Euclidean Distance

Overview

In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. In the graph, N numeric node properties (features) are specified to represent each node's position in an N-dimensional Euclidean space.

Concepts

Euclidean Distance

In 2-dimensional space, the formula to compute the Euclidean distance between points A(x1, y1) and B(x2, y2) is:

In 3-dimensional space, the formula to compute the Euclidean distance between points A(x1, y1, z1) and B(x2, y2, z2) is:

Generalized to N-dimensional space, the formula to compute the Euclidean distance is:

where xi1 represents the i-th dimensional coordinates of the first point, and xi2 represents the i-th dimensional coordinates of the second point.

Euclidean distance ranges from 0 to +∞; smaller values indicate greater similarity between the two nodes.

Normalized Euclidean Distance

This algorithm returns the normalized Euclidean distance using the formula 1 / (1 + d), which scales the result into the range (0, 1]:

Values closer to 1 indicate greater similarity. For example, if the Euclidean distance between two nodes is 94.38, the normalized similarity is 1 / (1 + 94.38) ≈ 0.01049.

Considerations

• The calculation of Euclidean distance between two nodes is independent of their connectivity in the graph — it uses node properties only.

Example Graph

GQL
INSERT (:product {_id: "product1", price: 50, weight: 160, width: 20, height: 152}),
       (:product {_id: "product2", price: 42, weight: 90, width: 30, height: 90}),
       (:product {_id: "product3", price: 24, weight: 50, width: 55, height: 70}),
       (:product {_id: "product4", price: 38, weight: 20, width: 32, height: 66})

Parameters

Supports three computation modes:

• All-pairs: When both ids and ids2 are empty, computes similarity between all node pairs in the graph.
• Pairing: When both ids and ids2 are specified, computes similarity between each node in ids and each node in ids2.
• Selection: When only ids is specified (no ids2), computes similarity between each node in ids and all other nodes. Use top_limit to limit results per source node.

Run Mode

Returns:

Euclidean similarity in pairing mode:

GQL
CALL algo.similarity({
  type: "euclidean",
  ids: ["product1"],
  ids2: ["product2", "product3", "product4"],
  node_property: ["price", "weight", "width", "height"]
}) YIELD node1, node2, similarity

Result:

Euclidean similarity in selection mode (top 1 per source node):

GQL
CALL algo.similarity({
  type: "euclidean",
  ids: ["product1", "product3"],
  node_property: ["price", "weight", "width", "height"],
  top_limit: 1
}) YIELD node1, node2, similarity

Result:

Stream Mode

Returns the same columns as run mode, streamed for memory efficiency.

GQL
CALL algo.similarity.stream({
  type: "euclidean",
  ids: ["product1"],
  node_property: ["price", "weight", "width", "height"],
  order: "desc"
}) YIELD node1, node2, similarity
RETURN node1, node2, similarity

Result:

Stats Mode

Returns:

GQL
CALL algo.similarity.stats({
  type: "euclidean",
  node_property: ["price", "weight", "width", "height"]
}) YIELD pairCount, minSimilarity, maxSimilarity, avgSimilarity

Result:

Write Mode

Computes results and writes them back to node properties. The write configuration is passed as a second argument map.

Write parameters:

Writable columns:

Returns:

GQL
CALL algo.similarity.write({
  type: "euclidean",
  ids: ["product1", "product2"],
  node_property: ["price", "weight", "width", "height"]
}, {
  db: {
    property: "sim_score"
  }
}) YIELD task_id, nodesWritten, computeTimeMs, writeTimeMs