# Graph Operations

Underlying each instance of the Toolkit there is a directed graph. This
object - accessed via the `getGraph`

method of a Toolkit - offers a complete set of methods for querying the state of the graph - from questions like "is node A connected to node B" through to "what is the shortest path from node A and node B?", or "what is the centrality of vertex C?".

The code snippets on this page all refer to this dataset:

`toolkit.load({`

data:{

nodes:[ { id:"1" }, { id:"2" }, { id:"3" }, { id:"4" }, { id:"5" } ],

edges:[

{ source:"1", target:"2", cost:10 },

{ source:"2", target:"3", cost:5 },

{ source:"3", target:"4", cost:10 },

{ source:"3", target:"5", cost:10 }

]

}

});

## Paths

### Distance between objects

To find the distance between two objects, use the `getDistance`

method:

`toolkit.getGraph().getDistance("1", "3")`

-> 15

toolkit.getGraph().getDistance("1", "4")

-> 25

toolkit.getGraph().getDistance("4", "5")

-> undefined

Note the last example returns `undefined`

: remember that edges in the Toolkit are, by default, directed. If you look again at the dataset you can see that there is no route from vertex 4 to vertex 5, since both of these Nodes are connected to the graphonly as targets of edges from vertex 3.

### Shortest Paths

To get the shortest path from some vertex to another vertex, use the `getPath`

method:

`toolkit.getPath({source:"1",target:"3"})`

-> e…s.Path {path: Object}

The return value from this method is a Path. This is a complex object with many helper functions; if you need a more simple return value you can access the graph's `findPath`

method directly:

`toolkit.getGraph().findPath("1","3")`

-> Object {dist: Object, previous: Object, edges: Object, path: Array[3], pathDistance: 15}

## Centrality

### Degree Centrality

The degree centrality of a node is the sum of the number of edges entering and exiting the node divided by the total
number of edges in the graph. Looking at the dataset we're using on this page you may have noticed that vertex 3 has more
connections than any other node, and this observation is reflected in the output of `getDegreeCentrality`

:

`toolkit.getGraph().getDegreeCentrality("3")`

-> 0.75

toolkit.getGraph().getDegreeCentrality("1")

-> 0.25

toolkit.getGraph().getDegreeCentrality("2")

-> 0.5

toolkit.getGraph().getDegreeCentrality("4")

-> 0.25

toolkit.getGraph().getDegreeCentrality("5")

-> 0.25

### Indegree Centrality

The indegree centrality of a node is the number of edges entering the node, divided by the number of edges in the graph:

`toolkit.getGraph().getIndegreeCentrality("3")`

-> 0.25

toolkit.getGraph().getIndegreeCentrality("1")

-> 0

toolkit.getGraph().getIndegreeCentrality("2")

-> 0.25

toolkit.getGraph().getIndegreeCentrality("4")

-> 0.25

toolkit.getGraph().getIndegreeCentrality("5")

-> 0.25

### Outdegree Centrality

The outdegree centrality of a node is the number of edges exiting the vertex, divided by the number of edges in the graph:

`toolkit.getGraph().getOutdegreeCentrality("3")`

-> 0.5

toolkit.getGraph().getOutdegreeCentrality("1")

-> 0.25

toolkit.getGraph().getOutdegreeCentrality("2")

-> 0.25

toolkit.getGraph().getOutdegreeCentrality("4")

-> 0

toolkit.getGraph().getOutdegreeCentrality("5")

-> 0

### Farness

The `farness`

of a vertex is the sum of its distance from all other vertices, where the distance from one vertex to another
is given by the associated cost of the edge joining the two vertices. As with degree centrality, this is divided by
the number of edges, to normalise the results. Note that here we have a graph that has vertices which cannot "reach"
every other vertex (because all edges are `directed`

), so the "farness" of every vertex is Infinity except vertex 1, which
can trace a path to every other vertex.

