Overview

Jaccard similarity, also known as Jaccard index, was proposed by Paul Jaccard in 1901. It is an indicator of node similarity defined based on the semi-structured information of the internet. It divides the size of the intersection of two sets by the size of their union with the purpose to indicate how similar the two sets are. In the graph, Jaccard similarity uses node to represent set, and neighbors of node to represent elements in set, and to calculate the proportion of common neighbors in all neighbors.

In application, elements in set typically are a series of properties of an entity. For instance, when calculating the similarity between two credit applications, elements are the phone number, email, device IP, company name and so on in the application form. In general graph applications, these kinds of information are often stored as properties of a node; however, when executing this algorithm, these information is designed as nodes and incorporated into the graph.

The range of values of Jaccard similarity is [0,1]; the larger the value, the more similar the two sets are.

Basic Concept

Set

A set consists of multiple elements; elements in a set are unordered and distinct; the number of elements in set A is the size of set A, denoted as |A|.

Set that consists of common elements of set A and set B is called the intersection of A and B, denoted as A⋂B; set consists of all elements of set A and set B is called the union of A and B, denoted as A⋃B.

In the image above, set A is {b,c,e,f,g}, set B is {a,d,b,g}, intersection A⋂B is {b,g}, union A⋃B is {a,b,c,d,e,f,g}.

Jaccard Similarity

Known sets A and B, Jaccard similarity between them can be expressed as:

Jaccard similarity between sets A and B in the previous example can be calculated upon this definition: 2 / 7 = 0.2857.

Neighbors

In the graph, Kx is the set of neighbors of node x to represent set A, Ky is the set of neighbors of node y to represent set B. Note that neither Kx nor Ky contains repeated value, nor x, nor y, so the following interferences need to be eliminated when finding neighbors by edge in the graph:

• Multiple edges between x/y and their neighbors
• Self-loop edges of x and y
• Edges between x and y

In the graph above, the red and green nodes have 2 common neighbors and 6 neighbors in total, their Jaccard similarity is 2 / 6 = 0.3333.

Special Case

Lonely Node, Disconnected Graph

There is rarely computational valuable lonely node (empty set) in practice, intersection that involves lonely node is empty, and Jaccard similarity is 0.

For two nodes belong to different connected components, their Jaccard similarity must be 0.

Self-loop Edge

Self-loop edge of a node does not increase the number of neighbors of the node.

Directed Edge

For directed edges, the algorithm ignores the direction of edges but calculates them as undirected edges.

Results and Statistics

The graph below shows which sports userA, userB, userC and userD (their UUID are 1, 2, 3 and 4 respectively) likes:

Algorithm results 1: Calculate the similarity between each user and other 3 users, return node and top_list; top_list is all nodes which has the similarity > 0 with the node, those nodes' UUID and the similarity between then are displayed (ordered by the descending similarity)

Algorithm results 2: Calculate the similarity between userC and other 3 users, return node1, node2 and similarity

Algorithm statistics: N/A

Command and Configuration

• Command: algo(similarity)
• Configurations for the parameter params():

This algorithm has two calculation modes:

1. Selection mode

• When ids2 and uuids2 are ignored or given empty value, for each node in ids/uuids, select all nodes that have similarity > 0 with it (ordered by the descending similarity).
• The configuration item limit is to control the number of nodes in ids/uuids to return.

2. Pairing mode

• When ids2/uuids2 is given solid value, pair node in ids/uuids with node in ids2/uuids2 (Cartesian product), similarities are calculated for each node pair.
• The configuration item limit is to control the number of nodes pairs to return.

Example: Calculate Jaccard similarity between any two nodes in set A (UUID = 1,2) and set B (UUID = 3,4)

algo(similarity).params({ 
  uuids: [1,2], 
  uuids2: [3,4],
  type: "jaccard"
}) as jacc
return jacc

Example: For each node in set (UUID = 1,2,3), calculate the most similar node to each node measured by Jaccard similarity

algo(similarity).params({ 
  uuids: [1,2,3], 
  type: "jaccard",
  top_limit: 1
}) as nodes
return nodes

Algorithm Execution

Task Writeback

1. File Writeback

Example: Calculate Jaccard similarity between node UUID = 3 and other nodes, write the algorithm results back to file named s3

algo(similarity).params({
  uuids: 3,
  uuids2: [1,2,4],
  type: "jaccard"
}).write({
  file:{ 
    filename: "s3"
  }
})

2. Property Writeback

Not supported by this algorithm.

3. Statistics Writeback

This algorithm has no statistics.

Direct Return

Example: Calculate Jaccard similarity between each node and other nodes, define algorithm results as alias named similarity, and return the results

algo(similarity).params({
  uuids: [1,2,3,4],
  type: "jaccard"
}) as s
return s

Streaming Return

Example: Calculate Jaccard similarity between each node and other nodes, only keep the one with the highest similarity, define algorithm results as alias named similarity, and return the results

algo(similarity).params({
  uuids: [1,2,3,4],
  type: "jaccard",
  top_limit: 1 
}).stream() as similarity
return similarity

Real-time Statistics

This algorithm has no statistics.