How Does Works Eccentricity Work in a Network?

Definition Eccentricity

The distance between two nodes in a connected network is the shortest path (minimal number of links) between these nodes. The eccentricity of a node is the maximum distance to the other nodes, the farthest node from it in the network.

In diffusion processes, this is an indicator of the effort to reach the periphery of a network from a source node.

Disconnected Network

In a disconnected network, the eccentricity of nodes in different components is considered to be undefined or infinite. A component is a connected subnetwork that is not part of any larger connected subnetwork.

Descriptive Statistics

Some descriptive statistics of a network are based on the eccentricity of nodes:

• diameter = maximum eccentricity
• radius = minimum eccentricity
• central node: eccentricity node = radius network
• absolute centre: node with the lowest eccentricity
• centre: set of all central nodes of a network

A less known metric nestled in between the diameter and radius is the average eccentricity [1]. This is the average eccentricity of all the nodes in this network. So, the radius ≤ average eccentricity ≤ diameter of a network.

Eccentricity Centrality

The reciprocal of eccentricity is called eccentricity centrality. This is used to make sure that more central nodes have a higher value because such nodes are the ones with the smallest eccentricity score. Like other centralities, the node with the highest score is now the most central node, and these values are comparable.

A synonym is (Harary) graph centrality.

Eccentricity vs. Closeness

In contrast to eccentricity, closeness centrality uses not only the maximum distance between the node and all other nodes but also the sum of the distances of this node and all other nodes to calculate the mean distance.

Cons Eccentricity

An important disadvantage of eccentricity is that this metric is sensitive to the diameter of the network. Adding or removing a node and/or link may change the eccentricity score.

Another disadvantage is that all nodes must be reachable, and the network must be connected.

Conclusion

Eccentricity is related to closeness centrality; both measure the distance between a node and all other nodes in the network: eccentricity is the maximum, and closeness is the average distance. Some descriptive statistics of a network are based on the eccentricity of nodes.

Do not confuse eccentricity with eccentricity-centrality. If you want to compare this statistic with other centralities, they must have the same orientation, and you should use the eccentricity centrality.