{B}$.

Using the notion of mutilated graphs and d-separation we can write the three rules of the do-calculus in SCGs.

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG and $\mathbb{X},\mathbb{Y},\mathbb{U},\mathbb{W}\subseteq\mathbb{S}$ be disjoint subsets of vertices.
The three rules of the do-calculus are:

\begin{equation*}
\begin{aligned}
\textbf{Rule 1:}& \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{x}},\mathbb{v}^{\mathbb{w}}} = \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{w}}}\\
&\text{if } \dsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}}}\\
\textbf{Rule 2:}& \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\interv{\mathbb{v}^{\mathbb{x}}},\mathbb{v}^{\mathbb{w}}} = \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{x}},\mathbb{v}^{\mathbb{w}}}\\
&\text{if } \dsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}\underaccent{\bar}{\mathbb{X}}}}\\
\textbf{Rule 3:}& \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\interv{\mathbb{v}^{\mathbb{x}}},\mathbb{v}^{\mathbb{w}}} = \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{w}}}\\
&\text{if } \dsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}\bar{\mathbb{X}(\mathbb{W})}}}\\
\end{aligned}
\end{equation*}

where $\mathbb{X}(\mathbb{W})$ is the set of vertices in $\mathbb{X}$ that are non-ancestors of any vertex in $\mathbb{W}$ in the mutilated graph $\mathcal{G}^s_{\bar{\mathbb{U}}}$.

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SCGs with not identifiable macro total effects. Each pair of red and blue vertices represents the total effect we are interested in.

In the following, we introduce a property of compatibility between mutilated graphs that will be useful for proving the soundness and the completness of the do-calculus in SCGs (Definition <ref>).

[Compatibility of Mutilated Graphs]

Let $\mathcal{G}=(\mathbb{V},\mathbb{E})$ be an FT-ADMG, $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ its compatible SCG and $\mathbb{A},\mathbb{B}\subseteq\mathbb{S}$.
The mutilated graph $\mathcal{G}^ s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$ is an SCG compatible with the mutilated FT-ADMG $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$.

Let $\mathcal{G}=(\mathbb{V},\mathbb{E})$ be an FT-ADMG, $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ its compatible SCG and $\mathbb{A},\mathbb{B}\subseteq\mathbb{S}$.

Firstly, let us show that every arrow in $\mathcal{G}^ s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$ corresponds to an arrow in $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$.
Let $X,Y\in\mathbb{S}$ such that $X\rightarrow Y$ is in $\mathbb{E}^s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$.
Because $X\rightarrow Y$ is in $\mathbb{E}^s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$, we know that $X\notin\mathbb{B}$ and $Y\notin \mathbb{A}$ and that $X\rightarrow Y$ is in $\mathbb{E}^s$.
Therefore, there exists $t_0\leq t_X \leq t_Y \leq t_{max}$ such that $X_{t_X}\rightarrow Y_{t_Y}$ is in $\mathbb{E}$ and $X_{t_X}\notin\mathbb{V}^{\mathbb{B}}$ and $Y_{t_Y}\notin \mathbb{V}^{\mathbb{A}}$ so $X_{t_X}\rightarrow Y_{t_Y}$ is in $\mathbb{E}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$.

Similarly, let $X,Y\in\mathbb{S}$ such that $X\longdashleftrightarrow Y$ is in $\mathbb{E}^s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$.
Because $X\longdashleftrightarrow Y$ is in $\mathbb{E}^s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$, we know that $X,Y\notin \mathbb{A}$ and that $X\longdashleftrightarrow Y$ is in $\mathbb{E}^s$.
Therefore, there exists $t_X , t_Y \in [t_0,t_{max}]$ such that $X_{t_X}\longdashleftrightarrow Y_{t_Y}$ is in $\mathbb{E}$ and $X_{t_X},Y_{t_Y}\notin \mathbb{V}^{\mathbb{A}}$ so $X_{t_X}\longdashleftrightarrow Y_{t_Y}$ is in $\mathbb{E}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$.

