Find Bonacich Power Centrality Scores of Network Positions
Source:R/centrality.R
power_centrality.Rd
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
isFALSE
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
Details
Bonacich's power centrality measure is defined by
CBP(α,β)=α(I−βA)−1A1, where β is an attenuation parameter (set
here by exponent
) and A is the graph adjacency
matrix. (The coefficient α 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
β→1/λA1 (the reciprocal of the largest
eigenvalue of A), this is to within a constant multiple of
the familiar eigenvector centrality score; for other values of β,
the behavior of the measure is quite different. In particular, β
gives positive and negative weight to even and odd walks, respectively, as
can be seen from the series expansion
CBP(α,β)=α∑∞k=0βkAk+11 which converges so long as |β|<1/λA1.
The magnitude of β 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.
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 I get a better algorithm. power_centrality()
will not symmetrize your
data before extracting eigenvectors; don't send this routine asymmetric
matrices unless you really mean to do so.
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.
See also
eigen_centrality()
and alpha_centrality()
Centrality measures
alpha_centrality()
,
betweenness()
,
closeness()
,
diversity()
,
eigen_centrality()
,
harmonic_centrality()
,
hub_score()
,
page_rank()
,
spectrum()
,
strength()
,
subgraph_centrality()
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