degreecentrality

Degree centrality is a centrality measure in network analysis that quantifies the number of direct connections a node has. In an undirected network, the degree centrality of a node i equals k_i, the number of edges incident to i. In directed networks, centrality can be decomposed into in-degree and out-degree centrality, counting incoming and outgoing edges, respectively. In weighted networks, degree centrality is often defined as the strength of the node, the sum of weights of incident edges.

Normalized degree centrality divides by the maximum possible degree (n-1 for undirected graphs) to yield a value in the range [0,1], enabling comparison across networks of different sizes. It can also be reported without normalization for absolute degree.

Interpretation of degree centrality is straightforward: a higher value indicates more direct connections and potentially greater immediate influence or accessibility. However, degree centrality captures only local structure and does not reflect a node's broader role in the network, such as acting as a bridge between communities or occupying a strategic position in network flows.

Computation can be described using an adjacency matrix A. For undirected graphs, the centrality of i is the sum_j A_{ij}; for directed graphs, in-degree centrality is sum_j A_{ji} and out-degree centrality is sum_j A_{ij}. In weighted graphs, sums use the edge weights w_{ij} instead of binary indicators.

Applications span social networks to identify popular individuals, infrastructure networks to locate hubs, biological networks to find highly connected proteins, and information networks to assess potential pathways of spread. Limitations include sensitivity to network size and density and the fact that high degree does not guarantee global importance.