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

Isolated Node, Disconnected Graph

There is rarely computational valuable isolated node (empty set) in practice, intersection that involves isolated 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.

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 shows the sports liked by userA, userB, userC and userD (UUIDs are 1, 2, 3 and 4 in order):

Task Writeback

1. File Writeback

Example: Calculate Jaccard similarity between userC and the sets of userA, userB and userD, write the algorithm results back to file

algo(similarity).params({
  ids: "userC",
  ids2: ["userA", "userB", "userD"],
  type: "jaccard"
}).write({
  file:{ 
    filename: "sc"
  }
})

Results: File sc

userC,userA,0.25
userC,userB,0.5
userC,userD,0

Example: For each user in the set of UUID = 1,2,3,4, select the nodes that have Jaccard similarity above 0 with the user, write the algorithm results back to file

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

Results: File list

userA,userC:0.250000;userB:0.200000;userD:0.166667;
userB,userC:0.500000;userD:0.250000;userA:0.200000;
userC,userB:0.500000;userA:0.250000;
userD,userB:0.250000;userA:0.166667;

2. Property Writeback

Not supported by this algorithm.

3. Statistics Writeback

This algorithm has no statistics.

Direct Return

Example: Calculate Jaccard similarity between user UUID = 1 and users UUID = 2,3,4, order results in the descending similarity

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

Results:

Example: Select the node with the highest Jaccard similarity with nodes UUID = 1,2 respectively

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

Results:

Streaming Return

Example: Calculate Jaccard similariy between user UUID = 3 and users UUID = 1,2,4, only return results that have similariy above 0

algo(similarity).params({ 
  uuids: [3], 
  uuids2: [1,2,4],
  type: "jaccard"
}).stream() as jacc
where jacc.similarity > 0
return jacc

Results:

Example: Select two nodes with the hightest Jaccard similarity with node UUID = 1

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

Results:

Real-time Statistics

This algorithm has no statistics.