L1centLOC {L1centrality}

Local L1 Centrality/Prestige

Description

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

Usage

L1centLOC(g, eta, alpha, mode)

## S3 method for class 'igraph'
L1centLOC(g, eta = NULL, alpha, mode = c("centrality", "prestige"))

## S3 method for class 'matrix'
L1centLOC(g, eta = NULL, alpha, mode = c("centrality", "prestige"))

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), 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

Details

Suppose that the given graph has n vertices. We choose about n\alpha vertices (L₁ centrality- or prestige-based neighborhood) for each vertex (see L1centNB()), and compute the L₁ centrality or prestige of the vertex conditioned on these vertices, i.e., derive the L₁ centrality or prestige locally. For details, refer to Kang and Oh (2024a) for undirected graphs, and Kang and Oh (2024b) for directed graphs.

Value

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").

Note

The function is valid only for connected graphs. If the graph is directed, it must be strongly connected.

References

S. Kang and H.-S. Oh. On a notion of graph centrality based on L₁ data depth. arXiv preprint arXiv:2404.13233, 2024a.

S. Kang and H.-S. Oh. L₁ prominence measures for directed graphs. Manuscript. 2024b.

See Also

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

Examples

weight <- igraph::V(MCUmovie)$worldwidegross
MCUmovie_cent <- L1cent(MCUmovie, eta = weight)
MCUmovie_loc_cent <- L1centLOC(MCUmovie, eta = weight, alpha = 5/32)
plot(MCUmovie_cent, MCUmovie_loc_cent[[1]],
     xlab="Global L1 centrality", ylab="Local L1 centrality (alpha = 5/32)",
     main="MCU movie network: global vs. local centrality")
graphics::text(MCUmovie_cent, MCUmovie_loc_cent[[1]], igraph::V(MCUmovie)$name)