Bonacich’s Centrality

Bonacich’s Approach to Centrality (also known as Eigenvector Centrality)

  • You are more central when there are more connections within your local network
  • Fewer connections within your local network means you are more powerful
    • Power comes from being connected to those that are powerless
  • The Measure
    • The centrality nodes in a network are given by: λ e = R e where R is matrix formulation of the network in question, e is an eigenvector of R, and λ is its associated eigenvalue
    • Additional variations:
      • Introduction of user-defined Β and α to measure centrality c such that c(α, Β) = α(I-ΒR)-1R*1 where c is a vector of node centralities, I is an identity matrix, and 1 is a column vector of 1’s.
      • Β reflects the degree to which a node’s power is related to the power of the nodes it is connected to
      • Intrepretation: A more positive Β means that other nodes centralities are taken more into account. A more negative Β means that a node’s power is reduced by the powerful nodes it is connected to.
      • α simply scales node centrality

More Information:

  1. R documentation
  2. Centrality and Centralization
  3. Degree centrality:  Bonacich’s approach
  4. Phillip Bonacich (1987) “Power and Centrality: A Family of Measures,” American Journal Of Sociology 92(5): 1170-1182.

References to use of this measure in literature:

  1. Peer Standing and Substance Use in Early-Adolescent Grade-Level Networks: A Short-Term Longitudinal Study
  2. Network Structure and Proxy Network Measures of HIV, Drug and Incarceration Risks for Active Drug Users
  3. An Empirical Assessment of Rural Community Support Networks for Individuals with Severe Mental Disorders

Betweenness Centrality

Betweenness Centrality

  • The higher a node’s centrality is the more “dependent” other nodes are on it
  • Based on shortest paths between nodes and the number of paths that pass through two points and the total number of paths
  • BC(i) = SUMs≠i≠ t∈Vμst(i) / μst where μst is the number of paths from s and t and μst(i) is the number of paths from s and t that pass through node i

Algorithm

Input: V, a vertex and G, a graph

  1. For all pairs of vertices (v1 and v2) in graph G, compute every shortest path between them
  2. Using v1 and v2, compute the fraction of paths between these vertices that pass through V
  3. Sum over all pairs of vertices

More Information and other Algorithms:

  1. Approximating Betweenness Centrality
References to use of this measure in literature:
  1. Collaboration and Integration of Community-Based Health and Human Services in a Nonprofit Managed Care System
  2. The Peer Context of Adolescent Substance Use: Findings from Social Network Analysis
  3. Peer Standing and Substance Use in Early-Adolescent Grade-Level Networks: A Short-Term Longitudinal Study