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# Euclidean Distance

✓ File Writeback ✕ Property Writeback ✓ Direct Return ✓ Stream Return ✕ Stats

## 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, specifying N numeric properties (features) of nodes to indicate the location of the node 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:

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

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

The Euclidean distance ranges from 0 to +∞; the smaller the value, the more similar the two nodes.

### Normalized Euclidean Distance

Normalized Euclidean distance scales the Euclidean distance into range from 0 to 1; the closer to 1, the more similar the two nodes.

Ultipa adopts the following formula to normalize the Euclidean distance:

## Considerations

• Theoretically, the calculation of Euclidean distance between two nodes does not depend on their connectivity.

## Syntax

• Command: `algo(similarity)`
• Parameters:
Name
Type
Spec
Default
Optional
Description
ids / uuids []`_id` / []`_uuid` / / No ID/UUID of the first group of nodes to calculate
ids2 / uuids2 []`_id` / []`_uuid` / / Yes ID/UUID of the second group of nodes to calculate
type string `euclideanDistance`, `euclidean` `cosine` No Type of similarity; `euclideanDistance` is to compute Euclidean Distance, `euclidean` is to compute Normalized Euclidean Distance
node_schema_property []`@<schema>?.<property>` Numeric type, must LTE / No Specify two or more node properties to form the vectors, all properties must belong to the same (one) schema
limit int ≥-1 `-1` Yes Number of results to return, `-1` to return all results
top_limit int ≥-1 `-1` Yes In the selection mode, limit the maximum number of results returned for each node specified in `ids`/`uuids`, `-1` to return all results with similarity > 0; in the pairing mode, this parameter is invalid

The algorithm has two calculation modes:

1. Pairing: when both `ids`/`uuids` and `ids2`/`uuids2` are configured, pairing each node in `ids`/`uuids` with each node in `ids2`/`uuids2` (ignore the same node) and computing pair-wise similarities.
2. Selection: when only `ids`/`uuids` is configured, for each target node in it, computing pair-wise similarities between it and all other nodes in the graph. The returned results include all or limited number of nodes that have similarity > 0 with the target node and is ordered by the descending similarity.

## Examples

The example graph has 4 products (edges are ignored), each product has properties price, weight, weight and height:

### File Writeback

Spec Content
filename `node1`,`node2`,`similarity`
``````algo(similarity).params({
uuids: [1],
uuids2: [2,3,4],
node_schema_property: ['price', 'weight', 'width', 'height'],
type: 'euclideanDistance'
}).write({
file:{
filename: 'ed'
}
})
``````

Results: File ed

``````product1,product2,94.3822
product1,product3,143.962
product1,product4,165.179
``````
``````algo(similarity).params({
uuids: [1,2,3,4],
node_schema_property: ['price', 'weight', 'width', 'height'],
type: 'euclidean'
}).write({
file:{
filename: 'ed_list'
}
})
``````

Results: File ed_list

``````product1,product2,0.010484
product1,product3,0.006898
product1,product4,0.006018
product2,product3,0.018082
product2,product4,0.013309
product2,product1,0.010484
product3,product4,0.024091
product3,product2,0.018082
product3,product1,0.006898
product4,product3,0.024091
product4,product2,0.013309
product4,product1,0.006018
``````

### Direct Return

Alias Ordinal
Type
Description Columns
0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
``````algo(similarity).params({
uuids: [1,2],
uuids2: [2,3,4],
node_schema_property: ['price', 'weight', 'width', 'height'],
type: 'euclideanDistance'
}) as distance
return distance
``````

Results: distance

node1 node2 similarity
1 2 94.3822017119753
1 3 143.96180048888
1 4 165.178691119648
2 3 54.3046959295419
2 4 74.1350119714025
``````algo(similarity).params({
uuids: [1,2],
type: 'euclidean',
node_schema_property: ['price', 'weight', 'width', 'height'],
top_limit: 1
}) as top
``````

Results: top

node1 node2 similarity
1 2 0.0104841362649574
2 3 0.0180816471945529

### Stream Return

Alias Ordinal
Type
Description Columns
0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
``````algo(similarity).params({
uuids: [3],
uuids2: [1,2,4],
node_schema_property: ['@product.price', '@product.weight', '@product.width'],
type: 'euclidean'
}).stream() as distance
where distance.similarity > 0.01
return distance
``````

Results: distance

node1 node2 similarity
3 2 0.0180816471945529
3 4 0.0240910110982062
``````algo(similarity).params({
uuids: [1,3],
node_schema_property: ['price', 'weight', 'width', 'height'],
type: 'euclideanDistance',
top_limit: 1
}).stream() as top