Secondly, let us show that every arrow in $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$ corresponds to an arrow in $\mathcal{G}^ s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$.
Let $X_{t_X},Y_{t_Y}\in\mathbb{V}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$ such that $X_{t_X}\rightarrow Y_{t_Y}$ is in $\mathbb{E}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$.
Because $X_{t_X}\rightarrow Y_{t_Y}$ is in $\mathbb{E}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$, we know that $X_{t_X}\notin\mathbb{V}^{\mathbb{B}}$ and $Y_{t_Y}\notin \mathbb{V}^{\mathbb{A}}$ and that $X_{t_X}\rightarrow Y_{t_Y}$ is in $\mathbb{E}$.
Therefore, $X\rightarrow Y$ is in $\mathbb{E}^s$ and $X\notin{\mathbb{B}}$ and $Y\notin {\mathbb{A}}$ so $X\rightarrow Y$ is in $\mathbb{E}^s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$.

Similarly, let $X_{t_X},Y_{t_Y}\in\mathbb{V}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$ such that $X_{t_X}\longdashleftrightarrow Y_{t_Y}$ is in $\mathbb{E}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$.
Because $X_{t_X}\longdashleftrightarrow Y_{t_Y}$ is in $\mathbb{E}_{\bar{\mathbb{V}^{\mathbb{A}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{B}}}}$, we know that $X_{t_X},Y_{t_Y}\notin \mathbb{V}^{\mathbb{A}}$ and that $X_{t_X}\longdashleftrightarrow Y_{t_Y}$ is in $\mathbb{E}$.
Therefore, $X\longdashleftrightarrow Y$ is in $\mathbb{E}^s$ and $X,Y\notin {\mathbb{A}}$ so $X\longdashleftrightarrow Y$ is in $\mathbb{E}^s_{\bar{\mathbb{A}}\underaccent{\bar}{\mathbb{B}}}$.

In the following we introduce the first main theorem of this section.

The three rules of the do-calculus are valid in SCGs.

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG and $\mathbb{X},\mathbb{Y},\mathbb{U},\mathbb{W}\subseteq\mathbb{S}$.

Suppose that $\dsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}}}$.
Using Property <ref> and Theorem <ref> we know that for every compatible FT-ADMG $\mathcal{G}$, $\dsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}}}$.
Thus, the usual rule 1 of the do-calculus given by Pearl, 1995 in the case of ADMGs states that $\probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{x}},\mathbb{v}^{\mathbb{w}}} = \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{w}}}$.

Suppose that $\dsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}\underaccent{\bar}{\mathbb{X}}}}$.
Using Property <ref> and Theorem <ref> we know that for every compatible FT-ADMG $\mathcal{G}$, $\dsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{ \mathbb{V}^{\mathbb{U}},\mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\underaccent{\bar}{\mathbb{V}^\mathbb{X}}}}$.
Thus, the usual rule 2 of the do-calculus given by Pearl, 1995 in the case of ADMGs states that $\probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\interv{\mathbb{v}^{\mathbb{x}}},\mathbb{v}^{\mathbb{w}}} = \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{x}},\mathbb{v}^{\mathbb{w}}}$.

Suppose that $\dsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}\underaccent{\bar}{\mathbb{X}(\mathbb{W}})}}$.
Using Property <ref> $\mathcal{G}^s_{\bar{\mathbb{U}}}$ is compatible with any $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}}$ and if $\exists X_{t_X}\in\mathbb{V}^{\mathbb{X}}$ such that $X_{t_X}\in \ancestors{\mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}}}$ then $X\in \ancestors{\mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}}}$. Therefore, $\forall X\in\mathbb{X},~ X\notin \ancestors{\mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}}} \implies \forall X_{t_X} \in \mathbb{V}^{\mathbb{X}},~ X_{t_X}\notin \ancestors{\mathbb{V}^{\mathbb{X}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}}}$.
Thus $\mathcal{G}^s_{\bar{\mathbb{U}}\bar{\mathbb{X}(\mathbb{W}})}$ and $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\bar{\mathbb{V}^{\mathbb{X}}(\mathbb{V}^{\mathbb{W}}})}$ are compatible and using Theorem <ref> we know that for every compatible FT-ADMG $\mathcal{G}$, $\dsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}},\mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\bar{\mathbb{V}^{\mathbb{X}}(\mathbb{V}^{\mathbb{W}}})}}$.
Thus, the usual rule 3 of the do-calculus given by Pearl, 1995 in the case of ADMGs states that $\probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\interv{\mathbb{v}^{\mathbb{x}}},\mathbb{v}^{\mathbb{w}}} = \probac{\mathbb{v}^{\mathbb{y}}}{\interv{\mathbb{v}^{\mathbb{u}}},\mathbb{v}^{\mathbb{w}}}$.

