Directed Acyclic Graph

Every connection in a DAG is a directed edge (an arrow) pointing from one node to another. This directionality explicitly encodes the flow of causation, dependency, or data. In a causal DAG, an arrow from X to Y asserts that X is a direct cause of Y relative to the other variables in the graph. This is fundamentally different from undirected graphs, which only represent symmetric associations.

The 'A' in DAG stands for acyclic, meaning it is impossible to start at a node and follow a sequence of directed edges to return to the same node. This property prevents circular reasoning and feedback loops. In causal inference, a cycle would imply that an event causes itself, which is a logical paradox. In computation, acyclicity ensures that topological ordering is possible, guaranteeing that tasks like data transformation or causal effect estimation can be completed in a finite number of steps.

Each node in a causal DAG represents a variable in the system being modeled. These can be observed quantities like 'Supplier Lead Time' or 'Order Volume,' or unobserved latent variables like 'Market Sentiment.' The graph's structure makes explicit which variables are parents (direct causes), children (direct effects), and ancestors or descendants of other variables, forming the basis for applying criteria like the Backdoor Criterion to identify confounding.

Every DAG has at least one topological ordering, which is a linear sequence of its nodes such that for every directed edge from node A to node B, A comes before B in the sequence. This property is critical for algorithms that process the graph sequentially. In a supply chain causal model, a topological sort ensures that root causes are analyzed before their downstream effects, enabling efficient propagation of interventions through the system.

d-separation (directional separation) is the graphical criterion for reading conditional independencies from a DAG. A path between two nodes is blocked if it contains a chain or fork where the middle node is conditioned on, or a collider where neither the collider nor its descendants are conditioned on. If all paths between two sets of nodes are blocked, they are d-separated and statistically independent. This is the core mechanism for testing whether a causal model is consistent with observed data.

The absence of cycles means a DAG cannot represent feedback loops directly. In a supply chain, a true feedback loop—like a stockout causing a demand surge that worsens the stockout—must be modeled by unrolling the cycle over time. This is achieved by creating separate nodes for each time step (e.g., Inventory_t and Inventory_{t+1}), transforming a cyclic process into an acyclic temporal representation that preserves the DAG's mathematical tractability.

The formal mathematical framework that pairs a Directed Acyclic Graph with a set of structural equations. While the DAG encodes qualitative causal assumptions, the SCM defines the precise functional relationships between nodes. Each variable is generated by a function of its direct causes and an exogenous noise term, representing the data-generating mechanism. This allows for rigorous counterfactual reasoning by mutilating specific equations.

A distortion introduced by conditioning on a collider variable—a node in a DAG that is a common effect of two other variables. While conditioning on a confounder opens a path, conditioning on a collider artificially creates a spurious association between its parents. This is a critical pitfall in causal analysis, often occurring when controlling for mediators or selecting samples based on outcomes.

A computational method that infers the DAG structure directly from observational data without human specification. Algorithms like the PC algorithm or Greedy Equivalence Search test conditional independencies to prune edges and orient causal directions. These methods output a Markov equivalence class—a set of DAGs that are statistically indistinguishable—rather than a single definitive graph.

A method for decomposing the total causal effect into a direct effect and an indirect effect operating through an intermediate mediator. In a DAG, the mediator sits on the causal path between treatment and outcome. This analysis quantifies how much of the effect is transmitted through a specific mechanism, essential for understanding why an intervention works.