diff --git a/src/matchbox/common/factories.py b/src/matchbox/common/factories.py
index 6e295d32..71be3427 100644
--- a/src/matchbox/common/factories.py
+++ b/src/matchbox/common/factories.py
@@ -1,45 +1,33 @@
 from collections import Counter
 from textwrap import dedent
+from typing import Any
 
 import numpy as np
 import pyarrow as pa
 import rustworkx as rx
 
+from matchbox.common.transform import graph_results
 
-def verify_components(table: pa.Table) -> dict:
+
+def verify_components(all_nodes: list[Any], table: pa.Table) -> dict:
     """
     Fast verification of connected components using rustworkx.
 
     Args:
+        all_nodes: list of identities of inputs being matched
         table: PyArrow table with 'left', 'right' columns
 
     Returns:
         dictionary containing basic component statistics
     """
-    graph = rx.PyGraph()
-
-    unique_nodes = set(table["left"].to_numpy()) | set(table["right"].to_numpy())
-    graph.add_nodes_from(range(len(unique_nodes)))
-
-    node_to_idx = {node: idx for idx, node in enumerate(unique_nodes)}
-    edges = [
-        (node_to_idx[left], node_to_idx[right])
-        for left, right in zip(
-            table["left"].to_numpy(),
-            table["right"].to_numpy(),
-            strict=True,
-        )
-    ]
-
-    graph.add_edges_from_no_data(edges)
-
+    graph, _, _ = graph_results(table, all_nodes)
     components = rx.connected_components(graph)
     component_sizes = Counter(len(component) for component in components)
 
     return {
         "num_components": len(components),
-        "total_nodes": len(unique_nodes),
-        "total_edges": len(edges),
+        "total_nodes": graph.num_nodes(),
+        "total_edges": graph.num_edges(),
         "component_sizes": component_sizes,
         "min_component_size": min(component_sizes.keys()),
         "max_component_size": max(component_sizes.keys()),
@@ -122,7 +110,7 @@ def calculate_min_max_edges(
     max_edges += _max_edges_component(
         left_after_min_mod, right_after_min_mod, deduplicate
     ) * (max_mod - min_mod)
-    # components where both side have maximum nodes
+    # components where both side have minimum nodes
     min_edges += _min_edges_component(left_div, right_div, deduplicate) * (
         num_components - max_mod
     )
@@ -173,6 +161,9 @@ def generate_dummy_probabilities(
     )
 
     mode = "dedupe" if deduplicate else "link"
+
+    if total_rows == 0:
+        raise ValueError("At least one edge must be generated")
     if total_rows < min_possible_edges:
         raise ValueError(
             dedent(f"""
@@ -204,7 +195,7 @@ def generate_dummy_probabilities(
     left_components = np.array_split(np.array(left_values), num_components)
     right_components = np.array_split(np.array(right_values), num_components)
     # For each left-right component pair, the right equals the left rotated by one
-    right_components = [np.roll(c, 1) for c in right_components]
+    right_components = [np.roll(c, -1) for c in right_components]
 
     all_edges = []
 
diff --git a/src/matchbox/common/transform.py b/src/matchbox/common/transform.py
index 1ebcc36b..72a10e6f 100644
--- a/src/matchbox/common/transform.py
+++ b/src/matchbox/common/transform.py
@@ -2,7 +2,7 @@
 import multiprocessing
 from collections import defaultdict
 from concurrent.futures import ProcessPoolExecutor
-from typing import Callable, Generic, Hashable, TypeVar
+from typing import Callable, Generic, Hashable, Iterable, TypeVar
 
 import numpy as np
 import pyarrow as pa
@@ -92,14 +92,23 @@ def to_clusters(results: ProbabilityResults) -> ClusterResults:
     )
 
 
-def attach_components_to_probabilities(probabilities: pa.Table) -> pa.Table:
+def graph_results(
+    probabilities: pa.Table, all_nodes: Iterable[int] | None = None
+) -> tuple[rx.PyDiGraph, np.ndarray, np.ndarray]:
     """
-    Takes an Arrow table of probabilities and adds a component column.
+    Convert probability table to graph representation.
 
