Centrality metrics identify the most important or influential nodes in a network. Different measures capture different aspects of importance based on network position and connectivity patterns.
Degree Centrality:
Counts the number of direct connections a node has
Formula: C_D(i) = k_i (where k_i is the degree of node i)
Identifies locally important nodes with many direct contacts
Best for: Finding well-connected individuals in social networks
Betweenness Centrality:
Measures how often a node lies on shortest paths between other nodes
Formula: C_B(i) = Σ(σ_st(i) / σ_st) for all node pairs s,t
Identifies brokers and bridges who control information flow
Best for: Finding gatekeepers and influential connectors
Closeness Centrality:
Measures the average shortest path distance to all other nodes
Formula: C_C(i) = n / Σd(i,j) where d(i,j) is distance from i to j
Identifies nodes that can quickly reach the entire network
Best for: Finding efficient broadcasters and rapid responders
Eigenvector Centrality:
Considers both quantity and quality of connections
High score if connected to other high-scoring nodes
Based on the principal eigenvector of the adjacency matrix
Best for: Finding nodes connected to other important nodes
Applications:
Organizational Networks: Identify key employees, informal leaders, and knowledge brokers
Social Networks: Find influencers and opinion leaders
Infrastructure: Locate critical nodes in transportation or communication networks
Biological Networks: Identify essential proteins or genes
