Centrality is a key concept in network theory and graph analysis, used to measure the importance or influence of a node (or vertex) in a network. There are several types of centrality, each with a different way of determining what makes a node "central" or important:

Degree Centrality: How many immediate neighbors a node has

This measures the number of direct connections a node has. The more connections, the more "central" or important the node is in the network. It’s calculated as:

$$ \text{Degree Centrality} = \frac{\text{Number of connections}}{\text{Total nodes} - 1} $$

Betweenness Centrality: Finding nodes that act as "bridges" in a network

• This measures the extent to which a node lies on the shortest paths between other nodes. A node with high betweenness centrality has significant control over the flow of information in the network.
• Formula:

where is the total number of shortest paths from node to node , and is the number of those paths that pass through node .

$$
\\text{Betweenness Centrality} = \\sum \\frac{\\sigma(s, t|v)}{\\sigma(s,t)}

$$

where

$σ(s,t∣v)$ is the number of those paths that pass through node $v$

$σ(s,t)$ is the total number of shortest paths from node  to node $t$,

Closeness Centrality: how quickly information can spread from a node

• This measures how close a node is to all other nodes in the network. Nodes with high closeness centrality can reach others more quickly. It’s calculated as the inverse of the average shortest path distance from the node to all other nodes:

  $$ Closeness Centrality =\frac{1}{\sum \text{Shortest path distances from a node to all other nodes}} $$

Eigenvector Centrality:

• This not only considers the number of connections a node has but also takes into account the importance of the nodes it is connected to. A node is more central if it is connected to other central nodes. The idea is that a connection to an important node is more valuable than a connection to a less important one.
• Solving for eigenvector centrality involves finding the principal eigenvector of the adjacency matrix representing the network.

PageRank:

• This is a variant of eigenvector centrality, originally developed by Google to rank web pages. It considers both the quantity and quality of links to a node (e.g., web page). Important nodes are those linked by many important nodes.