-    Expects an Arrow table of column, left, right, probability.
-
-    Returns a table with an additional column, component.
+    Args:
+        probabilities: PyArrow table with 'left', 'right' columns
+        all_nodes: superset of node identities figuring in probabilities table.
+            Used to optionally add isolated nodes to the graph.
+    Returns:
+        A tuple containing:
+        - Rustwork directed graph
+        - A list mapping the 'left' probabilities column to node indices in the graph
+        - A list mapping the 'right' probabilities column to node indices in the graph
     """
+
     # Create index to use in graph
     unique = pc.unique(
         pa.concat_arrays(
@@ -109,8 +118,9 @@ def attach_components_to_probabilities(probabilities: pa.Table) -> pa.Table:
             ]
         )
     )
-    left_indices = pc.index_in(probabilities["left"], unique)
-    right_indices = pc.index_in(probabilities["right"], unique)
+
+    left_indices = pc.index_in(probabilities["left"], unique).to_numpy()
+    right_indices = pc.index_in(probabilities["right"], unique).to_numpy()
 
     # Create and process graph
     n_nodes = len(unique)
@@ -119,9 +129,25 @@ def attach_components_to_probabilities(probabilities: pa.Table) -> pa.Table:
     graph = rx.PyGraph(node_count_hint=n_nodes, edge_count_hint=n_edges)
     graph.add_nodes_from(range(n_nodes))
 
-    edges = tuple(zip(left_indices.to_numpy(), right_indices.to_numpy(), strict=True))
+    if all_nodes is not None:
+        isolated_nodes = len(set(all_nodes) - set(unique.to_pylist()))
+        graph.add_nodes_from(range(isolated_nodes))
+
+    edges = tuple(zip(left_indices, right_indices, strict=True))
     graph.add_edges_from_no_data(edges)
 
+    return graph, left_indices, right_indices
+
+
+def attach_components_to_probabilities(probabilities: pa.Table) -> pa.Table:
+    """
+    Takes an Arrow table of probabilities and adds a component column.
+
+    Expects an Arrow table of column, left, right, probability.
+
+    Returns a table with an additional column, component.
+    """
+    graph, left_indices, _ = graph_results(probabilities)
     components = rx.connected_components(graph)
 
     # Convert components to arrays, map back to input to join, and reattach
@@ -130,10 +156,10 @@ def attach_components_to_probabilities(probabilities: pa.Table) -> pa.Table:
         np.arange(len(components)), [len(c) for c in components]
     )
 
-    node_to_component = np.zeros(len(unique), dtype=np.int64)
+    node_to_component = np.zeros(graph.num_nodes(), dtype=np.int64)
     node_to_component[component_indices] = component_labels
 
-    edge_components = pa.array(node_to_component[left_indices.to_numpy()])
+    edge_components = pa.array(node_to_component[left_indices])
 
     return probabilities.append_column("component", edge_components).sort_by(
         [("component", "ascending"), ("probability", "descending")]
diff --git a/src/matchbox/server/postgresql/benchmark/generate_tables.py b/src/matchbox/server/postgresql/benchmark/generate_tables.py
index f24b594e..c18dadb1 100644
--- a/src/matchbox/server/postgresql/benchmark/generate_tables.py
+++ b/src/matchbox/server/postgresql/benchmark/generate_tables.py
@@ -217,8 +217,8 @@ def generate_result_tables(
         indexed_parents, clusters["probability"].to_numpy()
     )
 
