Betweenness Centrality

Brandes defines several strategies for selecting source nodes. The GDS implementation is based on the random degree selection strategy, which selects nodes with a probability proportional to their degree. The idea behind this strategy is that such nodes are likely to lie on many shortest paths in the graph and thus have a higher contribution to the betweenness centrality score.

First off, we will estimate the cost of running the algorithm using the estimate procedure. This can be done with any execution mode. We will use the write mode in this example. Estimating the algorithm is useful to understand the memory impact that running the algorithm on your graph will have. When you later actually run the algorithm in one of the execution modes the system will perform an estimation. If the estimation shows that there is a very high probability of the execution going over its memory limitations, the execution is prohibited. To read more about this, see Automatic estimation and execution blocking.

For more details on estimate in general, see Memory Estimation.

As is discussed in Considerations and sampling we can configure the memory requirements using the concurrency configuration parameter.

Here we can note that the estimated memory requirements were lower than when running with the default concurrency setting. Similarly, using a higher value will increase the estimated memory requirements.

In the stream execution mode, the algorithm returns the centrality for each node. This allows us to inspect the results directly or post-process them in Cypher without any side effects. For example, we can order the results to find the nodes with the highest betweenness centrality.

For more details on the stream mode in general, see Stream.

We note that the 'Carol' node has the highest score, followed by the 'Frank' node. Studying the example graph we can see that these nodes are in bottleneck positions in the graph. The 'Carol' node connects the 'Alice' and 'Bob' nodes to all other nodes, which increases its score. In particular, the shortest path from 'Alice' or 'Bob' to any other reachable node passes through 'Carol'. Similarly, all shortest paths that lead to the 'Gale' node passes through the 'Frank' node. Since 'Gale' is reachable from each other node, this causes the score for 'Frank' to be high.

Conversely, there are no shortest paths that pass through either of the nodes 'Alice', 'Bob' or 'Gale' which causes their betweenness centrality score to be zero.

In the stats execution mode, the algorithm returns a single row containing a summary of the algorithm result. In particular, Betweenness Centrality returns the minimum, maximum and sum of all centrality scores. This execution mode does not have any side effects. It can be useful for evaluating algorithm performance by inspecting the computeMillis return item. In the examples below we will omit returning the timings. The full signature of the procedure can be found in the syntax section.

For more details on the stats mode in general, see Stats.

Comparing this to the results we saw in the stream example, we can find our minimum and maximum values from the table. It is worth noting that unless the graph has a particular shape involving a directed cycle, the minimum score will almost always be zero.

The mutate execution mode extends the stats mode with an important side effect: updating the named graph with a new node property containing the centrality for that node. The name of the new property is specified using the mandatory configuration parameter mutateProperty. The result is a single summary row, similar to stats, but with some additional metrics. The mutate mode is especially useful when multiple algorithms are used in conjunction.

For more details on the mutate mode in general, see Mutate.

The returned result is the same as in the stats example. Additionally, the graph 'myGraph' now has a node property betweenness which stores the betweenness centrality score for each node. To find out how to inspect the new schema of the in-memory graph, see Listing graphs.

The write execution mode extends the stats mode with an important side effect: writing the centrality for each node as a property to the Neo4j database. The name of the new property is specified using the mandatory configuration parameter writeProperty. The result is a single summary row, similar to stats, but with some additional metrics. The write mode enables directly persisting the results to the database.

For more details on the write mode in general, see Write.

The returned result is the same as in the stats example. Additionally, each of the seven nodes now has a new property betweenness in the Neo4j database, containing the betweenness centrality score for that node.

Betweenness Centrality can be very resource-intensive to compute. To help with this, it is possible to approximate the results using a sampling technique. The configuration parameters samplingSize and samplingSeed are used to control the sampling. We illustrate this on our example graph by approximating Betweenness Centrality with a sampling size of two. The seed value is an arbitrary integer, where using the same value will yield the same results between different runs of the procedure.

Here we can see that the 'Carol' node has the highest score, followed by a three-way tie between the 'Dan', 'Eve', and 'Frank' nodes. We are only sampling from two nodes, where the probability of a node being picked for the sampling is proportional to its outgoing degree. The 'Carol' node has the maximum degree and is the most likely to be picked. The 'Gale' node has an outgoing degree of zero and is very unlikely to be picked. The other nodes all have the same probability to be picked.

With our selected sampling seed of 0, we seem to have selected either of the 'Alice' and 'Bob' nodes, as well as the 'Carol' node. We can see that because either of 'Alice' and 'Bob' would add four to the score of the 'Carol' node, and each of 'Alice', 'Bob', and 'Carol' adds one to all of 'Dan', 'Eve', and 'Frank'.

To increase the accuracy of our approximation, the sampling size could be increased. In fact, setting the samplingSize to the node count of the graph (seven, in our case) will produce exact results.

Betweenness Centrality can also be run on undirected graphs. To illustrate this, we will project our example graph using the UNDIRECTED orientation.

Now we can run Betweenness Centrality on our undirected graph. The algorithm automatically figures out that the graph is undirected.

The central nodes now have slightly higher scores, due to the fact that there are more shortest paths in the graph, and these are more likely to pass through the central nodes. The 'Dan' and 'Eve' nodes retain the same centrality scores as in the directed case.