!CHAPTER Graph Functions

This chapter describes [various functions on a graph](../Graphs/README.md).
A lot of these accept a vertex (or edge) example as parameter as defined in the next section.


Examples will explain the API on the [the city graph](../Graphs/README.md#the-city-graph):

![Social Example Graph](../Graphs/cities_graph.png)


!SECTION Definition of examples

@startDocuBlock JSF_general_graph_example_description

!SECTION Get vertices from edges.

!SUBSECTION Get vertex *from* of an edge



Get the source vertex of an edge

`graph._fromVertex(edgeId)`

Returns the vertex defined with the attribute *_from* of the edge with *edgeId* as its *_id*.


**Parameters**
 * edgeId (required) *_id* attribute of the edge


**Examples**


    @startDocuBlockInline generalGraphGetFromVertex
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphGetFromVertex}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("social");
      graph._fromVertex("relation/aliceAndBob")
    ~ examples.dropGraph("social");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphGetFromVertex



!SUBSECTION Get vertex *to* of an edge



Get the target vertex of an edge

`graph._toVertex(edgeId)`

Returns the vertex defined with the attribute *_to* of the edge with *edgeId* as its *_id*.


**Parameters**


 * edgeId (required) *_id* attribute of the edge


**Examples**


    @startDocuBlockInline generalGraphGetToVertex
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphGetToVertex}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("social");
      graph._toVertex("relation/aliceAndBob")
    ~ examples.dropGraph("social");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphGetToVertex



!SECTION _neighbors



Get all neighbors of the vertices defined by the example

`graph._neighbors(vertexExample, options)`

The function accepts an id, an example, a list of examples or even an empty
example as parameter for vertexExample.
The complexity of this method is **O(n\*m^x)** with *n* being the vertices defined by the
parameter vertexExamplex, *m* the average amount of neighbors and *x* the maximal depths.
Hence the default call would have a complexity of **O(n\*m)**;


**Parameters**

 * vertexExample (optional) See [Definition of examples](#definition-of-examples)
 * options (optional) An object defining further options. Can have the following values:
  * *direction*: The direction of the edges. Possible values are *outbound*, *inbound* and *any* (default).
  * *edgeExamples*: Filter the edges, see [Definition of examples](#definition-of-examples)
  * *neighborExamples*: Filter the neighbor vertices, see [Definition of examples](#definition-of-examples)
  * *edgeCollectionRestriction* : One or a list of edge-collection names that should be
      considered to be on the path.
  * *vertexCollectionRestriction* : One or a list of vertex-collection names that should be
      considered on the intermediate vertex steps.
  * *minDepth*: Defines the minimal number of intermediate steps to neighbors (default is 1).
  * *maxDepth*: Defines the maximal number of intermediate steps to neighbors (default is 1).


**Examples**


A route planner example, all neighbors of capitals.

    @startDocuBlockInline generalGraphModuleNeighbors1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleNeighbors1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._neighbors({isCapital : true});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleNeighbors1

A route planner example, all outbound neighbors of Hamburg.

    @startDocuBlockInline generalGraphModuleNeighbors2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleNeighbors2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._neighbors('germanCity/Hamburg', {direction : 'outbound', maxDepth : 2});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleNeighbors2



!SECTION _commonNeighbors



Get all common neighbors of the vertices defined by the examples.

`graph._commonNeighbors(vertex1Example, vertex2Examples, optionsVertex1, optionsVertex2)`

This function returns the intersection of *graph_module._neighbors(vertex1Example, optionsVertex1)*
and *graph_module._neighbors(vertex2Example, optionsVertex2)*.
For parameter documentation see [_neighbors](#neighbors).

The complexity of this method is **O(n\*m^x)** with *n* being the maximal amount of vertices
defined by the parameters vertexExamples, *m* the average amount of neighbors and *x* the
maximal depths.
Hence the default call would have a complexity of **O(n\*m)**;


**Examples**


A route planner example, all common neighbors of capitals.