`toolkit.getGraph().getFarness("3")`

-> Infinity

toolkit.getGraph().getFarness("1")

-> 18.75

toolkit.getGraph().getFarness("2")

-> Infinity

toolkit.getGraph().getFarness("4")

-> Infinity

toolkit.getGraph().getFarness("5")

-> Infinity

### Closeness

This is the inverse of a vertex's `farness`

:

`toolkit.getGraph().getCloseness("3")`

-> 0

toolkit.getGraph().getCloseness("1")

-> 0.05333333333333334

toolkit.getGraph().getCloseness("2")

-> 0

toolkit.getGraph().getCloseness("4")

-> 0

toolkit.getGraph().getCloseness("5")

-> 0

...so the only vertex that has a useful value for closeness is vertex 1. At the bottom of this page we present a revised
version of the dataset in which edges are all marked `directed:false`

, and then we show that the output of `getFarness`

and `getCloseness`

provides a usable value for every vertex.

### Betweenness Centrality

The betweenness centrality of a vertex measures how central the vertex is in the graph. It is the number of shortest paths between any two vertices in the graphthat pass through the given Nodes. The jsPlumb Toolkit computes this by first computing all the shortest paths in the graphusing the Floyd-Warshall algorithm.

`toolkit.getGraph().getBetweenness("3")`

-> 1.6666666666666667

toolkit.getGraph().getBetweenness("1")

-> 0

toolkit.getGraph().getBetweenness("2")

-> 0.6666666666666666

toolkit.getGraph().getBetweenness("4")

-> 0

toolkit.getGraph().getBetweenness("5")

-> 0

We see from the output here that vertex 3 is the most "central" vertex in this graph, with vertex 2 next. vertices 1, 4 and 5 are not "central" as no paths go through these Nodes.

### Graph diameter

Use the `getDiameter`

method to find the graph's diameter - the length
of the "longest shortest path" in the graph.

In a graph that contains at least one pair of vertices for which there is no available path, this value, strictly speaking, is Infinity. Our dataset on this page is one such graph, but it doesn't seem like an unreasonable dataset, right? So the Toolkit allows you to specify that you're happy to ignore the case that there are one or more pairs of vertices for which no Path exists:

`toolkit.getGraph().getDiameter()`

->Infinity

toolkit.getGraph().getDiameter(true)

-> 25

The second result - 25 - is the distance of the path from vertex 1 to vertex 4 or 5, which is to say, the "longest shortest path" in the graph:

`toolkit.getGraph().getDistance("1","5")`

-> 25

### Undirected graphs

If we take the data from above and make every edge bidirectional, we get vastly different results for the centrality methods:

`toolkit.load({`

data:{

nodes:[ { id:"1" }, { id:"2" }, { id:"3" }, { id:"4" }, { id:"5" } ],

edges:[

{ source:"1", target:"2", cost:10, directed:false },

{ source:"2", target:"3", cost:5, directed:false },

{ source:"3", target:"4", cost:10, directed:false },

{ source:"3", target:"5", cost:10, directed:false }

]

}

});

For instance, here are the values for `getFarness`

:

`toolkit.getGraph().getFarness("3")`

-> 10

toolkit.getGraph().getFarness("1")

-> 18.75

toolkit.getGraph().getFarness("2")

-> 11.25

toolkit.getGraph().getFarness("4")

-> 17.5

toolkit.getGraph().getFarness("5")

-> 17.5

Note how the value for vertex 1 - the only vertex in the previous graph that could "reach" all the other vertices - is the same as in the previous graph.

Here are the results for `getCloseness`

:

`toolkit.getGraph().getCloseness("3")`

-> 0.1

toolkit.getGraph().getCloseness("1")

-> 0.05333333333333334

toolkit.getGraph().getCloseness("2")

-> 0.08888888888888889

toolkit.getGraph().getCloseness("4")

-> 0.05714285714285714

toolkit.getGraph().getCloseness("5")

-> 0.05714285714285714