In the previous module, we talked about the degree of connectivity
to a given node in a network and this leads us to the broader concept of centrality.
Centrality is really a measure that tells us how influential or significant a node is within the overall network.
This concept of significance will have different meanings depending on the type of network
we are analyzing. So in some ways, centrality indices are asking the question 'what characterizes an important node?'
From this measurement of centrality, we can get some idea of the nodes importance within the overall network. The degree of a node's connectivity
that we previously looked at is probably the simplest and most basic measure of centrality.
We can measure the degree of a node by looking at the total number of other nodes it's connected to
versus the total that it could possibly be connected to. But this measurement of degree only really captures
what is happening locally around the node.
It doesn't really tell us where the node lies in the network,
which is needed to get a proper understanding of its overall degree centrality and influence.
This concept of centrality is quite a bit more complex than that of degree and may often depend on the context. But we'll be presenting
some of the most important parameters for trying to capture the significance of any given node in a network.
The significance of a node can be thought of in two ways:
firstly, how much of the network's resources flow through this node and
secondly, how critical is the node to that flow. As in, can it be replaced?
So a bridge with international transportation network may be very
significant because it carries a very large percentage of the traffic or because it's the only bridge between two important locations.
So this helps us to understand
significance on a conceptual level. But we need to define some concrete parameters to capture and quantify this intuition.
We will present four of the most significant metrics for doing this here.
Firstly, as we've already discussed a node's degree of connectivity is a primary metric that defines its degree of
significance within its local environment.
Secondly, we have what is called closeness centrality measures
They 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 in the network?
Betweenness is a third metric that 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 measurements that are trying to describe
how significant you are based upon the significance of the nodes you are connected to.
Again, which one of these works best will depend on the context.
So to talk about closeness
then closeness
may be defined as the
Reciprocal of Farness where the farness of a given node is defined as the sum of its distance 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 is a measure of the node's capacity to affect all the other nodes in the network.
Betweenness, as mentioned,
is really talking about how critical a node is to a network in its
functioning as a 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 amount of connections that 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 low 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 into it
So now that we have a basic understanding of the different metrics for talking about a node centrality within a network
we can take a look at a graph and see how each represents a different perspective and set of results to this question.
Here is the same set of networks
with node ranking depicted in colors. Dark blue for a
low-ranking through to red with a high ranking we can note how the color changes
around the network for each set of metrics we're applying, thus indicating the different information
that our different metrics are capturing. And from this we can see the degree of relativity in trying to capture a node's degree of centrality.