Degree centrality

Degree centrality is the simplest measure of node importance in a network: the number of direct connections a node has. In an undirected graph, it is the number of edges incident to the node; in a directed graph, it separates into \'\'\'in-degree\'\'\' (connections received) and \'\'\'out-degree\'\'\' (connections made). Despite its simplicity, degree centrality remains one of the most predictive measures in network analysis, particularly for processes like epidemic spread where direct contact is the dominant transmission mechanism.

The measure\'s limitations are as obvious as its strengths. Degree centrality treats all connections as equal, ignoring the importance of the nodes at the other end. A connection to a peripheral node counts the same as a connection to the network\'s core. This is why eigenvector centrality and \'\'\'PageRank\'\'\' were developed — to weight connections by the importance of neighbors. Degree centrality also fails to capture global structure: a node can have high degree yet be isolated from the network\'s main component if all its connections are local.

Degree centrality is most useful as a baseline — a null model against which more sophisticated measures are compared. When eigenvector centrality and degree centrality produce radically different rankings, the discrepancy itself is informative: it reveals a network in which local connectivity does not predict global influence, suggesting the presence of \'\'\'core-periphery structure\'\'\' or \'\'\'assortative mixing\'\'\' that decouples local from global properties.