PageRank Centrality

PageRank Centrality is a centrality measure that evaluates the importance of a node in a network by considering the quality and quantity of the connections that the node has with other nodes in the network. It is based on the concept of "voting" or "endorsement" by other nodes, where a node is considered more important if it is endorsed by other important nodes.

In mathematical terms, PageRank Centrality is calculated using the eigenvector of the adjacency matrix of the network, and can be expressed as:

$$PR(v) = \frac{1-d}{N} + d\sum_{\substack{ u\in B_v }} \frac{PR(u)}{L(u)}$$

where PR(w) is the PageRank Centrality of node v, I is the total number of nodes in the network, Bo is the set of nodes that are linked to node w, L(r) is the number of links outgoing from node u, and d is a damping factor that represents the probability of a user to randomly jump to another node in the network

The core idea of PageRank Centrality is to evaluate the importance of a node by calculating its in-degree and the out-degree of its neighbouring nodes. Nodes with high PageRank Centrality are usually the most important nodes in the social network because they are pointed to by many other nodes, and the nodes that point to them also have high PageRank Centrality. This creates a "group endorsement" effect similar to "the more nodes pointing to this node, the more important it is". Unlike In/Out Degree centrality, PageRank Centrality considers not only the degree of a node, but also its in-degree, out-degree, and connections with other nodes. Nodes with high PageRank Centrality are linked to many other important nodes, and they have a greater influence in the entire network, making it easier to spread information and influence.

In Bond's social network analysis, PageRank Centrality can be used to discover important figures and sources of influence in the social network. By analysing PageRank Centrality, we can better understand the flow of information and influence, and optimise the design and operation of the social network. Additionally, PageRank Centrality can be used to identify spam and false information in the social network, thereby improving its quality and reliability.