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      Graph Centrality

      Overview

      Graph centrality is used to measure the maximum shortest distance from node to other nodes in its connected component. This concept, along with other concepts (such as closeness centrality, graph diameter, etc.), can be considered jointly to determine whether a node is literally located at the very center of the graph.

      The range of values of Graph Centrality is [0,1]; the larger the value, the closer the node is to the center.

      Basic Concept

      Graph Centrality

      Graph centrality is defined as the reciprocal of the maximum shortest distance from a node to other nodes in its connected component. Please read the chapter Closeness Centrality for the introduction to the Shortest Distance.

      where x is the node to be calculated, y is any node (x is excluded) in the connected component that x is in, d(x,y) is the shortest distance from x to y.

      In the graph above, the shortest distances from each node to the red and green nodes have been marked, and the graph centrality of the red and green nodes are 0.3333 and 0.25. If only calculates the closeness centrality of the two nodes - the red node is 8/(3+3+3+2+1+1+2+1)=0.5, the green node is 8/(1+1+1+1+2+3+4+3)=0.5 - the two values are the same; thus when closeness centrality is the same, graph centrality can serve as a subsidiary basis to determine which node is closer to the center.

      Special Case

      Lonely Node, Disconnected Graph

      Lonely node is not connected with any other node, so its graph centrality is 0. Lonely node does not participate in any graph centrality calculation.

      Nodes in one connected component must NOT participate in the graph centrality calculation of nodes in other connected components.

      Self-loop Edge

      It is the shortest distance between nodes that graph centrality calculates, self-loop edge does not constitute the shortest path, thus it does not participate in the calculation.

      Directed Edge

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

      Results and Statistics

      Take the graph of 10 nodes below as an example, run the Graph Centrality algorithm against all nodes:

      Algorithm results: Calculate graph centrality for each node, return _id, centrality or _uuid, centrality according to the execution method

      _uuid _id centrality
      1 A 0.25000000
      2 B 0.20000000
      3 C 0.20000000
      4 D 0.20000000
      5 E 0.33333334
      6 F 0.33333334
      7 G 0.25000000
      8 H 0.20000000
      9 I 0.25000000
      10 J 0.0000000

      Algorithm statistics: N/A

      Command and Configuration

      • Command: algo(graph_centrality)
      • Configurations for the parameter params():
      Name
      Type
      Default
      Specification
      Description
      ids / uuids []_id / []_uuid / / IDs or UUIDs of nodes to be calculated; all nodes to be calculated if not set
      limit int -1 >=-1 Number of results to return; return all results if sets to -1 or not set
      order string / ASC/DESC, case insensitive To sort the returned results; no sorting is applied if not set

      Example: Calculate graph centrality of all nodes, return all results

      algo(graph_centrality).params({ 
        limit: -1
      }) as gc
      return gc
      

      Algorithm Execution

      Task Writeback

      1. File Writeback

      Configuration Data in Each Row
      filename _id,centrality

      Example: Calculate graph centrality of all nodes, write the algorithm results back to file named res.csv

      algo(graph_centrality).params().write({
        file:{ 
          filename: "res.csv"
        }
      })
      

      2. Property Writeback

      Configuration Writeback Content Type Data Type
      property centrality Node property float

      Example: Calculate graph centrality of all nodes, write the centrality back to node property named graphC

      algo(graph_centrality).params().write({
        db:{ 
          property: "graphC"
        }
      })
      

      3. Statistics Writeback

      This algorithm has no statistics.

      Direct Return

      Alias Ordinal
      Type
      Description
      Column Name
      0 []perNode Node and its graph centrality _uuid, centrality

      Example: Calculate graph centrality of all nodes, define algorithm results as alias named results, and return the 3 results with the highest centrality

      algo(graph_centrality).params({
        order: "desc",
        limit: 3
      }) as results
      return results
      

      Streaming Return

      Alias Ordinal
      Type
      Description
      Column Name
      0 []perNode Node and its graph centrality _uuid, centrality

      Example: Calculate graph centrality of all nodes, define algorithm results as alias named results, and return the results with centrality equals to 0

      algo(graph_centrality).params().stream() as results
      where results.centrality == 0
      return results
      

      Real-time Statistics

      This algorithm has no statistics.

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