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v4.0
    v4.0

    Preferential Attachment

    Overview

    Preferential attachment refers to a common phenomenon in complex network that the more connections a node has, the more likely it establishes new connections. If both nodes have a large number of connections, there is great possibility that they will be connected. In 2002, this method is used by A. Barabási and R. Albert in their proposed BA model for producing random scale-free networks:

    Basic Concept

    Preferential Attachment

    Preferential attachment measurement uses the product of the number of neighbors of two nodes to determine their closeness, which is calculated by the following formula:

    where N(x) and N(y) are neighbor sets of x and y respectively. The larger the value of PA(x,y) is, the closer the two nodes are, value of 0 indicates that the two nodes are not close.

    Taking the above graph as an example, the blue node has 3 neighbors, the red node has 4 neighbors, so the score of their preferential attachment is 3 * 4 = 12.

    Special Case

    Lonely Node, Disconnected Graph

    Lonely node does not have any neighbor node, the algorithm does not calculate the Preferential Attachment between lonely node and any other node, either it considers the Preferential Attachment of two nodes which are located in different connected components.

    Self-loop Edge

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

    Directed Edge

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

    Results and Statistics

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

    Algorithm results: Calculate the Preferential Attachment between node 3 and other nodes, return node1, node2 and num

    node1 node2 num
    3 1 3
    3 2 12
    3 4 12
    3 5 6
    3 6 9
    3 7 3

    Algorithm statistics: N/A

    Command and Configuration

    • Command: algo(topological_link_prediction)
    • Configurations for the parameter params():
    Name
    Type
    Default
    Specification Description
    ids / uuids []_id / []_uuid / Mandatory IDs or UUIDs of the first set of nodes to be calculated, only need to configure one of them; every node in ids/uuids will be paired with every node in ids2/uuids2 for calculation
    ids2 / uuids2 []_id / []_uuid / Mandatory IDs or UUIDs of the second set of nodes to be calculated, only need to configure one of them; every node in ids/uuids will be paired with every node in ids2/uuids2 for calculation
    type string Adamic_Adar Adamic_Adar / Common_Neighbors / Preferential_Attachment / Resource_Allocation / Total_Neighbors Measurement of the closeness of the node pair; Adamic_Adar means to calculate AA index, Common_Neighbors means to calculate the number of common neighbors, Preferential_Attachment means to calculate the score of preferential attachment, Resource_Allocation means to calculate the score of resource allocation, Total_Neighbors means to calculate the number of total neighbors
    limit int -1 >=-1 Number of results to return; return all results if sets to -1 or not set

    Algorithm Execution

    Task Writeback

    1. File Writeback

    Configuration Data in Each Row
    filename node1,node2,num

    Example: Calculate the Preferential Attachment of node UUID = 3 and all other nodes, write the algorithm results back to file named pa

    algo(topological_link_prediction).params({
      uuids: [3],
      uuids2: [1,2,4,5,6,7],
      type: "Preferential_Attachment"
      }).write({
      file:{ 
        filename: "pa"
      }
    })
    

    2. Property Writeback

    Not supported by this algorithm.

    3. Statistics Writeback

    This algorithm has no statistics.

    Direct Return

    Alias Ordinal Type
    Description
    Column Name
    0 []perNodePair Closeness of node pair node1, node2, num

    Example: Calculate the Preferential Attachment of node UUID = 3 and UUID = 4, define algorithm results as alias named pa and return the results

    algo(topological_link_prediction).params({
      uuids: [3],
      uuids2: [4],
      type: "Preferential_Attachment"
    }) as pa 
    return pa
    

    Streaming Return

    Alias Ordinal Type
    Description
    Column Name
    0 []perNodePair Closeness of node pair node1, node2, num

    Example: Calculate the Preferential Attachment of node UUID = 1 and UUID = 5,6,7, return the results in the descending closeness score

    algo(topological_link_prediction).params({
      uuids: [1],
      uuids2: [5,6,7],
      type: "Preferential_Attachment"
    }).stream() as pa 
    return pa order by pa.num desc
    

    Real-time Statistics

    This algorithm has no statistics.

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