Centrality Measures

Centrality measures are quantitative indices that identify which nodes in a network are structurally important. The concept of importance is not monolithic: a node can be important for different reasons, and different centrality measures capture different structural roles. The choice of centrality measure is therefore a theoretical commitment about what kind of process is operating on the network — diffusion, flow, control, or influence.

Degree centrality is the simplest measure: the number of edges incident on a node. It captures local activity — how many direct neighbors a node has — and is the appropriate measure for processes that operate through direct contact, such as the spread of information in a network where transmission requires acquaintance.

Betweenness centrality measures the fraction of shortest paths between all pairs of nodes that pass through a given node. A node with high betweenness is a bridge: it controls the flow of information or resources between otherwise separated parts of the network. Betweenness is the appropriate measure for processes involving brokerage, gatekeeping, or structural holes. Removing a high-betweenness node fragments the network; this is why betweenness is used to identify critical infrastructure vulnerabilities.

Eigenvector centrality and PageRank extend degree centrality by weighting connections by the importance of the neighbors. A node connected to many important nodes is more important than a node connected to many peripheral nodes. These measures are appropriate for processes where influence or status is recursively defined: a prestigious node is one that is connected to other prestigious nodes. Google's original PageRank algorithm applied this logic to the web link graph, treating a link as an endorsement whose value depends on the endorser's own endorsement count.

The critical methodological point: centrality measures are not interchangeable. A node that is central by degree may be peripheral by betweenness. A terrorist network, for example, is often designed so that operational nodes have high betweenness while leaders have low degree — a deliberate decoupling of structural importance from visibility. Using the wrong centrality measure for the process under study produces misleading conclusions about which nodes "matter."