Elements of Access: Betweenness

Degree is often good for measuring local circumstance, but adequately characterizing centrality is a bit more complicated.  When trying to figure out centrality in terms of how connected and influential a node or link is, it is useful to get a sense of relative network flow through a particular node or link.

RED NODE INDICATING HIGHEST BETWEENNESS VALUE FOR THIS NETWORK
RED NODE INDICATING HIGHEST BETWEENNESS VALUE FOR THIS NETWORK

Betweenness measures attempt to capture this relative flow by quantifying the number of times a node or link is on a shortest path between two other nodes.  The first step would be to calculate the shortest path between every origin and every destination.  Next, we count the number of times that a particular node or link shows up on a shortest path.  The resulting number represents the relative role of a node or link as a connector between clusters of nodes or links.  In the above street network, the intersection highlighted in red must be included in over half of the shortest paths.  We call this count Betweenness, which is essentially an attempt to quantify how necessary a node or link is to get from one side of the network to the other.  The Panama Canal, for instance, is a key maritime link connecting the Atlantic and Pacific Oceans.  Without it, ships would have to route around Cape Horn at the southernmost tip of Chile or through the Straits of Magellan.  For a ship traveling from New York to San Francisco, the Panama Canal – due to its high Betweenness value – cuts more than 7,500 miles from the journey.  In terms of other transportation issues, Betweenness usually relates to metrics such as accessibility and traffic congestion.

 

In addition to revealing relative importance, Betweenness also indicates how irreplaceable a node or link may be to a network.  In other words, what happens if we remove a certain node or link from the network?  Very high betweenness values can indicate a critical connection between various groups of nodes or links.  In some cases, this represents a vulnerability where we would want to add redundancies to the network.

 

In transportation networks, if we assume all travelers take the shortest path and treat each traveler as having a unique origin and destination, Betweenness is the same as the flow (number of travelers) on the link.  We call this Flow-weighted Betweenness.