    @startDocuBlockInline generalGraphModuleCommonNeighbors1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCommonNeighbors1}
    var examples = require("@arangodb/graph-examples/example-graph.js");
    var graph = examples.loadGraph("routeplanner");
    graph._commonNeighbors({isCapital : true}, {isCapital : true});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCommonNeighbors1

A route planner example, all common outbound neighbors of Hamburg with any other location
which have a maximal depth of 2 :

    @startDocuBlockInline generalGraphModuleCommonNeighbors2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCommonNeighbors2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
    | graph._commonNeighbors(
    |   'germanCity/Hamburg',
    |   {},
    |   {direction : 'outbound', maxDepth : 2},
        {direction : 'outbound', maxDepth : 2});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCommonNeighbors2



!SECTION _countCommonNeighbors



Get the amount of common neighbors of the vertices defined by the examples.

`graph._countCommonNeighbors(vertex1Example, vertex2Examples, optionsVertex1, optionsVertex2)`

Similar to [_commonNeighbors](#commonneighbors) but returns count instead of the elements.


**Examples**


A route planner example, all common neighbors of capitals.

    @startDocuBlockInline generalGraphModuleCommonNeighborsAmount1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCommonNeighborsAmount1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      var example = { isCapital: true };
      var options = { includeData: true };
      graph._countCommonNeighbors(example, example, options, options);
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCommonNeighborsAmount1

A route planner example, all common outbound neighbors of Hamburg with any other location
which have a maximal depth of 2 :

    @startDocuBlockInline generalGraphModuleCommonNeighborsAmount2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCommonNeighborsAmount2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      var options = { direction: 'outbound', maxDepth: 2, includeData: true };
      graph._countCommonNeighbors('germanCity/Hamburg', {}, options, options);
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCommonNeighborsAmount2



!SECTION _commonProperties



Get the vertices of the graph that share common properties.

`graph._commonProperties(vertex1Example, vertex2Examples, options)`

The function accepts an id, an example, a list of examples or even an empty
example as parameter for vertex1Example and vertex2Example.

The complexity of this method is **O(n)** with *n* being the maximal amount of vertices
defined by the parameters vertexExamples.


**Parameters**


* vertex1Examples (optional) Filter the set of source vertices, see [Definition of examples](#definition-of-examples)

* vertex2Examples (optional) Filter the set of vertices compared to, see [Definition of examples](#definition-of-examples)
* options (optional) An object defining further options. Can have the following values:
  * *vertex1CollectionRestriction* : One or a list of vertex-collection names that should be
      searched for source vertices.
  * *vertex2CollectionRestriction* : One or a list of vertex-collection names that should be
      searched for compare vertices.
  * *ignoreProperties* : One or a list of attribute names of a document that should be ignored.


**Examples**


A route planner example, all locations with the same properties:

    @startDocuBlockInline generalGraphModuleProperties1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleProperties1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._commonProperties({}, {});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleProperties1

A route planner example, all cities which share same properties except for population.

    @startDocuBlockInline generalGraphModuleProperties2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleProperties2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._commonProperties({}, {}, {ignoreProperties: 'population'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleProperties2



!SECTION _countCommonProperties



Get the amount of vertices of the graph that share common properties.

`graph._countCommonProperties(vertex1Example, vertex2Examples, options)`

Similar to [_commonProperties](#commonproperties) but returns count instead of
the objects.


**Examples**


A route planner example, all locations with the same properties:

    @startDocuBlockInline generalGraphModuleAmountProperties1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAmountProperties1}
    var examples = require("@arangodb/graph-examples/example-graph.js");
    var graph = examples.loadGraph("routeplanner");
    graph._countCommonProperties({}, {});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAmountProperties1

A route planner example, all German cities which share same properties except for population.

    @startDocuBlockInline generalGraphModuleAmountProperties2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAmountProperties2}
    var examples = require("@arangodb/graph-examples/example-graph.js");
    var graph = examples.loadGraph("routeplanner");
    | graph._countCommonProperties({}, {}, {vertex1CollectionRestriction : 'germanCity',
      vertex2CollectionRestriction : 'germanCity' ,ignoreProperties: 'population'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAmountProperties2



!SECTION _paths



The _paths function returns all paths of a graph.

`graph._paths(options)`

This function determines all available paths in a graph.

