Cosine Similarity

Overview

In cosine similarity, data objects in a dataset are treated as vectors, and it uses the cosine value of the angle between two vectors to indicate the similarity between them. In the graph, specifying N numeric properties (features) of nodes to form N-dimensional vectors, two nodes are considered similar if their vectors are similar.

Cosine similarity ranges from -1 to 1; 1 means that the two vectors have the same direction, -1 means that the two vectors have the opposite direction.

In 2-dimensional space, the cosine similarity between vectors A(a₁, a₂) and B(b₁, b₂) is computed as:

In 3-dimensional space, the cosine similarity between vectors A(a₁, a₂, a₃) and B(b₁, b₂, b₃) is computed as:

The following diagram shows the relationship between vectors A and B in 2D and 3D spaces, as well as the angle θ between them:

Generalize to N-dimensional space, the cosine similarity is computed as:

Considerations

• Theoretically, the calculation of cosine similarity between two nodes does not depend on their connectivity.
• The value of cosine similarity is independent of the length of the vectors, but only the direction of the vectors.

Syntax

• Command: algo(similarity)
• Parameters:

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

UQL
algo(similarity).params({
  uuids: [1], 
  uuids2: [2,3,4],
  node_schema_property: ['price', 'weight', 'width', 'height']
}).write({
  file:{ 
    filename: 'cs_result'
  }
})

Results: File cs_result

File
product1,product2,0.986529
product1,product3,0.878858
product1,product4,0.816876

UQL
algo(similarity).params({
  uuids: [1,2,3,4],
  node_schema_property: ['price', 'weight', 'width', 'height'],
  type: 'cosine'
}).write({
  file:{ 
    filename: 'list'
  }
})

Results: File list

File
product1,product2,0.986529
product1,product3,0.878858
product1,product4,0.816876
product2,product1,0.986529
product2,product3,0.934217
product2,product4,0.881988
product3,product2,0.934217
product3,product4,0.930153
product3,product1,0.878858
product4,product3,0.930153
product4,product2,0.881988
product4,product1,0.816876

Direct Return

UQL
algo(similarity).params({
  uuids: [1,2], 
  uuids2: [2,3,4],
  node_schema_property: ['price', 'weight', 'width', 'height'],
  type: 'cosine'
}) as cs
return cs

Results: cs

UQL
algo(similarity).params({
  uuids: [1,2],
  type: 'cosine',
  node_schema_property: ['price', 'weight', 'width', 'height'],
  top_limit: 1
}) as top
return top

Results: top

Stream Return

UQL
algo(similarity).params({
  uuids: [3], 
  uuids2: [1,2,4],
  node_schema_property: ['@product.price', '@product.weight', '@product.width'],
  type: 'cosine'
}).stream() as cs
where cs.similarity > 0.8
return cs

Results: cs

UQL
algo(similarity).params({
  uuids: [1,3],
  node_schema_property: ['price', 'weight', 'width', 'height'],
  type: 'cosine',
  top_limit: 1
}).stream() as top
return top

Results: top