Using the soundness of the do-calculus in SCGs (Theorem <ref>), we can easily use the rules of do-calculus to find out that the total effect $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$
is identifiable in all SCGs in Figure <ref>.

In the following we introduce the second main theorem of this section.

If one of the rule of the do-calculus does not apply for a given SCG, then there exists a compatible FT-ADMG for which the corresponding rule does not apply.

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG and $\mathbb{X},\mathbb{Y},\mathbb{U},\mathbb{W}\subseteq\mathbb{S}$.

Suppose that rule 1 does not apply ie, $\notdsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}}}$.
Then, using Theorem <ref> there exists an FT-ADMG $\tilde{\mathcal{G}}$ compatible with $\mathcal{G}^s_{\bar{\mathbb{U}}}$ in which $\notdsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\tilde{\mathcal{G}}}$.
Notice that there exists an FT-ADMG $\mathcal{G}$ compatible with $\mathcal{G}^s$ such that $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}} = \tilde{\mathcal{G}}$.
Therefore, $\mathcal{G}$ is an FT-ADMG compatible with $\mathcal{G}^s$ in which $\notdsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}}}$ and thus

the usual rule 1 of the do-calculus given by Pearl, 1995 in the case of ADMGs does not apply.

Suppose that rule 2 does not apply ie, $\notdsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}\underaccent{\bar}{\mathbb{X}}}}$.
Then, using Theorem <ref> there exists an FT-ADMG $\tilde{\mathcal{G}}$ compatible with $\mathcal{G}^s_{\bar{\mathbb{U}}\underaccent{\bar}{\mathbb{X}}}$ in which $\notdsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\tilde{\mathcal{G}}}$.
Notice that there exists an FT-ADMG $\mathcal{G}$ compatible with $\mathcal{G}^s$ such that $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{X}}}} = \tilde{\mathcal{G}}$.
Therefore, $\mathcal{G}$ is an FT-ADMG compatible with $\mathcal{G}^s$ in which $\notdsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\underaccent{\bar}{\mathbb{V}^{\mathbb{X}}}}}$ and thus

the usual rule 2 of the do-calculus given by Pearl, 1995 in the case of ADMGs does not apply.

Suppose that rule 3 does not apply ie, $\notdsepc{\mathbb{Y}}{\mathbb{X}}{\mathbb{U}, \mathbb{W}}{\mathcal{G}^s_{\bar{\mathbb{U}}\bar{\mathbb{X}(\mathbb{W})}}}$.
Then, using Theorem <ref> there exists an FT-ADMG $\tilde{\mathcal{G}}$ compatible with $\mathcal{G}^s_{\bar{\mathbb{U}}\bar{\mathbb{X}(\mathbb{W})}}$ in which $\notdsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\tilde{\mathcal{G}}}$.
Using the same idea as in the proof of Theorem <ref>, notice that there exists an FT-ADMG $\mathcal{G}$ compatible with $\mathcal{G}^s$ such that $\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\bar{\mathbb{V}^{\mathbb{X}}(\mathbb{V}^{\mathbb{W}})}} = \tilde{\mathcal{G}}$.
Therefore, $\mathcal{G}$ is an FT-ADMG compatible with $\mathcal{G}^s$ in which $\notdsepc{\mathbb{V}^{\mathbb{Y}}}{\mathbb{V}^{\mathbb{X}}}{\mathbb{V}^{\mathbb{U}}, \mathbb{V}^{\mathbb{W}}}{\mathcal{G}_{\bar{\mathbb{V}^{\mathbb{U}}}\bar{\mathbb{V}^{\mathbb{X}}(\mathbb{V}^{\mathbb{W}})}}}$ and thus

the usual rule 3 of the do-calculus given by Pearl, 1995 in the case of ADMGs does not apply.

Using the completeness of the do-calculus in SCGs (Theorem <ref>), we can determine that the total effect $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$  is not identifiable in all SCGs depicted in Figure <ref> and in the SCG in Figure <ref> by examining all possible iterations of the rules of the do-calculus.

§.§ Strongly Connected Hedges and Non-identifiability

In ADMGs, there exists a sub-graphical structure, called an Hedge [Shpitser and Pearl, 2006], which is employed to  graphically characterize non-identifiability as shown in .
To properly define it for SCGs, it is essential to first familiarize oneself with the two related definitions which we have adapted and provided below specifically for the context of SCGs:

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG. A subset of vertices $\mathbb{V} \subseteq \mathbb{S}$ such that $\forall V,V' \in \mathbb{V},~\exists V^1,\cdots,V^n\in \mathbb{V}$ with $V^1=V$, $V^n=V'$ and $\forall i\in[1,n-1]~V^i\longdashleftrightarrow V^{i+1}$ is called a C-component.

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG. A subgraph[Following Shpitser and Pearl, 2006, we consider that a subgraph of $\mathcal{G}^s$ is a graph containing a subset of the vertices of $\mathcal{G}^s$ and a subset of the edges between those vertices in $\mathcal{G}^s$.] of  $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ which is acyclic, where every vertex has at most one child (, is a forest), and is a C-component is called a C-forest.

Consider an SCG $\mathcal{G}^s=(\mathbb{S}, \mathbb{E}^s)$ and two sets of vertices $\mathbb{X}, \mathbb{Y}\subset \mathbb{S}$.
Let $\mathbb{F}$ and $\mathbb{F}'$ be R-rooted C-forests in $\mathcal{G}^s$ such that $\mathbb{X} \cap \mathbb{F} \ne\emptyset$, $\mathbb{X} \cap \mathbb{F}' =\emptyset$, $\mathbb{F}' \subseteq \mathbb{F}$, and $R\subset \ancestors{Y}{\mathcal{G}^s_{\underaccent{\bar}{\mathbb{X}}}}$. Then $\mathbb{F}$ and $\mathbb{F}'$ form an Hedge for $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ in $\mathcal{G}^s$.

An Hedge turned out to be too weak to cover non-identifiability in SCGs.
For example, the SCG in Figure <ref> contains no Hedge but the macro total effect is not identifiable due to the cycle between $X$ and $Y$.
Consequently, we introduce a new notion designed to graphically characterize non-identifiability of macro total effects in SCGs.

[black, circle, draw, inner sep=0]
nodes=draw,rounded corners,minimum height=0.6cm,minimum width=0.6cm
latent/.append style=white, fill=black

[fill=red!30] (X) at (0,0) $X$ ;
[fill=blue!30] (Y) at (2.4,0) $Y$;
(Z) at (1.2,0) $W$;

[transform canvas=yshift=-.25em]
[->,>=latex] (X) – (Z);

[transform canvas=yshift=.25em]
[<-,>=latex] (X) – (Z);

[transform canvas=yshift=-.25em]
[->,>=latex] (Z) – (Y);

[transform canvas=yshift=.25em]
[<-,>=latex] (Z) – (Y);

[<->,>=latex, dashed] (X) to [out=90,in=90, looseness=1] (Y);

[->,>=latex] (X) to [out=165,in=120, looseness=2] (X);
[->,>=latex] (Y) to [out=15,in=60, looseness=2] (Y);

[->,>=latex] (Z) to [out=60,in=105, looseness=2] (Z);

[<->,>=latex, dashed] (X) to [out=-155,in=-110, looseness=2] (X);
[<->,>=latex, dashed] (Y) to [out=-25,in=-70, looseness=2] (Y);
[<->,>=latex, dashed] (Z) to [out=-25,in=-70, looseness=2] (Z);
[<->,>=latex, dashed, sorbonnered] (X) to [out=45,in=135, looseness=1] (Z);
[<->,>=latex, dashed, sorbonnered] (Z) to [out=45,in=135, looseness=1] (Y);

SC-projection of  Figure <ref>.

