Directed Acyclic Graphs Across Omics Layers

Why Direction Changes a Network Model

A multi-omics network can describe relationships among features without assigning a direction to them. A directed acyclic graph (DAG) makes a more specific modeling choice: each edge points from one variable to another, and the full joint distribution is represented using conditional distributions associated with those arrows. In a layer-ordered model, for example, an arrow may run from a transcript measurement to a protein measurement and then to a metabolite measurement.

The direction is part of the statistical model, not a decorative annotation. It determines which variables are used in each node’s conditional distribution. But drawing an arrow does not, by itself, establish that changing the source variable would cause a change in the destination. A direction may reflect a prespecified ordering or a modeling choice; causal interpretation requires additional assumptions and evidence.

Definition: A directed graph consists of nodes and arrows connecting some pairs of nodes. A node $u$ is a parent of node $v$ when the graph contains $u \to v$; $v$ is then a child of $u$. A directed graph is acyclic if it contains no directed cycle, such as $A \to B \to C \to A$. A directed acyclic graph is a DAG.

For a multi-omics DAG, each node represents a measured feature or other modeled variable. An arrow $u \to v$ says that $u$ is included among the parents used to specify the conditional distribution for $v$. A topological ordering is an ordering of the nodes in which every arrow points from an earlier node to a later one. Every DAG has at least one such ordering. If layers are ordered in advance, arrows can be constrained to go from an earlier layer to a later one; additional within-layer arrows must also respect an ordering that avoids cycles.

Node-Conditional Factorization

Suppose the modeled variables are $X_1,\ldots,X_p$, with a DAG $G$. Let $\operatorname{pa}(j)$ denote the set of parent indices of node $j$. A distribution that factorizes according to $G$ has a joint density or probability mass function equal to the product of one conditional distribution per node.

Definition: The distribution of $X_1,\ldots,X_p$ factorizes according to the DAG $G$ when $$ p(x_1,\ldots,x_p)=\prod_{j=1}^{p} p\bigl(x_j \mid x_{\operatorname{pa}(j)}\bigr). $$ For a node with no parents, the corresponding term is its marginal distribution $p(x_j)$. Each conditional distribution must be normalized over the possible values of its node, for every parent configuration with positive probability.

This factorization is a statement about a joint probability model. It does not require every variable to be continuous or Gaussian: a node-conditional distribution can be chosen to suit the type of measurement. For example, a binary node may use a Bernoulli conditional distribution, while a continuous node may use a Gaussian conditional distribution. The factorization specifies how these node-level distributions combine; their particular forms are additional modeling choices.

Why the Product Has This Form

The chain rule expresses any joint distribution using a chosen ordering $v_1,\ldots,v_p$ of the nodes:

p(x_{v_1},\ldots,x_{v_p}) = \prod_{i=1}^{p}p\bigl(x_{v_i}\mid x_{v_1},\ldots,x_{v_{i-1}}\bigr).

Choose a topological ordering, so all parents of $v_i$ occur before $v_i$. To obtain the DAG factorization, impose the DAG modeling assumption that, given its parents, a node’s conditional distribution does not additionally depend on the earlier nodes that are not its parents. In symbols, for each $i$,

p\bigl(x_{v_i}\mid x_{v_1},\ldots,x_{v_{i-1}}\bigr) = p\bigl(x_{v_i}\mid x_{\operatorname{pa}(v_i)}\bigr).

Substituting this equality into every factor of the chain rule gives

p(x_{v_1},\ldots,x_{v_p}) = \prod_{i=1}^{p}p\bigl(x_{v_i}\mid x_{\operatorname{pa}(v_i)}\bigr).

This argument justifies the product when the stated DAG assumption holds. Conversely, one can define a model by specifying normalized node-conditional distributions and taking their product. A product of this form is a normalized joint distribution: in reverse topological order, summing over a node removes its normalized conditional factor, leaving the product for the remaining nodes; repeating this step gives total probability one.