The complexity of this method is **O(n\*n\*m)** with *n* being the amount of vertices in
the graph and *m* the average amount of connected edges;


**Parameters**


* options (optional) An object containing options, see below:
  * *direction*        : The direction of the edges. Possible values are *any*,
    *inbound* and *outbound* (default).
  * *followCycles* (optional) : If set to *true* the query follows cycles in the graph,
    default is false.
  * *minLength* (optional)     : Defines the minimal length a path must
    have to be returned (default is 0).
  * *maxLength* (optional)     : Defines the maximal length a path must
     have to be returned (default is 10).


**Examples**


Return all paths of the graph "social":

    @startDocuBlockInline generalGraphModulePaths1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModulePaths1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var g = examples.loadGraph("social");
      g._paths();
    ~ examples.dropGraph("social");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModulePaths1

Return all inbound paths of the graph "social" with a maximal
length of 1 and a minimal length of 2:

    @startDocuBlockInline generalGraphModulePaths2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModulePaths2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var g = examples.loadGraph("social");
      g._paths({direction : 'inbound', minLength : 1, maxLength :  2});
    ~ examples.dropGraph("social");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModulePaths2


!SECTION _shortestPath



The _shortestPath function returns all shortest paths of a graph.

`graph._shortestPath(startVertexExample, endVertexExample, options)`

This function determines all shortest paths in a graph.
The function accepts an id, an example, a list of examples
or even an empty example as parameter for
start and end vertex. If one wants to call this function to receive nearly all
shortest paths for a graph the option *algorithm* should be set to
[Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm)
to increase performance.
If no algorithm is provided in the options the function chooses the appropriate
one (either [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm)
or [Dijkstra](http://en.wikipedia.org/wiki/Dijkstra's_algorithm)) according to its parameters.
The length of a path is by default the amount of edges from one start vertex to
an end vertex. The option weight allows the user to define an edge attribute
representing the length.

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Parameters**


* startVertexExample (optional) An example for the desired start Vertices (see [Definition of examples](#definition-of-examples)).
* endVertexExample (optional) An example for the desired end Vertices (see [Definition of examples](#definition-of-examples)).
* options (optional) An object containing options, see below:
  * *direction*                        : The direction of the edges as a string.
                                         Possible values are *outbound*, *inbound* and *any* (default).
  * *edgeCollectionRestriction*        : One or multiple edge
                                         collection names. Only edges from these collections will be considered for the path.
  * *startVertexCollectionRestriction* : One or multiple vertex
                                         collection names. Only vertices from these collections will be considered as
                                          start vertex of a path.
  * *endVertexCollectionRestriction*   : One or multiple vertex
                                         collection names. Only vertices from these collections will be considered as
                                         end vertex of a path.
  * *edgeExamples*                     : A filter example for the
                                         edges in the shortest paths
                                         (see [example](#definition-of-examples)).
  * *algorithm*                        : The algorithm to calculate
                                         the shortest paths. If both start and end vertex examples are empty
                                         [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm) is
                                         used, otherwise the default is [Dijkstra](http://en.wikipedia.org/wiki/Dijkstra's_algorithm)
  * *weight*                           : The name of the attribute of
                                         the edges containing the length as a string.
  * *defaultWeight*                    : Only used with the option *weight*.
                                         If an edge does not have the attribute named as defined in option *weight* this default
                                         is used as length.
                                         If no default is supplied the default would be positive Infinity so the path could
                                         not be calculated.


**Examples**


A route planner example, shortest path from all german to all french cities:

    @startDocuBlockInline generalGraphModuleShortestPaths1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleShortestPaths1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var g = examples.loadGraph("routeplanner");
    | g._shortestPath({}, {}, {weight : 'distance', endVertexCollectionRestriction : 'frenchCity',
      startVertexCollectionRestriction : 'germanCity'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleShortestPaths1

A route planner example, shortest path from Hamburg and Cologne to Lyon:

    @startDocuBlockInline generalGraphModuleShortestPaths2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleShortestPaths2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var g = examples.loadGraph("routeplanner");
    | g._shortestPath([{_id: 'germanCity/Cologne'},{_id: 'germanCity/Munich'}], 'frenchCity/Lyon',
      {weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleShortestPaths2



!SECTION _distanceTo



The _distanceTo function returns all paths and there distance within a graph.

