L1centrality (version 0.4.0)

L1centLOC: Local L1 Centrality/Prestige

Description

Computes local L₁ centrality or local L₁ prestige at each alpha level for every vertex. For undirected graphs, the two measures are identical.

Usage

L1centLOC(g, eta, alpha, mode, weight_transform)
# S3 method for igraph
L1centLOC(
  g,
  eta = NULL,
  alpha,
  mode = c("centrality", "prestige"),
  weight_transform = NULL
)
# S3 method for matrix
L1centLOC(
  g,
  eta = NULL,
  alpha,
  mode = c("centrality", "prestige"),
  weight_transform = NULL
)
# S3 method for L1centLOC
print(x, ...)
# S3 method for L1centLOC
plot(x, y = NULL, add = FALSE, threshold = NULL, ...)
# S3 method for L1centLOC
summary(object, ...)

Value

L1centLOC() returns an object of class L1centLOC. It is a list of numeric vectors. The length of the list is equivalent to the length of alpha, and the names of the list are the values of alpha. Each component of the list is a numeric vector whose length is equivalent to the number of vertices in the graph g. Specifically, the ith component of the list is a vector of local L₁ centrality at level alpha[i] for each vertex (if mode = "centrality") or local L₁ prestige at level alpha[i] for each vertex (if mode = "prestige").

print.L1centLOC() prints local L₁ centrality or local L₁ prestige values at each locality level alpha and returns them invisibly.

plot.L1centLOC() draws a following plot.

• y is not supplied and alpha is of length one: A Lorenz curve (the group heterogeneity plot) and returns an invisible copy of a Gini coefficient (the group heterogeneity index). threshold is ignored.

• y is supplied and alpha is of length one: A scatter plot of x versus y. threshold is ignored.

• alpha's length is larger than one: A plot of alpha versus local L₁ prominence values (in a uniform margin) for each vertex. If threshold is set, vertices that have their maximum and minimum local L₁ prominence value difference above the threshold are indicated in colored lines. y is ignored.

summary.L1centLOC() returns an object of class table. It is a summary of the prominence values with the five-number summary, mean, and the Gini coefficient, at each level of alpha.

Arguments

g

An igraph graph object or a distance matrix. The graph must be connected. For a directed graph, it must be strongly connected. Equivalently, all entries of the distance matrix must be finite. Here, the (i,j) component of the distance matrix is the geodesic distance from the ith vertex to the jth vertex.

eta

An optional nonnegative multiplicity (weight) vector for (vertex) weighted networks. The sum of its components must be positive. If set to NULL (the default), all vertices will have the same positive weight (multiplicity) of 1, i.e., g is treated as a vertex unweighted graph. The length of the eta must be equivalent to the number of vertices.

alpha

A number or a numeric vector of locality levels. Values must be between 0 and 1.

mode

A character string. For an undirected graph, either choice gives the same result.

• centrality (the default): L₁ centrality (prominence of each vertex in terms of making a choice) is used for analysis.

• prestige: L₁ prestige (prominence of each vertex in terms of receiving a choice) is used for analysis.

weight_transform

An optional function to transform the edge weights when g is an igraph object and an edge weight attribute exists. This argument is ignored when g is a distance matrix.

x

An L1centLOC object, obtained as a result of the function L1centLOC().

...

Further arguments passed to or from other methods.

y

An optional argument providing the coordinates for a scatter plot. It could be an object of class L1cent or L1centLOC, or a numeric vector.

add

A logical value. This argument is considered only when drawing a Lorenz curve.

• TRUE: add the Lorenz curve to an already existing plot.

• FALSE (the default): draw the Lorenz curve to a new graphic device.

threshold

A number between 0 and 1. Vertices that have their maximum and minimum local L₁ prominence value difference above the threshold are indicated in colored lines.

object

An L1centLOC object, obtained as a result of the function L1centLOC().

Details

Suppose that the given graph has n vertices. We choose about \(n\alpha\) vertices (L₁ centrality- or L₁ prestige-based neighborhood) for each vertex (see L1centNB()), and compute the L₁ centrality or L₁ prestige of the vertex conditioned on these vertices, i.e., derive the L₁ centrality or L₁ prestige locally. For details, refer to Kang and Oh (2025a) for undirected graphs, and Kang and Oh (2025b) for directed graphs.

References

S. Kang and H.-S. Oh. On a notion of graph centrality based on L₁ data depth. Journal of the American Statistical Association, 1--13, 2025a.

S. Kang and H.-S. Oh. L₁ prominence measures for directed graphs. The American Statistician, 1--16, 2025b.

See Also

L1cent() for L₁ centrality/prestige, L1centNB() for L₁ centrality/prestige-based neighborhood.

Examples

weight <- igraph::V(MCUmovie)$worldwidegross
MCUmovie_cent <- L1cent(MCUmovie, eta = weight)
MCUmovie_loc_cent <- L1centLOC(MCUmovie, eta = weight, alpha = 5/32)
plot(MCUmovie_cent, MCUmovie_loc_cent,
     main = "MCU movie network: global vs. local centrality")