Basic  

Overview

Adamic-Adar index is a node similarity metric defined based on the structured information of the internet, this is how it differs from Jaccard similarity (semi-structured information). AA index uses the weights of the common neighbor nodes of two nodes as the similarity of the two nodes, it was proposed by Lada A. Adamic and Eytan Adar in 2003.

Related materials are as below:

• Friends and Neighbors on the Web
• Adamic-Adar Index

Basic Concept

Node Weight

In AA index, the weight of a node x is defined as the reciprocal of the logarithm based on 10 of the number of neighbors Γ(x) of x. Note that x is not included in its neighbors and neighbors are deduplicated:

Known from the definition, the weight of the yellow node in the graph below is: 1/(log4) = 1.6610, the weight of the green node is: 1/(log3) = 2.0959.

AA Similarity

AA similarity between nodes x and y is the sum of weights w(z) of their common neighbors z. Note that x and y are not included in the common neighbors and common neighbors are deduplicated:

Still taking the previous graph as an example, AA similarity of the blue and red nodes is the sum of weights of the yellow and green nodes, which is 1/(log4) + 1/(log3) = 3.7569.

Special Case

Lonely Node, Disconnected Graph

There is no edge between lonely node and any other nodes in the graph, Ultipa AA Index algorithm does not calculate AA index between lonely node and any other node, nor does it calculate AA index between two nodes in different connected components.

Self-loop Edge

AA Index algorithm ignores all self-loop edges when calculating neighbor nodes.

Directed Edge

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

Results and Statistics

Take the graph below as an example, run the AA Index algorithm in the graph:

Algorithm results: Calculate AA index for each edge (its the start node and end node), return edge_uuid and adamic_adar_value

Algorithm statistics: N/A

Command and Configuration

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

Ultipa AA Index algorithm calculates the AA similarity of all adjacent node pairs (connected by edge) in the whole graph.

Algorithm Execution

Task Writeback

1. File Writeback

Example: Calculate AA index of all adjacent node pairs, write the algorithm results back to file named aa

algo(adamic_adar).params().write({
  file:{ 
    filename: "aa"
  }
})

2. Property Writeback

Example: Calculate AA index of all adjacent node pairs, write the AA index back to edge property named aaIndex

algo(adamic_adar).params().write({
  db:{ 
    property: "aaIndex"
  }
})

3. Statistics Writeback

This algorithm has no statistics.

Direct Return

Example: Calculate AA index of all adjacent node pairs, define algorithm results as alias named aa, and return the results

algo(adamic_adar).params() as aa
return aa

Streaming Return

Example: Calculate AA index of all adjacent node pairs, define algorithm results as alias named aa, return the results ordered by the the ascending UUID of edges

algo(adamic_adar).params().stream() as aa
return aa order by aa.edge_uuid

Real-time Statistics

This algorithm has no statistics.