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nodes=draw,rounded corners,minimum height=0.6cm,minimum width=0.6cm
anomalous/.append style=fill=easyorange
rc/.append style=fill=easyorange

[fill=red!30] (X) at (0,0) $X$ ;
[fill=blue!30] (Y) at (1.2,0) $Y$;

[transform canvas=yshift=-.25em]
[->,>=latex] (X) – (Y);

[transform canvas=yshift=.25em]
[<-,>=latex] (X) – (Y);

[->,>=latex] (X) to [out=180,in=135, looseness=2] (X);

[<->,>=latex, dashed] (X) to [out=-155,in=-110, looseness=2] (X);
[<->,>=latex, dashed] (Y) to [out=-25,in=-70, looseness=2] (Y);

[<->,>=latex, dashed, sorbonnered] (X) to [out=90,in=90, looseness=1] (Y);

SC-projection of  Figure <ref>.

[black, circle, draw, inner sep=0]
nodes=draw,rounded corners,minimum height=0.6cm,minimum width=0.6cm
anomalous/.append style=fill=easyorange
rc/.append style=fill=easyorange

[fill=red!30] (X) at (0,0) $X$ ;
[fill=blue!30] (Y) at (1.2,0) $Y$;

[->,>=latex] (X) – (Y);
[<->,>=latex, dashed] (X) to [out=90,in=90, looseness=1] (Y);

[->,>=latex] (X) to [out=180,in=135, looseness=2] (X);
[->,>=latex] (Y) to [out=15,in=60, looseness=2] (Y);

[<->,>=latex, dashed] (X) to [out=-155,in=-110, looseness=2] (X);
[<->,>=latex, dashed] (Y) to [out=-25,in=-70, looseness=2] (Y);

SC-projection of  Figure <ref>.

[black, circle, draw, inner sep=0]
nodes=draw,rounded corners,minimum height=0.6cm,minimum width=0.6cm
anomalous/.append style=fill=easyorange
rc/.append style=fill=easyorange

(W) at (0,1.2) $W$ ;
[fill=red!30] (X) at (0,0) $X$ ;
[fill=blue!30] (Y) at (1.2,0) $Y$;

[->,>=latex] (X) – (Y);
[transform canvas=xshift=-.25em]
[->,>=latex] (X) – (W);

[transform canvas=xshift=.25em]
[<-,>=latex] (X) – (W);

[<->,>=latex, dashed] (W) to [out=0,in=90, looseness=1] (Y);

[->,>=latex] (W) to [out=180,in=135, looseness=2] (W);
[->,>=latex] (X) to [out=220,in=175, looseness=2] (X);
[->,>=latex] (Y) to [out=15,in=60, looseness=2] (Y);

[<->,>=latex, dashed] (X) to [out=-25,in=-70, looseness=2] (X);
[<->,>=latex, dashed] (Y) to [out=-25,in=-70, looseness=2] (Y);
[<->,>=latex, dashed] (W) to [out=20,in=65, looseness=2] (W);

[<->,>=latex, dashed, sorbonnered] (W) to [out=190,in=155, looseness=1] (X);

SC-projection of  Figure <ref>.

SC-projections of the SCGs in Figures <ref> and <ref>. Each pair of red and blue vertices represents the total effect we are interested in, and the red edges indicate those added through the SC-projection.

Consider an SCG $\mathcal{G}^s=\{\mathbb{S}, \mathbb{E}^s\}$.
The SC-projection $\mathcal{H}^s$ of $\mathcal{G}^s$ is the graph that includes all vertices and edges from $\mathcal{G}^s$, plus a bidirected dashed edge between each pair $X,Y\in \mathbb{S}$ such that $\scc{X}{\mathcal{G}^s} = \scc{Y}{\mathcal{G}^s}$ and $X\ne Y$.

Consider an SCG $\mathcal{G}^s=\{\mathbb{S}, \mathbb{E}^s\}$ and its SC-projection $\mathcal{H}^s$.
An Hedge for $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ in $\mathcal{H}^s$ is an SC-Hedge for $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ in $\mathcal{G}^s$.

If there exists a SC-Hedge  for $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ in $\mathcal{G}^s$ then $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ is not identifiable.

