Jaccard Similarity

Overview

Jaccard similarity, or Jaccard index, was proposed by Paul Jaccard in 1901. It’s a metric of similarity for two sets of data. In the graph, collecting the neighbors of a node into a set, two nodes are considered similar if their neighborhood sets are similar.

Jaccard similarity ranges from 0 to 1; 1 means that two sets are exactly the same, 0 means that the two sets do not have any element in common.

Concepts

Jaccard Similarity

Given two sets A and B, the Jaccard similarity between them is computed as:

In the following example, set A = {b,c,e,f,g}, set B = {a,d,b,g}, their intersection A⋂B = {b,g}, their union A⋃B = {a,b,c,d,e,f,g}, hence the Jaccard similarity between A and B is 2 / 7 = 0.285714.

When applying Jaccard Similarity to compare two nodes in a graph, we use the 1-hop neighborhood set to represent each target node. The 1-hop neighborhood set:

• contains no repeated nodes;
• excludes the two target nodes.

In this graph, the 1-hop neighborhood set of nodes u and v is:

• N_u = {a,b,c,d,e}
• N_v = {d,e,f}

Therefore, the Jaccard similarity between nodes u and v is 2 / 6 = 0.333333.

In practice, you may need to convert some node properties into node schemas in order to calculate the similarity index that is based on common neighbors, just as the Jaccard Similarity. For instance, when considering the similarity between two applications, information like phone number, email, device IP, etc. of the application might have been stored as properties of @application node schema; they need to be designed as nodes and incorporated into the graph in order to be used for comparison.

Weighted Jaccard Similarity

The Weighted Jaccard Similarity is an extension of the classic Jaccard Similarity that takes into account the weights associated with elements in the sets being compared.

The formula for Weighted Jaccard Similarity is given by:

In this weighted graph, the union of the 1-hop neighborhood sets N_u and N_v is {a,b,c,d,e,f}. Set each element in the union set to the sum of the edge weights between the target node and the corresponding node, or 0 if there are no edges between them:

Therefore, the Weighted Jaccard Similarity between nodes u and v is (0+0+0+0.5+0.5+0) / (1+1+1+1+1.6+1) = 0.151515.

Please ensure that the sum of the edge weights between the target node and the neighboring node is greater than or equal to 0.

Considerations

• The Jaccard Similarity algorithm ignores the direction of edges but calculates them as undirected edges.
• The Jaccard Similarity algorithm ignores any self-loop.

Example Graph

To create this graph:

// Runs each row separately in order in an empty graphset
create().node_schema("user").node_schema("sport").edge_schema("like")
create().edge_property(@like, "weight", int32)
insert().into(@user).nodes([{_id:"userA"}, {_id:"userB"}, {_id:"userC"}, {_id:"userD"}])
insert().into(@sport).nodes([{_id:"running"}, {_id:"tennis"}, {_id:"baseball"}, {_id:"swimming"}, {_id:"badminton"}, {_id:"iceball"}])
insert().into(@like).edges([{_from:"userA", _to:"tennis", weight:2}, {_from:"userA", _to:"baseball", weight:1}, {_from:"userA", _to:"swimming", weight:3}, {_from:"userA", _to:"badminton", weight:2}, {_from:"userB", _to:"running", weight:1}, {_from:"userB", _to:"swimming", weight:3}, {_from:"userC", _to:"swimming", weight:2}, {_from:"userD", _to:"running", weight:1}, {_from:"userD", _to:"badminton", weight:2}, {_from:"userD", _to:"iceball", weight:2}])

Creating HDC Graph

To load the entire graph to the HDC server hdc-server-1 as hdc_sim_nbr:

CALL hdc.graph.create("hdc-server-1", "hdc_sim_nbr", {
  nodes: {"*": ["*"]},
  edges: {"*": ["*"]},
  direction: "undirected",
  load_id: true,
  update: "static",
  query: "query",
  default: false
})

hdc.graph.create("hdc_sim_nbr", {
  nodes: {"*": ["*"]},
  edges: {"*": ["*"]},
  direction: "undirected",
  load_id: true,
  update: "static",
  query: "query",
  default: false
}).to("hdc-server-1")

Parameters

Algorithm name: similarity

The algorithm has two calculation modes:

1. Pairing: When both ids/uuids and ids2/uuids2 are configured, each node in ids/uuids is paired with each node in ids2/uuids2 (excluding self-pairing), and pairwise similarities are computed.
2. Selection: When only ids/uuids is configured, pairwise similarities are computed between each target node and all other nodes in the graph. The results include all or a limited number of nodes with a similarity > 0 to the target node, ordered in descending similarity.

File Writeback

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

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

Result:

_id1,_id2,similarity
userC,userA,0.25
userC,userB,0.5
userC,userD,0

Full Return

CALL algo.similarity("hdc_sim_nbr", {
  params: {
    return_id_uuid: "id",
    ids: ["userA","userB"], 
    ids2: ["userB","userC","userD"],
    type: "jaccard"
  },
  return_params: {}
}) YIELD jacc
RETURN jacc

exec{
  algo(similarity).params({
    return_id_uuid: "id",
    ids: ["userA","userB"], 
    ids2: ["userB","userC","userD"],
    type: "jaccard"
  }) as jacc
  return jacc
} on hdc_sim_nbr

Result:

Stream Return

CALL algo.similarity("hdc_sim_nbr", {
  params: {
    return_id_uuid: "id",
    ids: ["userA"], 
    type: "jaccard",
    edge_weight_property: "weight",
    top_limit: 2    
  },
  return_params: {
  	stream: {}
  }
}) YIELD jacc
RETURN jacc

exec{
  algo(similarity).params({
    return_id_uuid: "id",
    ids: ["userA"], 
    type: "jaccard",
    edge_weight_property: "weight",
    top_limit: 2  
  }).stream() as jacc
  return jacc
} on hdc_sim_nbr

Result: