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v4.0
v4.0

# Euclidean Distance

## Overview

Euclidean distance is named after the ancient Greek mathematician Euclid, this is the most commonly used distance measurement which measures the absolute distance between two nodes in a multi-dimensional space, that is, the shortest straight-line distance between them. The Euclidean distance between two nodes in graph is calculated by using N properties of node to form two N-dimensional vectors.

## Basic Concept

### Vector

Vector is one of the basic concepts in Advanced Mathematics, vectors in low dimensional spaces are relatively easy to understand and express. The following diagram shows the relationship between vectors A, B and coordinate axes in 2- and 3-dimensional spaces respectively, as well as the angle `θ` between them: When comparing two nodes in graph, N properties of node are used to form the two N-dimensional vectors.

### Euclidean Distance

In 2-dimensional space, the formula to calculate the Euclidean distance is: In 3-dimensional space, the formula to calculate the Euclidean distance is: Generalize to n-dimensional space, the formula to calculate the Euclidean distance is: where xi1 represents the i-th dimensional coordinates of the first node, xi2 represents the i-th dimensional coordinates of the second node.

The range of Euclidean distance values is [0,+∞]; the smaller the value, the more similar the two nodes are.

### Normalized Euclidean Distance

Normalized Euclidean distance is an improvement on Euclidean distance. The range of normalized Euclidean distance values is [0,1], the larger the value, the more similar the two nodes are.

Ultipa adopts the following formula to normalize Euclidean distance: ## Special Case

### Isolated Node, Disconnected Graph

Theoretically, the calculation of Euclidean distance between two nodes does not depend on the existence of edges in the graph. Regardless of whether the two nodes to be calculated are isolated nodes or whether they are in the same connected component, it does not affect the calculation of their Euclidean distance.

### Self-loop Edge

The calculation of Euclidean distance has nothing to do with edges.

### Directed Edge

The calculation of Euclidean distance has nothing to do with edges.

## Command and Configuration

• Command: `algo(similarity)`
• Configurations for the parameter `params()`:
Name
Type
Default
Specification
Description
ids / uuids []`_id` / []`_uuid` / Mandatory IDs or UUIDs of the first set of nodes to be calculated
ids2 / uuids2 []`_id` / []`_uuid` / Optional IDs or UUIDs of the second set of nodes to be calculated
type string cosine jaccard / overlap / cosine / pearson / euclideanDistance / euclidean Measurement of the similarity:
jaccard: Jaccard Similarity
overlap: Overlap Similarity
cosine: Cosine Similarity
pearson: Pearson Correlation Coefficient
euclideanDistance: Euclidean Distance
euclidean: Normalized Euclidean Distance
node_schema_property []`@<schema>?.<property>` / Numeric node property; LTE needed; schema can be either carried or not When `type` is cosine / pearson / euclideanDistance / euclidean, must specify two or more node properties to form the vector; when `type` is jaccard / overlap, this parameter is invalid
limit int -1 >=-1 Number of results to return; return all results if sets to -1
top_limit int -1 >=-1 Only available in the selection mode, limit the length of selection results (`top_list`) of each node, return the full `top_list` if sets to -1

## Calculation Mode

This algorithm has two calculation modes:

1. Pairing mode: when two sets of valid nodes are configured, pair each node in the first set with each node in the second set (Cartesian product), similarities are calculated for all node pairs.
2. Selection mode: when only one set (the first) of valid nodes are configured, for each node in the set, calculate its similarities with all other nodes in the graph, return the results if the similarity > 0, order the results the descending similarity.

## Examples

### Example Graph

The example graph has product1, product2, product3 and product4 (UUIDs are 1, 2, 3 and 4 in order; edges are ignored), product node has properties price, weight, weight and height: #### 1. File Writeback

Calculation Mode
Configuration
Data in Each Row
Pairing mode filename `node1`,`node2`,`similarity`
Selection mode filename `node`,`top_list`

Example: Calculate Euclidean distance between product UUID = 1 and products UUID = 2,3,4 through properties price, weight, width and height, write the algorithm results back to file

``````algo(similarity).params({
uuids: ,
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
``````

Example: Calculate normalized Euclidean distance between products UUID = 1,2,3,4 and all other products in the graph respectively through properties price, weight, width and height, write the algorithm results back to file

``````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;
``````

#### 2. Property Writeback

Not supported by this algorithm.

#### 3. Statistics Writeback

This algorithm has no statistics.

### Direct Return

Calculation Mode
Alias Ordinal
Type Description Column Name
Pairing mode 0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
Selection mode 0 []perNode Node and its selection results `node`, `top_list`

Example: Calculate Euclidean distance between product UUID = 1 and products UUID = 2,3,4 through properties price, weight, width and height, order results in the descending distance

``````algo(similarity).params({
uuids: ,
uuids2: [2,3,4],
node_schema_property: [price,weight,width,height],
type: "euclideanDistance"
}) as distance
return distance
order by distance.similarity desc
``````

Results:

node1 node2 similarity
1 4 165.178691119648
1 3 143.96180048888
1 2 94.3822017119753

Example: Select the product with the highest normalized Euclidean distance with products UUID = 1,2 respectively through properties price, weight, width and height,

``````algo(similarity).params({
uuids: [1,2],
type: "euclidean",
node_schema_property: [price,weight,width,height],
top_limit: 1
}) as top
``````

Results:

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

### Streaming Return

Calculation Mode
Alias Ordinal
Type Description Column Name
Pairing mode 0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
Selection mode 0 []perNode Node and its selection results `node`, `top_list`

Example: Calculate normalized Euclidean distance between product UUID = 3 and products UUID = 1,2,4 through properties price, weight, width and height, only return results that have similariy above 0.01

``````algo(similarity).params({
uuids: ,
uuids2: [1,2,4],
node_schema_property: [price,weight,width,height],
type: "euclidean"
}).stream() as distance
where distance.similarity > 0.01
return distance
``````

Results:

node1 node2 similarity
3 2 0.0180816471945529
3 4 0.0240910110982062

Example: Select the product with the farthest euclidean Distance with products UUID = 1,3 respectively

``````algo(similarity).params({
uuids: [1,3],
node_schema_property: [price,weight,width,height],
type: "euclideanDistance",
top_limit: 1
}).stream() as top