Worked Example: A Transcript, Protein, and Metabolite

Consider a hypothetical model with a binary transcript feature $T$, a binary protein feature $P$, and a binary metabolite feature $M$. Use the ordered DAG $T \to P$, $T \to M$, and $P \to M$. Its factorization is

p(t,p,m)=p(t)\,p(p\mid t)\,p(m\mid t,p).

Suppose the model specifies $P(T=1)=0.4$, $P(P=1\mid T=1)=0.7$, and $P(M=1\mid T=1,P=1)=0.8$. The complementary probabilities are $P(P=0\mid T=1)=0.3$ and $P(M=0\mid T=1,P=1)=0.2$, so the relevant conditional distributions are normalized. For the joint event $T=1,P=1,M=0$, substitute the specified values:

P(T=1,P=1,M=0) =0.4 \times 0.7 \times 0.2 =0.056.

Thus this model assigns probability $0.056$, or 5.6%, to that joint outcome. The value follows from the stated node-conditionals and graph. It is not an estimate of a real biological mechanism, and the arrow directions alone do not justify a causal interpretation.

Layer Order and the Meaning of an Edge

A prespecified layer order can rule out some arrows before any model is fitted. If the chosen order is transcript, protein, metabolite, then arrows from transcript to protein, protein to metabolite, or transcript to metabolite respect the order. An arrow from metabolite back to transcript violates it. A proposed set of arrows must also be checked as a whole: even when individual edges appear plausible, together they may form a directed cycle.

Direction also distinguishes a DAG from an undirected network. An undirected edge $T\text{—}P$ has no arrowhead and does not specify which variable is conditioned on the other in a node-conditional factorization. An arrow $T \to P$, by contrast, places $T$ in the parent set for the conditional distribution of $P$. The undirected graphical models covered in the earlier material on Gaussian graphical models use a different representation: their edges describe conditional association, rather than a parent-to-child ordering. An edge in either representation should not automatically be read as a causal effect.

Worked Example: The Same Pair With Opposite Factorizations

Let $T$ and $P$ be binary transcript and protein features in a hypothetical two-node model with the DAG $T \to P$. Suppose $P(T=1)=0.3$, $P(P=1\mid T=1)=0.6$, and $P(P=1\mid T=0)=0.1$. Using complementary conditional probabilities gives all four joint probabilities:

\begin{aligned} P(T=1,P=1)&=0.3(0.6)=0.18,\\ P(T=1,P=0)&=0.3(0.4)=0.12,\\ P(T=0,P=1)&=0.7(0.1)=0.07,\\ P(T=0,P=0)&=0.7(0.9)=0.63. \end{aligned}

The values sum to $0.18+0.12+0.07+0.63=1.00$. The same joint distribution can also be written in reverse order using the chain rule. For example, $P(P=1)=0.18+0.07=0.25$, and therefore

P(T=1\mid P=1)=\frac{0.18}{0.25}=0.72, \qquad P(T=1\mid P=0)=\frac{0.12}{0.75}=0.16.

So the joint distribution has both factorizations $p(t)p(p\mid t)$ and $p(p)p(t\mid p)$. This does not make the two arrows equivalent as scientific statements: the chosen DAG specifies which conditional model and ordering are being used. An undirected edge between $T$ and $P$ would indicate no such arrow direction; it would not select either factorization as a parent-child claim.

Checking a Layered DAG in Practice

A useful check is to write the graph’s factorization before interpreting any edge. That makes it possible to see whether each node has the intended parent set and whether the implied conditional distributions match the proposed layer order. Consider a hypothetical structure with two transcript nodes $A,B$, a protein node $C$, and a metabolite node $D$. Let the arrows be $A \to C$, $B \to C$, $C \to D$, and $A \to D$. The graph is acyclic and respects the order transcripts, protein, metabolite. Its factorization is

p(a,b,c,d)=p(a)\,p(b)\,p(c\mid a,b)\,p(d\mid a,c).