`graph._distanceTo(startVertexExample, endVertexExample, options)`

This function is a wrapper of [graph._shortestPath](#shortestpath).
It does not return the actual path but only the distance between two vertices.


**Examples**


A route planner example, shortest distance from all german to all french cities:

    @startDocuBlockInline generalGraphModuleDistanceTo1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleDistanceTo1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var g = examples.loadGraph("routeplanner");
    | g._distanceTo({}, {}, {weight : 'distance', endVertexCollectionRestriction : 'frenchCity',
      startVertexCollectionRestriction : 'germanCity'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleDistanceTo1

A route planner example, shortest distance from Hamburg and Cologne to Lyon:

    @startDocuBlockInline generalGraphModuleDistanceTo2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleDistanceTo2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var g = examples.loadGraph("routeplanner");
    | g._distanceTo([{_id: 'germanCity/Cologne'},{_id: 'germanCity/Munich'}], 'frenchCity/Lyon',
      {weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleDistanceTo2



!SECTION _absoluteEccentricity



Get the
[eccentricity](http://en.wikipedia.org/wiki/Distance_%28graph_theory%29)
of the vertices defined by the examples.

`graph._absoluteEccentricity(vertexExample, options)`

The function accepts an id, an example, a list of examples or even an empty
example as parameter for vertexExample.

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Parameters**

* vertexExample (optional) Filter the vertices, see [Definition of examples](#definition-of-examples)
* options (optional) An object defining further options. Can have the following values:
  * *direction*: The direction of the edges. Possible values are *outbound*, *inbound* and *any* (default).
  * *edgeCollectionRestriction* : One or a list of edge-collection names that should be
      considered to be on the path.
  * *startVertexCollectionRestriction* : One or a list of vertex-collection names that should be
      considered for source vertices.
  * *endVertexCollectionRestriction* : One or a list of vertex-collection names that should be
      considered for target vertices.
  * *edgeExamples*: Filter the edges to be followed, see [Definition of examples](#definition-of-examples)
  * *algorithm*: The algorithm to calculate the shortest paths, possible values are
      [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm) and
      [Dijkstra](http://en.wikipedia.org/wiki/Dijkstra's_algorithm).
  * *weight*: The name of the attribute of the edges containing the weight.
  * *defaultWeight*: Only used with the option *weight*.
      If an edge does not have the attribute named as defined in option *weight* this default
      is used as weight.
      If no default is supplied the default would be positive infinity so the path and
      hence the eccentricity can not be calculated.


**Examples**


A route planner example, the absolute eccentricity of all locations.

    @startDocuBlockInline generalGraphModuleAbsEccentricity1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsEccentricity1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
    | db._query("RETURN GRAPH_ABSOLUTE_ECCENTRICITY("
    |   + "'routeplanner', {})"
      ).toArray();
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsEccentricity1

A route planner example, the absolute eccentricity of all locations.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleAbsEccentricity2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsEccentricity2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._absoluteEccentricity({}, {weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsEccentricity2

A route planner example, the absolute eccentricity of all cities regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleAbsEccentricity3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsEccentricity3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
    | graph._absoluteEccentricity({}, {startVertexCollectionRestriction : 'germanCity',
      direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsEccentricity3



!SECTION _eccentricity



Get the normalized
[eccentricity](http://en.wikipedia.org/wiki/Distance_%28graph_theory%29)
of the vertices defined by the examples.

`graph._eccentricity(vertexExample, options)`

Similar to [_absoluteEccentricity](#absoluteeccentricity) but returns a normalized result.

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Examples**


A route planner example, the eccentricity of all locations.

    @startDocuBlockInline generalGraphModuleEccentricity2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleEccentricity2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._eccentricity();
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleEccentricity2

A route planner example, the weighted eccentricity.

    @startDocuBlockInline generalGraphModuleEccentricity3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleEccentricity3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._eccentricity({weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleEccentricity3



!SECTION _absoluteCloseness



Get the
[closeness](http://en.wikipedia.org/wiki/Centrality#Closeness_centrality)
of the vertices defined by the examples.

