laplace_centr: Calculate Laplacian centrality for a single graph.

In abnormally-distributed/rsfcNet: Analyze resting state functional connectivity networks

Description

This calculates the Laplacian centrality metric for a graph. This function works much better on a network with thresholding applied or on sparse graphs such that some entries have edge weights of zero. This function can take a long time to run on dense graphs.

Usage

laplace_centr(graph, prog.bar = TRUE)

Arguments

graph	A network as an igraph object.
prog.bar	Should a progress bar be displayed? Defaults to TRUE.
weights	Edge weights should you wish to provide modified input (ie, absolute value, thresholding etc)

Details

To first understand Laplacian centrality the concept of the Laplacian matrix must be understood. The Laplacian matrix (or graph Laplacian) is the adjacency (or correlation or other weighted connectivity) matrix subtracted from the degree (or strength) matrix. In the strength matrix all entries are zero except for the diagonal, which has as an entry the number corresponding to the degree or strength of node_i.

Mathematically this is represented as

L(G) = X(G) - W(G)

or

L(G) = D(G) - A(G)

The energy of L(G) is used in physics to measure things such as the diffusion of energy through a system and is shown in the formula below:

E_L(G)=∑_{i=1}^nx_i^2+2∑_{i<j}w_{i\text{,}j}^2

Laplacian centrality extends the idea by studying what happens to the ability of energy (or information) to difuse through a network if a node is removed. The centrality score for each node is calculated by the expected drop in energy for the graph Laplacian when node_i is removed. A method of calculating this is given by the formula below:

Δ E_{n}=s_{G}^{2}(n)+s_{G}(n)+2∑_{n _{i}\in N(n)}s_{G}(n_{i})

Built around code translated from python. Original python code available at http://igraph.wikidot.com/python-recipes#toc3

Value

A vector of the Laplacian centrality scores.

Author(s)

Brandon Vaughan

References

Pauls, S.D., & Remondini, D. (2012). A measure of centrality based on the spectrum of the Laplacian. Physical review. E, Statistical, nonlinear, and soft matter physics, 85 6 Pt 2, 066127.

Qi, Xingqin, et al. (2012). Laplacian centrality: A new centrality measure for weighted networks. Information Sciences 194: 240-253.

See Also

laplace_centr_mult leverage_centr leverage_centr_mult

Examples

# LPL= laplace_centr(graph)

abnormally-distributed/rsfcNet documentation built on March 8, 2020, 5:32 p.m.