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():

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:

Task Writeback

1. File Writeback

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: [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

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

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: [1], 
  uuids2: [2,3,4],
  node_schema_property: [price,weight,width,height],
  type: "euclideanDistance"
}) as distance
return distance
order by distance.similarity desc

Results:

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
return top

Results:

Streaming Return

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: [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:

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
return top

Results:

Real-time Statistics

This algorithm has no statistics.