Consider an SCG $\mathcal{G}^s=(\mathbb{S},\mathbb{E})$, its SC-projection $\mathcal{H}^s$, and a SC-Hedge $\mathbb{F},\mathbb{F}'$ for $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$.
Let us prove Theorem <ref> by induction on the number of bidirected dashed edges in the C-forest $\mathbb{F}$ that are in $\mathcal{H}^s$ but not in $\mathcal{G}^s$ (, which are artificially induced by cycles).
Firstly, if $\mathbb{F},\mathbb{F}'$ is an Hedge in $\mathcal{G}^s$ then there exists a compatible FT-ADMG in which for any $\exists t\in [t_0,t_{max}],~\mathbb{F}_{t} = \{F_{t} \mid F \in \mathbb{F}\},\mathbb{F}'_{t} = \{F_{t} \mid F \in \mathbb{F}'\}$ is an Hedge for $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ and thus $\probac{\mathbb{v}^\mathbb{y}}{\interv{\mathbb{v}^\mathbb{x}}}$ is not identifiable.
Secondly, if there exists $X\in\mathbb{X}$ and $Y\in\mathbb{Y}$ such that $\scc{X}{\mathcal{G}^s} = \scc{Y}{\mathcal{G}^s}$ then there exists a compatible FT-ADMG in which $\exists t\in [t_0,t_{max}]$ there is a directed path from $Y_{t}$ to $X_{t}$. This path must be blocked but every vertex on this path is a descendant of $Y_{t}$ and thus cannot be adjusted on without inducing a bias.
Lastly, assume that Theorem <ref> is true for any SC-Hedge with $k$ bidirected dashed edges which are in $\mathcal{H}^s$ but not in $\mathcal{G}^s$ and that $\mathbb{F}$ has $k+1$ such edges. Then, one cannot identify the macro total effect by the adjustment formula [Shpitser et al., 2010] due to the ambiguity [Assaad et al., 2024] induced by the cycle on $X$ or due to the bias induced by the latent confounder of $X$ that cannot be removed without using any rule of the do-calculus. Moreover, any decomposition of the effect using the do-calculus will necessitate the identification of other macro total effects. These effects have sub-C-forests of $\mathbb{F},\mathbb{F}'$ as SC-Hedges and at least one of theses sub-C-forests has at most $k$ artificial bidirected dashed edges induced by cycles. The associated total macro effect is therefore unidentifiable by induction.

For example, each SCG depicted in Figures <ref> and <ref> features an SC-Hedge. This is apparent from their SC-projections shown in Figure <ref>, where an Hedge is clearly present. This implies, based on Theorem <ref>, that in these cases, the total effect is not identifiable.

§ RELATED WORKS

The idea of representing complex low-level causal structures as partially specified high-level graphs has gained some interest in the last years.
For instance,

Iwasaki and Simon, 1994, Chalupka et al., 2016, Anand et al., 2023, Wahl et al., 2024, Ferreira and Assaad, 2024, Assaad et al., 2024 have explored clustering of low-level variables. Notably, Anand et al., 2023 introduced cluster-ADMGs, which generalize acyclic directed mixed graphs by allowing flexible partitioning of variables while maintaining acyclicity. They extended d-separation and the do-calculus to accommodate macro causal effects in these graphs. Even if SCGs, discussed in our paper, are somehow related (in both graphs vertices represent clusters) to cluster-ADMGs they are inherently different since SCGs can contain cycles.

It should be noted that Richardson, 1997, Forré and Mooij, 2017, Forré and Mooij, 2018, Forré and Mooij, 2020 extended d-separation and do-calculus to DMGs, potentially applicable to macro queries in cluster-DMGs. Although SCGs are a type of cluster-DMG, the applicability of these findings to SCGs remains uncertain due to the specific temporal information within each cluster in an SCGs, which might affect the completeness for macro queries but not necessarily their soundness. Our research suggests that this temporal knowledge does not improve the identifiability of macro queries.

