Centrality is a measure that tells us how influential or significant a node is within the overall network.1 This concept of significance will have different meanings depending on the type of network we are analyzing. In some ways, centrality indices are answers to the question “What characterizes an important node?” From this measurement of centrality, we can get some idea of the node’s position within the overall network. The degree of a node’s connectivity is probably the simplest and most basic measure of centrality. We can measure the degree of a node by looking at the number of other nodes it is connected to vs. the total it could possibly be connected to. But this measurement of degree only really captures what is happening locally around that node. It does not really tell us where the node lies in the network, which is needed to get a proper understanding of its degree centrality and overall influence.
The concept of centrality is quite a bit more complex than that of its degree of connectivity and may often depend on the context, but we will present some of the most important parameters for trying to capture the significance of any given node within a network. The significance of a node can be thought of in two ways, firstly how much of the networks resources flow through the node and secondly how critical is the node to that flow, i.e. can it be replaced? So a bridge within a nation’s transportation network may be very significant because it carries a very large percentage of the traffic or because it is the only bridge between two important locations.
The four most significant metrics for quantifying this are; Firstly, a node’s degree of connectivity, which is a primary metric that defined its degree of significance within its local environment. Secondly, we have what are called closeness centrality measures that try to capture how close a node is to any other node in the network, that is, how quickly or easily can the node reach other nodes. Betweenness is a third metric we might use, which is trying to capture the node’s role as a connector or bridge between other groups of nodes. Lastly, we have prestige measures that are trying to describe how significant you are based upon how significant the nodes you are connected to are. Again, which one of these works best will be context dependent.
Closeness may be defined as the reciprocal of farness, where the farness of a given node is defined as the sum of its distances to all other nodes. Thus, the more central a node is the lower its total distance to all other nodes. Closeness can be regarded as a measure of how long it will take to spread something, such as information, from the node of interest to all other nodes sequentially. We can understand how this correlates to the node’s significance in that it is a measurement of the node’s capacity to affect all the other elements in the network.
Betweenness, as mentioned, is really talking about how critical a node is to a network in its functioning as a unique bridging point between other nodes in the network. Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. In this formulation, vertices that have a high probability of occurring on a randomly chosen shortest path between two vertices have a high betweenness value.
Our last measure is trying to capture how connected the nodes that a given node is connected to are. So instead of looking at the total amount of connections you have, it is more interested in the value of those connections. One way of capturing this is called Eigenvector centrality. Eigenvector centrality assigns relative scores to all nodes in the network based on the concept that connections to highly connected nodes contribute more than connections to nodes with lower degrees of connectivity. Eigenvector centrality is one measure used by web search engines to try and rank the relative importance of a website by looking at the importance of the websites that link to it.