Find Bonacich Power Centrality Scores of Network Positions

power_centrality() takes a graph (dat) and returns the Boncich power centralities of positions (selected by nodes). The decay rate for power contributions is specified by exponent (1 by default).

Usage

power_centrality(
  graph,
  nodes = V(graph),
  loops = FALSE,
  exponent = 1,
  rescale = FALSE,
  tol = 1e-07,
  sparse = TRUE
)

Arguments

graph: the input graph.
nodes: vertex sequence indicating which vertices are to be included in the calculation. By default, all vertices are included.
loops: boolean indicating whether or not the diagonal should be treated as valid data. Set this true if and only if the data can contain loops. loops is FALSE by default.
exponent: exponent (decay rate) for the Bonacich power centrality score; can be negative
rescale: if true, centrality scores are rescaled such that they sum to 1.
tol: tolerance for near-singularities during matrix inversion (see solve())
sparse: Logical scalar, whether to use sparse matrices for the calculation. The ‘Matrix’ package is required for sparse matrix support

Value

A vector, containing the centrality scores.

Details

Bonacich's power centrality measure is defined by \(C_{BP}\left(\alpha,\beta\right)=\alpha\left(\mathbf{I}-\beta\mathbf{A}\right)^{-1}\mathbf{A}\mathbf{1}\), where \(\beta\) is an attenuation parameter (set here by exponent) and \(\mathbf{A}\) is the graph adjacency matrix. (The coefficient \(\alpha\) acts as a scaling parameter, and is set here (following Bonacich (1987)) such that the sum of squared scores is equal to the number of vertices. This allows 1 to be used as a reference value for the “middle” of the centrality range.) When \(\beta \rightarrow \)\( 1/\lambda_{\mathbf{A}1}\) (the reciprocal of the largest eigenvalue of \(\mathbf{A}\)), this is to within a constant multiple of the familiar eigenvector centrality score; for other values of \(\beta\), the behavior of the measure is quite different. In particular, \(\beta\) gives positive and negative weight to even and odd walks, respectively, as can be seen from the series expansion \(C_{BP}\left(\alpha,\beta\right)=\alpha \sum_{k=0}^\infty \beta^k \)\( \mathbf{A}^{k+1} \mathbf{1}\) which converges so long as \(|\beta| \)\( < 1/\lambda_{\mathbf{A}1}\). The magnitude of \(\beta\) controls the influence of distant actors on ego's centrality score, with larger magnitudes indicating slower rates of decay. (High rates, hence, imply a greater sensitivity to edge effects.)

Interpretively, the Bonacich power measure corresponds to the notion that the power of a vertex is recursively defined by the sum of the power of its alters. The nature of the recursion involved is then controlled by the power exponent: positive values imply that vertices become more powerful as their alters become more powerful (as occurs in cooperative relations), while negative values imply that vertices become more powerful only as their alters become weaker (as occurs in competitive or antagonistic relations). The magnitude of the exponent indicates the tendency of the effect to decay across long walks; higher magnitudes imply slower decay. One interesting feature of this measure is its relative instability to changes in exponent magnitude (particularly in the negative case). If your theory motivates use of this measure, you should be very careful to choose a decay parameter on a non-ad hoc basis.

For directed networks, the Bonacich power measure can be understood as similar to status in the network where higher status nodes have more edges that point from them to others with status. Node A's centrality depends on the centrality of all the nodes that A points toward, and their centrality depends on the nodes they point toward, etc. Note, this means that a node with an out-degree of 0 will have a Bonacich power centrality of 0 as they do not point towards anyone. When using this with directed network it is important to think about the edge direction and what it represents.

Note

This function was ported (i.e. copied) from the SNA package.

Warning

Singular adjacency matrices cause no end of headaches for this algorithm; thus, the routine may fail in certain cases. This will be fixed when we get a better algorithm.

References

Bonacich, P. (1972). “Factoring and Weighting Approaches to Status Scores and Clique Identification.” Journal of Mathematical Sociology, 2, 113-120.

Bonacich, P. (1987). “Power and Centrality: A Family of Measures.” American Journal of Sociology, 92, 1170-1182.

Author

Carter T. Butts (http://www.faculty.uci.edu/profile.cfm?faculty_id=5057), ported to igraph by Gabor Csardi csardi.gabor@gmail.com

Examples


# Generate some test data from Bonacich, 1987:
g.c <- make_graph(c(1, 2, 1, 3, 2, 4, 3, 5), dir = FALSE)
g.d <- make_graph(c(1, 2, 1, 3, 1, 4, 2, 5, 3, 6, 4, 7), dir = FALSE)
g.e <- make_graph(c(1, 2, 1, 3, 1, 4, 2, 5, 2, 6, 3, 7, 3, 8, 4, 9, 4, 10), dir = FALSE)
g.f <- make_graph(
  c(1, 2, 1, 3, 1, 4, 2, 5, 2, 6, 2, 7, 3, 8, 3, 9, 3, 10, 4, 11, 4, 12, 4, 13),
  dir = FALSE
)
# Compute power centrality scores
for (e in seq(-0.5, .5, by = 0.1)) {
  print(round(power_centrality(g.c, exp = e)[c(1, 2, 4)], 2))
}
#> [1] 0.00 1.58 0.00
#> [1] 0.73 1.45 0.36
#> [1] 0.97 1.34 0.49
#> [1] 1.09 1.27 0.54
#> [1] 1.15 1.23 0.58
#> [1] 1.2 1.2 0.6
#> [1] 1.22 1.17 0.61
#> [1] 1.25 1.16 0.62
#> [1] 1.26 1.14 0.63
#> [1] 1.27 1.13 0.64
#> [1] 1.28 1.12 0.64

for (e in seq(-0.4, .4, by = 0.1)) {
  print(round(power_centrality(g.d, exp = e)[c(1, 2, 5)], 2))
}
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54
#> [1] 1.62 1.08 0.54

for (e in seq(-0.4, .4, by = 0.1)) {
  print(round(power_centrality(g.e, exp = e)[c(1, 2, 5)], 2))
}
#> [1] -1.00  1.67 -0.33
#> [1] 0.36 1.81 0.12
#> [1] 1.00 1.67 0.33
#> [1] 1.30 1.55 0.43
#> [1] 1.46 1.46 0.49
#> [1] 1.57 1.40 0.52
#> [1] 1.63 1.36 0.54
#> [1] 1.68 1.33 0.56
#> [1] 1.72 1.30 0.57

for (e in seq(-0.4, .4, by = 0.1)) {
  print(round(power_centrality(g.f, exp = e)[c(1, 2, 5)], 2))
}
#> [1] -1.72  1.53 -0.57
#> [1] -0.55  2.03 -0.18
#> [1] 0.44 2.05 0.15
#> [1] 1.01 1.91 0.34
#> [1] 1.33 1.78 0.44
#> [1] 1.52 1.67 0.51
#> [1] 1.65 1.59 0.55
#> [1] 1.74 1.53 0.58
#> [1] 1.80 1.48 0.60