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Bidirected Edges

Bidirected edges (↔) represent unmeasured common causes (latent confounders) in causal diagrams.

The MixedGraph Type

MixedGraph supports both directed (→) and bidirected (↔) edges:

using Graphs, DAGMakie, CairoMakie

# Create empty mixed graph
mg = MixedGraph(3)

# Add directed edges
add_directed_edge!(mg, 1, 2)  # X → Y
add_directed_edge!(mg, 2, 3)  # Y → Z

# Add bidirected edge (unmeasured confounding)
add_bidirected_edge!(mg, 1, 3)  # X ↔ Z

fig, ax, p = dagplot(mg, nlabels = ["X", "Y", "Z"])
fig
Example block output

Creating Mixed Graphs

From Scratch

using Graphs, DAGMakie, CairoMakie

mg = MixedGraph(2)
add_directed_edge!(mg, 1, 2)
add_bidirected_edge!(mg, 1, 2)
fig, ax, p = dagplot(mg; nlabels = ["X", "Y"])
fig
Example block output

From Edge Lists

using Graphs, DAGMakie, CairoMakie

mg = mixed_graph(3,
    [(1, 2), (2, 3)],    # Directed edges
    [(1, 3)]              # Bidirected edges
)
fig, ax, p = dagplot(mg; nlabels = ["X", "Y", "Z"])
fig
Example block output

From Existing DiGraph

using Graphs, DAGMakie, CairoMakie

g = SimpleDiGraph(3)
add_edge!(g, 1, 2)
add_edge!(g, 2, 3)

mg = MixedGraph(g, [(1, 3)])  # Add bidirected X ↔ Z
fig, ax, p = dagplot(mg; nlabels = ["X", "Y", "Z"])
fig
Example block output

Customising Bidirected Edges

fig, ax, p = dagplot(mg,
    nlabels = ["X", "Y", "Z"],
    bidirected_color = :red,
    bidirected_width = 1.5,
    bidirected_style = :dash,
    bidirected_curvature = 0.4,
    bidirected_arrow_size = 10,
)
fig
Example block output

Common Confounded Patterns

DAGMakie provides convenience functions for common confounded structures. Each uses a pedagogical default layout so bidirected arcs are not obscured; pass layout=... to override.

Simple Confounding

fig, ax, p = dagplot_confounded(["X", "Y"])
fig
Example block output

Frontdoor Criterion

fig, ax, p = dagplot_frontdoor(["X", "M", "Y"])
fig
Example block output

Instrumental Variable with Confounding

fig, ax, p = dagplot_iv_confounded(["Z", "X", "Y"])
fig
Example block output

M-Bias

The classic five-node form with explicit latents (the letter M). Conditioning on the collider $M$ opens $X ← U₁ → M ← U₂ → Y$. The ADMG projection of the same structure is $X ↔ M ↔ Y$.

fig, ax, p = dagplot_m_bias()  # U₁, U₂, X, M, Y
fig
Example block output

Graph Constructors

Get the underlying mixed graph objects:

mg, labels = confounded_graph(["X", "Y"])
mg, labels = frontdoor_graph(["X", "M", "Y"])
mg, labels = iv_confounded_graph(["Z", "X", "Y"])
mg, labels = m_bias_graph()                 # U₁, U₂, X, M, Y (explicit latents)
spec = m_bias_spec()                        # same DAG with Latent / Collider styling

Querying Bidirected Edges

# Check if bidirected edge exists
has_bidirected_edge(mg, 1, 2)  # true/false

# Get all bidirected edges
bi_edges = bidirected_edges(mg)  # Set of (i, j) tuples

# Count bidirected edges
n = num_bidirected_edges(mg)