L1centNB {L1centrality}

L1 Centrality/Prestige-Based Neighborhood

Description

Derives L₁ centrality- or prestige-based neighborhood of each vertex. For undirected graphs, the two neighborhood are identical.

Usage

L1centNB(g, eta, mode)

## S3 method for class 'igraph'
L1centNB(g, eta = NULL, mode = c("centrality", "prestige"))

## S3 method for class 'matrix'
L1centNB(g, eta = NULL, 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.

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

For an undirected graph, if the graph is symmetrized (in a way defined in Kang and Oh (2024a)) w.r.t. a vertex v, vertex v becomes the graph median (Kang and Oh, 2024a), i.e., v has L₁ centrality 1. Based on this property, we define the L₁ centrality-based neighborhood of vertex v as vertices that have large L₁ centrality on the symmetrized graph w.r.t. vertex v.

For a directed graph, a vertex of interest, say v, is made to a centrality and prestige median vertex by the procedure described in Kang and Oh (2024b). We call the resulting graph as the modified graph w.r.t. v. L₁ centrality(prestige)-based neighborhood of vertex v is a set of vertices that have large L₁ centrality(prestige) on the modified graph w.r.t. vertex v.

Value

A list of numeric vectors. The length of the list is equivalent to the number of vertices in the graph g, and the names of the list are vertex names. 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 the L₁ centrality of each vertex, for the modified graph g w.r.t. the ith vertex (if mode = "centrality") or the L₁ prestige of each vertex, for the modified graph g w.r.t. the ith 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, L1centLOC() and L1centEDGE() internally uses L1centNB().

Examples

NB <- L1centNB(MCUmovie, eta = igraph::V(MCUmovie)$worldwidegross)
# Top 6 L1 centrality-based neighbors of "Iron Man"
utils::head(sort(NB$"Iron Man", decreasing = TRUE))