Overview

Graph centrality of a node is measured by the maximum shortest distance from the node to all other reachable nodes. This measurement, along with other measurements like closeness centrality and graph diameter, can be considered jointly to determine whether a node is literally located at the very center of the graph.

Graph centrality takes on values between 0 to 1, nodes with higher scores are closer to the center.

Concepts

Shortest Distance

The shortest distance of two nodes is the number of edges contained in the shortest path between them. Please refer to Closeness Centrality for more details.

Graph Centrality

Graph centrality score of a node defined by this algorithm is the inverse of the maximum shortest distance from the node to all other reachable nodes. The formula is:

where x is the target node, y is any node that connects with x along edges (x itself is excluded), d(x,y) is the shortest distance between x and y.

In this graph, the green number and red number next to each node is the shortest distance between the node and the green node and red node. Graph centrality scores of the green and red nodes are 1/4 = 0.25 and 1/3 = 0.3333 respectively.

Regarding closeness centrality, the green node has score 8/(1+1+1+1+2+3+4+3) = 0.5, the red node has score 8/(3+3+3+2+1+1+2+1) = 0.5. When two nodes have the same closeness centrality score, graph centrality can be viewed as the subsidiary basis to determine which node is closer to the center.

Considerations

• The graph centrality score of isolated nodes is 0.
• The Graph Centrality algorithm ignores the direction of edges but calculates them as undirected edges.

Example Graph

To create this graph:

// Runs each row separately in order in an empty graphset
create().node_schema("user").edge_schema("vote")
insert().into(@user).nodes([{_id:"A"},{_id:"B"},{_id:"C"},{_id:"D"},{_id:"E"},{_id:"F"},{_id:"G"},{_id:"H"},{_id:"I"},{_id:"J"}])
insert().into(@vote).edges([{_from:"A", _to:"B"}, {_from:"A", _to:"C"}, {_from:"A", _to:"D"}, {_from:"E", _to:"A"}, {_from:"E", _to:"F"}, {_from:"F", _to:"G"}, {_from:"F", _to:"I"}, {_from:"G", _to:"H"}, {_from:"H", _to:"I"}])

Creating HDC Projection

To project the entire graph to the HDC server hdc-server-1 as hdc_graph_centrality:

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

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

Parameters

Algorithm name: graph_centrality

File Writeback

CALL algo.graph_centrality.write("hdc_graph_centrality", {
  params: {
    return_id_uuid: "id"
  },
  return_params: {
    file: {
      filename: "graph_centrality"
    }
  }
})

algo(graph_centrality).params({
  project: "hdc_graph_centrality",
  return_id_uuid: "id"
}).write({
  file: {
    filename: "graph_centrality"
  }
})

Result:

_id,graph_centrality
J,0
D,0.2
F,0.333333
H,0.2
B,0.2
A,0.25
E,0.333333
C,0.2
I,0.25
G,0.25

DB Writeback

Writes the graph_centrality values from the results to the specified node property. The property type is float.

CALL algo.graph_centrality.write("hdc_graph_centrality", {
  params: {},
  return_params: {
    db: {
      property: 'gc'
    }
  }
})

algo(graph_centrality).params({
  project: "hdc_graph_centrality"
}).write({
  db:{ 
    property: 'gc'
  }
})

Full Return

CALL algo.graph_centrality("hdc_graph_centrality", {
  params: {
    return_id_uuid: "id",
    ids: ["A","B","C"],
    order: "asc"
  },
  return_params: {}
}) YIELD gc
RETURN gc

exec{
  algo(graph_centrality).params({
    return_id_uuid: "id",
    ids: ["A","B","C"],
    order: "asc"
  }) as gc
  return gc
} on hdc_graph_centrality

Result:

Stream Return

CALL algo.graph_centrality("hdc_graph_centrality", {
  params: {
    return_id_uuid: "id"
  },
  return_params: {type: "stream"}
}) YIELD gc
FILTER gc.graph_centrality > 0.25
RETURN gc

exec{
  algo(graph_centrality).params({
    return_id_uuid: "id"
  }).stream() as gc
  where gc.graph_centrality > 0.25
  return gc
} on hdc_graph_centrality

Result: