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)*where c is a vector of node centralities, I is an identity matrix, and 1 is a column vector of 1’s.^{-1}R*1 - 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

- Introduction of user-defined Β and α to measure centrality c such that

- The centrality nodes in a network are given by:

More Information:

- R documentation
- Centrality and Centralization
- Degree centrality: Bonacich’s approach
- 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:

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