-    set_children = set(indexed_children)
     source_entries = left_ids if right_ids is None else left_ids + right_ids
+    set_children = set(indexed_children)
     top_clusters = [c for c in final_clusters + source_entries if c not in set_children]
 
     probabilities_table = pa.table(
diff --git a/test/common/test_factories.py b/test/common/test_factories.py
index 967f3851..e48aa3e3 100644
--- a/test/common/test_factories.py
+++ b/test/common/test_factories.py
@@ -48,6 +48,13 @@ def test_calculate_min_max_edges(
 @pytest.mark.parametrize(
     ("parameters"),
     [
+        {
+            "left_count": 5,
+            "right_count": None,
+            "prob_range": (0.6, 0.8),
+            "num_components": 3,
+            "total_rows": 2,
+        },
         {
             "left_count": 1000,
             "right_count": None,
@@ -77,7 +84,13 @@ def test_calculate_min_max_edges(
             "total_rows": calculate_min_max_edges(1000, 1000, 10, False)[1],
         },
     ],
-    ids=["dedupe_min", "dedupe_max", "link_min", "link_max"],
+    ids=[
+        "dedupe_no_edges",
+        "dedupe_min",
+        "dedupe_max",
+        "link_min",
+        "link_max",
+    ],
 )
 def test_generate_dummy_probabilities(parameters: dict[str, Any]):
     len_left = parameters["left_count"]
@@ -103,19 +116,19 @@ def test_generate_dummy_probabilities(parameters: dict[str, Any]):
         num_components=n_components,
         total_rows=total_rows,
     )
-    report = verify_components(table=probabilities)
+    report = verify_components(table=probabilities, all_nodes=rand_vals)
     p_left = probabilities["left"].to_pylist()
     p_right = probabilities["right"].to_pylist()
 
     assert report["num_components"] == n_components
 
-    # Link
+    # Link job
     if right_values:
-        assert set(p_left) == set(left_values)
-        assert set(p_right) == set(right_values)
+        assert set(p_left) <= set(left_values)
+        assert set(p_right) <= set(right_values)
     # Dedupe
     else:
-        assert set(p_left) | set(p_right) == set(left_values)
+        assert set(p_left) | set(p_right) <= set(left_values)
 
     assert (
         pc.max(probabilities["probability"]).as_py() / 100
diff --git a/uv.lock b/uv.lock
index 6f2472e6..a3c3f534 100644
--- a/uv.lock
+++ b/uv.lock
@@ -945,7 +945,6 @@ dependencies = [
     { name = "connectorx" },
     { name = "duckdb" },
     { name = "httpx" },
-    { name = "ipywidgets" },
     { name = "matplotlib" },
     { name = "pandas" },
     { name = "psycopg2" },
@@ -970,6 +969,7 @@ server = [
 dev = [
     { name = "docker" },
     { name = "ipykernel" },
+    { name = "ipywidgets" },
     { name = "pre-commit" },
     { name = "pytest" },
     { name = "pytest-cov" },
@@ -989,7 +989,6 @@ requires-dist = [
     { name = "duckdb", specifier = ">=1.1.1" },
     { name = "fastapi", extras = ["standard"], marker = "extra == 'server'", specifier = ">=0.115.0,<0.116.0" },
     { name = "httpx", specifier = ">=0.28.0" },
-    { name = "ipywidgets", specifier = ">=8.1.5" },
     { name = "matplotlib", specifier = ">=3.9.2" },
     { name = "pandas", specifier = ">=2.2.3" },
     { name = "pg-bulk-ingest", marker = "extra == 'server'", specifier = ">=0.0.54" },
@@ -1009,6 +1008,7 @@ requires-dist = [
 dev = [
     { name = "docker", specifier = ">=7.1.0" },
     { name = "ipykernel", specifier = ">=6.29.5" },
+    { name = "ipywidgets", specifier = ">=8.1.5" },
     { name = "pre-commit", specifier = ">=3.8.0" },
     { name = "pytest", specifier = ">=8.3.3" },
     { name = "pytest-cov", specifier = ">=5.0.0" },