`graph._absoluteCloseness(vertexExample, options)`

The function accepts an id, an example, a list of examples or even an empty
example as parameter for *vertexExample*.

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Parameters**

* vertexExample (optional) Filter the vertices, see [Definition of examples](#definition-of-examples)
* options (optional) An object defining further options. Can have the following values:
  * *direction*: The direction of the edges. Possible values are *outbound*, *inbound* and *any* (default).
  * *edgeCollectionRestriction* : One or a list of edge-collection names that should be
                                  considered to be on the path.
  * *startVertexCollectionRestriction* : One or a list of vertex-collection names that should be
                                         considered for source vertices.
  * *endVertexCollectionRestriction* : One or a list of vertex-collection names that should be
                                       considered for target vertices.
  * *edgeExamples*: Filter the edges to be followed, see [Definition of examples](#definition-of-examples)
  * *algorithm*: The algorithm to calculate the shortest paths, possible values are
                 [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm) and
                 [Dijkstra](http://en.wikipedia.org/wiki/Dijkstra's_algorithm).
  * *weight*: The name of the attribute of the edges containing the weight.
  * *defaultWeight*: Only used with the option *weight*.
                     If an edge does not have the attribute named as defined in option *weight* this default
                     is used as weight.
                     If no default is supplied the default would be positive infinity so the path and
                     hence the closeness can not be calculated.


**Examples**


A route planner example, the absolute closeness of all locations.

    @startDocuBlockInline generalGraphModuleAbsCloseness1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsCloseness1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._absoluteCloseness({});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsCloseness1

A route planner example, the absolute closeness of all locations.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleAbsCloseness2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsCloseness2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._absoluteCloseness({}, {weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsCloseness2

A route planner example, the absolute closeness of all German Cities regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleAbsCloseness3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsCloseness3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
    | graph._absoluteCloseness({}, {startVertexCollectionRestriction : 'germanCity',
      direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsCloseness3



!SECTION _closeness



Get the normalized
[closeness](http://en.wikipedia.org/wiki/Centrality#Closeness_centrality)
of graphs vertices.

`graph._closeness(options)`

Similar to [_absoluteCloseness](#absolutecloseness) but returns a normalized value.

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Examples**


A route planner example, the normalized closeness of all locations.

    @startDocuBlockInline generalGraphModuleCloseness1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCloseness1}
    var examples = require("@arangodb/graph-examples/example-graph.js");
    var graph = examples.loadGraph("routeplanner");
    graph._closeness();
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCloseness1

A route planner example, the closeness of all locations.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleCloseness2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCloseness2}
    var examples = require("@arangodb/graph-examples/example-graph.js");
    var graph = examples.loadGraph("routeplanner");
    graph._closeness({weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCloseness2

A route planner example, the closeness of all cities regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleCloseness3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleCloseness3}
    var examples = require("@arangodb/graph-examples/example-graph.js");
    var graph = examples.loadGraph("routeplanner");
    graph._closeness({direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleCloseness3



!SECTION _absoluteBetweenness



Get the
[betweenness](http://en.wikipedia.org/wiki/Betweenness_centrality)
of all vertices in the graph.

`graph._absoluteBetweenness(vertexExample, options)`

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Parameters**

* vertexExample (optional) Filter the vertices, see [Definition of examples](#definition-of-examples)
* options (optional) An object defining further options. Can have the following values:
  * *direction*: The direction of the edges. Possible values are *outbound*, *inbound* and *any* (default).
  * *weight*: The name of the attribute of the edges containing the weight.
  * *defaultWeight*: Only used with the option *weight*.
                     If an edge does not have the attribute named as defined in option *weight* this default
                     is used as weight.
                     If no default is supplied the default would be positive infinity so the path and
                     hence the betweeness can not be calculated.


**Examples**


A route planner example, the absolute betweenness of all locations.

