L1centEDGE {L1centrality}

Multiscale Edge Representation

Description

Derives a multiscale edge representation. Each vertex is connected to its local median, which is found in its L₁ centrality-based neighborhood.

Usage

L1centEDGE(g, eta, alpha)

## S3 method for class 'igraph'
L1centEDGE(g, eta = NULL, alpha)

## S3 method for class 'matrix'
L1centEDGE(g, eta = NULL, alpha)

Arguments

g

An igraph graph object or a distance matrix. The graph must be undirected and connected. Equivalently, the distance matrix must be symmetric, and all entries must be finite.

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.

Details

In a global perspective, any given graph can be represented as a star-shaped (directed) graph, with each vertex making a connection to the median vertex. Based on this idea, a graph can be represented as a directed graph, with each vertex making a connection to the local median vertex. The local median vertex of, say, v_i, is defined as a median vertex among the L₁ centrality-based neighborhood of v_i. By varying the level of locality, the given graph can be visually inspected at multiple scales.

Value

A list of ‘edge lists’. The length of the list is equivalent to the length of alpha, and the names of the list are the values of alpha. The ith component of the list is a 2-column matrix, and each row defines one directed edge, i.e., it is an edge list. The second column is the local (level alpha[i]) median of the vertex at the first column. There may be more than one edge from each vertex, since there may be more than one local median.

Note

The function is valid only for undirected and connected graphs.

References

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

See Also

L1cent(), L1centNB(), L1centLOC(). Using the output, one can use igraph::graph_from_edgelist() for creating an igraph object. See the example code below.

Examples

library(igraph)
MCU_edge <- L1centEDGE(MCUmovie, eta = V(MCUmovie)$worldwidegross, alpha = 5/32)
graph <- graph_from_edgelist(MCU_edge[[1]], directed = TRUE)
plot(graph)