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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>` Must LTE / No Two or more numeric node properties must be specified to represent the nodes
limit int >=-1 `-1` Yes Number of results to return, `-1` to return all results
top_limit int >=-1 `-1` Yes Limit the length of `top_list`, `-1` to return the full `top_list`

This algorithm has two calculation modes:

1. Pairing: when `ids/uuids` and `ids2/uuids2` are both configured, pairing nodes in the first group with nodes in the second group (Cartesian product) to compute pair-wise similarities.
2. Selection: when only `ids/uuids` is configured, for each node in the group, computing pair-wise similarities between it and all other nodes in the graph in order to select the most similar nodes, the returned `top_list` includes all nodes that have similarity > 0 with it 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

Calculation Mode Spec Content
Pairing filename `node1`,`node2`,`similarity`
Selection filename `node`,`top_list`
``````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;product3:0.006898;product4:0.006018;
product2,product3:0.018082;product4:0.013309;product1:0.010484;
product3,product4:0.024091;product2:0.018082;product1:0.006898;
product4,product3:0.024091;product2:0.013309;product1:0.006018;
``````

### Direct Return

Calculation Mode
Alias Ordinal
Type
Description Columns
Pairing 0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
Selection 0 []perNode Node and its selection results `node`, `top_list`
``````algo(similarity).params({
uuids: [1],
uuids2: [2,3,4],
node_schema_property: [price,weight,width,height],
type: "euclideanDistance"
}) as distance
return distance
order by distance.similarity desc
``````

Results: distance

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

Results: top

node top_list
1 2:0.010484,
2 3:0.018082,

### Stream Return

Calculation Mode
Alias Ordinal
Type
Description Columns
Pairing 0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
Selection 0 []perNode Node and its selection results `node`, `top_list`
``````algo(similarity).params({
uuids: [3],
uuids2: [1,2,4],
node_schema_property: [price,weight,width,height],
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