    @startDocuBlockInline generalGraphModuleAbsBetweenness1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsBetweenness1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._absoluteBetweenness({});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsBetweenness1

A route planner example, the absolute betweenness of all locations.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleAbsBetweenness2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsBetweenness2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._absoluteBetweenness({weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsBetweenness2

A route planner example, the absolute betweenness of all cities regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleAbsBetweenness3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleAbsBetweenness3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._absoluteBetweenness({direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleAbsBetweenness3



!SECTION _betweenness



Get the normalized
[betweenness](http://en.wikipedia.org/wiki/Betweenness_centrality)
of graphs vertices.

`graph_module._betweenness(options)`

Similar to [_absoluteBetweeness](#absolutebetweenness) but returns normalized values.


**Examples**


A route planner example, the betweenness of all locations.

    @startDocuBlockInline generalGraphModuleBetweenness1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleBetweenness1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._betweenness();
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleBetweenness1

A route planner example, the betweenness of all locations.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleBetweenness2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleBetweenness2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._betweenness({weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleBetweenness2

A route planner example, the betweenness of all cities regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleBetweenness3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleBetweenness3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._betweenness({direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleBetweenness3



!SECTION _radius



Get the
[radius](http://en.wikipedia.org/wiki/Eccentricity_%28graph_theory%29)
of a graph.

`graph._radius(options)`

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Parameters**

* options (optional) An object defining further options. Can have the following values:
  * *direction*: The direction of the edges. Possible values are *outbound*, *inbound* and *any* (default).
  * *algorithm*: The algorithm to calculate the shortest paths, possible values are
      [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm) and
      [Dijkstra](http://en.wikipedia.org/wiki/Dijkstra's_algorithm).
  * *weight*: The name of the attribute of the edges containing the weight.
  * *defaultWeight*: Only used with the option *weight*.
                     If an edge does not have the attribute named as defined in option *weight* this default
                     is used as weight.
                     If no default is supplied the default would be positive infinity so the path and
                     hence the radius can not be calculated.


**Examples**


A route planner example, the radius of the graph.

    @startDocuBlockInline generalGraphModuleRadius1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleRadius1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._radius();
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleRadius1

A route planner example, the radius of the graph.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleRadius2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleRadius2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._radius({weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleRadius2

A route planner example, the radius of the graph regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleRadius3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleRadius3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._radius({direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleRadius3



!SECTION _diameter



Get the
[diameter](http://en.wikipedia.org/wiki/Eccentricity_%28graph_theory%29)
of a graph.

`graph._diameter(graphName, options)`

The complexity of the function is described
[here](../Aql/GraphOperations.md#the-complexity-of-the-shortest-path-algorithms).


**Parameters**

* options (optional) An object defining further options. Can have the following values:
  * *direction*: The direction of the edges. Possible values are *outbound*, *inbound* and *any* (default).
  * *algorithm*: The algorithm to calculate the shortest paths, possible values are
      [Floyd-Warshall](http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm) and
      [Dijkstra](http://en.wikipedia.org/wiki/Dijkstra's_algorithm).
  * *weight*: The name of the attribute of the edges containing the weight.
  * *defaultWeight*: Only used with the option *weight*.
                      If an edge does not have the attribute named as defined in option *weight* this default
                      is used as weight.
                      If no default is supplied the default would be positive infinity so the path and
                      hence the radius can not be calculated.


**Examples**


A route planner example, the diameter of the graph.

    @startDocuBlockInline generalGraphModuleDiameter1
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleDiameter1}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._diameter();
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleDiameter1

A route planner example, the diameter of the graph.
This considers the actual distances.

    @startDocuBlockInline generalGraphModuleDiameter2
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleDiameter2}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._diameter({weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleDiameter2

A route planner example, the diameter of the graph regarding only
outbound paths.

    @startDocuBlockInline generalGraphModuleDiameter3
    @EXAMPLE_ARANGOSH_OUTPUT{generalGraphModuleDiameter3}
      var examples = require("@arangodb/graph-examples/example-graph.js");
      var graph = examples.loadGraph("routeplanner");
      graph._diameter({direction : 'outbound', weight : 'distance'});
    ~ examples.dropGraph("routeplanner");
    @END_EXAMPLE_ARANGOSH_OUTPUT
    @endDocuBlock generalGraphModuleDiameter3