Notice that $B$ is not in the conditional distribution for $D$, because it is not a parent of $D$ in this graph. This is a feature of the specified factorization, not a claim that every unmodeled biological relationship is absent.

Worked Example: Evaluating a Four-Node Layered Model

For the graph just defined, suppose $A$ and $B$ are binary, with $P(A=1)=0.5$ and $P(B=1)=0.4$. Let $P(C=1\mid A=1,B=0)=0.6$, and let $P(D=1\mid A=1,C=1)=0.9$. Their complementary probabilities are $P(C=0\mid A=1,B=0)=0.4$ and $P(D=0\mid A=1,C=1)=0.1$. Each listed binary conditional distribution sums to one.

For the outcome $A=1,B=0,C=1,D=0$, the factorization gives

\begin{aligned} P(A=1,B=0,C=1,D=0) &=P(A=1)P(B=0)P(C=1\mid A=1,B=0)P(D=0\mid A=1,C=1)\\ &=0.5 \times 0.6 \times 0.6 \times 0.1\\ &=0.018. \end{aligned}

The model assigns probability $0.018$, or 1.8%, to that outcome. If the edge $D \to A$ were added, the existing path $A \to C \to D$ would close a directed cycle. The resulting graph would not be a DAG, so it would not have the topological order required for this factorization.

Diagnostics, Alternatives, and Limitations

Before using a proposed DAG, check that every node has a defined conditional distribution, that each conditional distribution is normalized, and that the arrows form no directed cycle. For a layer-ordered model, check every arrow against the specified order. Then verify the written factorization against the actual parent sets; a missing or extra parent changes the model. When fitting a model to data, further checks are needed to assess whether the chosen conditional forms describe the observations adequately. The graph’s structure alone cannot establish that fit.

An undirected network is often a more natural alternative when the goal is to represent conditional associations without imposing an ordering. In Gaussian graphical models, for example, edges are interpreted through conditional dependence given the other modeled variables. A DAG instead supplies node-conditional distributions arranged according to arrows. Neither representation is universally preferable: the choice should match the scientific question and the assumptions one can defend.

A DAG is limited by its specified node set, edge set, ordering, and conditional distribution forms. A graph that respects a plausible layer order may still omit relevant variables or represent the data poorly. Also, a graph can include an edge whose conditional distribution happens not to depend on that parent for particular parameter values; an arrow alone does not guarantee a nonzero association. Most importantly, a DAG is not, by itself, evidence that the arrows represent causal effects. Those claims require additional assumptions beyond the factorization.

Common pitfall: “There is an arrow from transcript to protein, so the transcript causally changes the protein.” The arrow specifies a parent-child relationship in the chosen DAG factorization; direction alone does not establish causation. Report it as a directed modeling relationship unless the additional assumptions needed for causal interpretation have been justified.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a DAG, what does it mean for $U$ to be a parent of $V$?
A graph contains $T \to P$ and $P \to M$. What is the directed path, and would adding $M \to T$ preserve acyclicity?
For the DAG $A \to B$, $A \to C$, write the node-conditional factorization.
Why does an undirected edge between two omics features not specify the same information as an arrow?
Does a transcript-to-protein arrow alone establish a causal effect? State the reason.

Solutions. (1) $U$ is a parent of $V$ when the graph contains $U \to V$; $U$ is included in the parent set conditioning the distribution for $V$. (2) The path is $T \to P \to M$. Adding $M \to T$ creates the cycle $T \to P \to M \to T$, so the graph is no longer acyclic. (3) The factorization is $p(a,b,c)=p(a)p(b\mid a)p(c\mid a)$. (4) An undirected edge has no arrowhead and does not identify a parent or select a node-conditional direction. (5) No. The arrow specifies a directed relationship in the chosen model; causal interpretation requires additional assumptions and evidence.