In dynamic systems, Assaad et al., 2023, Ferreira and Assaad, 2024, Assaad et al., 2024 considered SCGs but for micro queries. They showed that the total effect and the direct effect is identifiable under some mild conditions if there are no hidden confounding which is different from macro causal effects as they are always identifiable when there is no hidden confounding.
As far as we know there exists still no sound and complete result for d-separation and the do-calculus for SCG in the more general case for micro queries, ,
the results presented in this paper do not apply to micro queries. For example, in the context of conditional independencies, even if $\dsepc{X}{Y}{\mathbb{W}}{\mathcal{G}^s}$ in the SCG, we cannot infer that $\dsepc{X_{t-1}}{Y_t}{\tilde{\mathbb{V}}^{\mathbb{W}}}{\mathcal{G}}$ where $\mathcal{G}$ is compatible with $\mathcal{G}^s$ and $\tilde{\mathbb{V}}^{\mathbb{W}}$ does not include all variables represented by $\mathbb{V}^{\mathbb{W}}$.
In the context of total effects, Assaad et al., 2024 demonstrated that the total effect $\probac{y_t}{\interv{x_{t-1}}}$ is identifiable even if there is a bidirectional arrow $X \rightleftarrows Y$ in the SCG, provided there are no other cycles involving $X$ and $Y$ as in Figure <ref>.
However, the do-calculus rules described in this paper cannot identify micro queries. It is not complete in the sense that it does not identify every identifiable queries.

In the context of macro queries within dynamic systems, Reiter et al., 2024 explored the identifiability of causal effects from one spectrum (or frequency) to another using SCGs, without delving into the graphical conditions like the ones presented in this paper. Furthermore, in parallel (and independently) to this work, Boeken and Mooij, 2024 examined a more general scenario where time is treated as continuous, and instantaneous cycles are possible in the full-time graph.

In a different view, Rubenstein et al., 2017, Beckers and Halpern, 2019 studied more general notions of abstraction, namely exact ($\tau-\omega$)-transformations and $\tau$-abstractions.
If one is only focused in the macro effects then the SCG can be seen as a constructive $\tau$-abstraction where $\tau$ is the identity in $\mathbb{R}^{|[t_0,t_{max}]|}$ (assuming the micro variables take values in $\mathbb{R}$).
Which means that all contributions of this paper can contribute to the development of the theory of more general notion of abstraction.
However, this is no longer true if one is interested in the micro effects.

§ CONCLUSION

In this paper,

we established the soundness and completeness of d-separation and the do-calculus for respectively identifying macro conditional independencies and macro total effects in SCGs.
By doing so, we bridged the gap between SCGs and many real-world applications in epidemiology.

There are three main limitations to this work. The first limitation is that the completeness result in Theorem <ref> does not take into account that there might exists different iterations of the rules of the do-calculus in different FT-ADMGs that can give the same final identification of the the total effect.
A second related limitation is that we provided a graphical characterization for the non-identifiability of macro total effects, however this characterization is not necessarily complete[An Hedge turned out not to correspond to one-to-one with non-identifiability, as pointed out by
Shpitser, 2023, but it is needed in some way for the complete characterization.], therefore in future works we want to see if a complete characterization might involve SC-Hedges in some way.
The third limitation is that  our results require a complete observation in time of the entire event system, which is often not feasible in practice. For example, it is challenging to ensure that data collection begins at the onset of each epidemic outbreak. Without this assumption, past instances of observed variables might act as unobserved confounders.

In these situations, it would be worthwhile to explore the potential use of instrumental variables [Pearl, 2009] to address these challenges. When instrumental variables are unavailable, our results may still be useful for partially identifying macro total effects using SCGs by deriving bounds for the effect under a minimal set of assumptions [Robins, 1989, Manski, 1990, Pearl, 2009, Zhang and Bareinboim, 2021].

§.§ Acknowledgements

We thank Fabrice Carrat from IPLESP for his valuable discussions regarding the application of this work in epidemiology. Additionally, we thank Philip Boeken and Joris Mooij from KdVI for their valuable input on dynamic structural causal models and cyclic graphs. Last but not least, we are thankful to Jonas Wahl from TU Berlin for his insightful discussions regarding cluster graphs and summary causal graphs in the frequency domain.
This work was supported by the CIPHOD project (ANR-23-CPJ1-0212-01).

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