kpcent: Compute Group Centraltiy in a Network

In keyplayer: Locating Key Players in Social Networks

Compute Group Centraltiy in a Network

Description

kpcent reports the group-level centrality scores.

Usage

kpcent(
  adj.matrix,
  nodes,
  type,
  M = Inf,
  T = ncol(adj.matrix),
  method,
  binary = FALSE,
  cmode,
  large = TRUE,
  geodist.precomp = NULL
)

Arguments

adj.matrix	Matrix indicating the adjacency matrix of the network or the probability matrix in the case of calculating diffusion centrality.
nodes	Integer indicating the column index of the chosen player in the adjacenncy matrix. If there are multiple players, use c(index1,index2,...)
type	type="betweenness" for betweenness centrality. type="closeness" for closeness centrality. type="degree" for degree centraslity. type="diffusion" for diffusion centrality. type="evcent" for evcent (eigenvector) centrality. type="fragment" for fragment centrality. type="mreach.degree" for mreach.degree centrality. type="mreach.closeness" for mreach.closeness centrality.
M	Positive number indicating the maximum geodistance between two nodes, above witch the two nodes are considered disconnected. The default is Inf. The option is applicable to mreach.degree, mreach.closeness, and fragmentation centralities.
T	Integer indicating the maximum number of iterations of communication process. For diffusion centrality only. By default, T is the network size.
method	Indication of which grouping criterion should be used. "min" indicates the "minimum" criterion and is the default for betweenness, closeness, fragmentation, and M-reach centralities. "max" indicates the "maximum" criterion and is the default for degree and eigenvector centralities. "add" indicates the "addition" criterion. "union" indicates the "union" criterion and is the default for diffusion centrality. See Details section for explanations on grouping method.
binary	If TRUE, the adjacency matrix is binarized. If FALSE, the edge values are considered. By default, binary=FALSE
cmode	String indicating the type of centrality being evaluated. The option is applicable to degree and M-reach centralities. "outdegree", "indegree", and "total" refer to indegree, outdegree, and (total) degree respectively. "all" reports all the above measures. The default is to report the total degree. Note for closeness centrality, we use the Gil-Schmidt power index when large=FALSE. See closeness for explanation. When large=TRUE, the function reports the standard closeness score.
large	Logical scalar, whether the computation method for large network is implemented. If TRUE (the default), the method implmented in igraph is used; otherwise the method implemented in sna is used.
geodist.precomp	Geodistance precomputed for the graph to be analyzed (optional).

Details

The basic idea of measuring the group-level centrality is to treat a group of nodes as a large pseudo-node. We propose several methods to measure the tie status between this pseudo node and other nodes, responding to several common edge value interpretations (An and Liu, 2015).

Minimum Criterion: the edge value between a group and an outside node is measured as the minimal value among all the (nonzero) edge values between any node in the group and the outside node. Suggested if edge values are interpreted as distances.
Example: suppose node A to C has distance 2 and B to C has distance 1, then according to the minimum criterion, the distance between C and the merged set AB is 1. Note that if B and C are not connected, the algorithm takes the distance between A and C to describe the distance between AB and C.

Maximun Criterion: the edge value between a group and an outside node is measured as the maximal value among all the (nonzero) edge values between any node in the group and the outside node. Suggested if edge values are interpreted as non-cummulative strengths.
Example: we keep using the above example, but the figure now indicates the strength of tie. According to the maximum criterion, the strength of tie between AB and C is 2.

Addition Criterion: the edge value between a group and an outside node is measured as the sum of all the edge values between any node in the group and the outside node. Suggested if edge values are as cummulative strengths.
Example: according to the addition criterion, the strength of tie between AB and C is 3

Union Criterion: the edge value between a group and an outside node is measured as the probability that there is at least one path connecting the group with the outside node. Suggested if edge values are as probability.
Example: suppose A has probability 0.2 to reach C and B has probability 0.5 to reach C, then C can be reached from merged AB with probability 1-(1-0.2)*(1-0.5)=0.6 according to the union criterion.

Value

A vector indicating the centrality score of a group.

Author(s)

Weihua An [e-mail]; Yu-Hsin Liu [e-mail]

References

An, Weihua. (2015). "Multilevel Meta Network Analysis with Application to Studying Network Dynamics of Network Interventions." Social Networks 43: 48-56.

An, Weihua and Yu-Hsin Liu (2016). "keyplayer: An R Package for Locating Key Players in Social Networks." The R Journal, 8(1): 257-268.

Banerjee, A., A. Chandrasekhar, E. Duflo, and M. Jackson (2013): "Diffusion of Microfinance," Science, Vol. 341. p.363

Banerjee, A., A. Chandrasekhar, E. Duflo, and M. Jackson (2014): "Gossip: Identifying Central Individuals in a Social Network," Working Paper.

Borgatti, Stephen P. (2006). "Identifying Sets of Key Players in a Network." Computational, Mathematical and Organizational Theory, 12(1):21-34.

Butts, Carter T. (2014). sna: Tools for Social Network Analysis. R package version 2.3-2. https://cran.r-project.org/package=sna

Csardi, G and Nepusz, T (2006). "The igraph software package for complex network research." InterJournal, Complex Systems 1695. https://igraph.org/

See Also

contract kpset

Examples

# Create a 5x5 weighted and directed adjacency matrix,
# where edge values represent the strength of tie
W <- matrix(
  c(0,1,3,0,0,
    0,0,0,4,0,
    1,1,0,2,0,
    0,0,0,0,3,
    0,2,0,0,0),
    nrow=5, ncol=5, byrow = TRUE)

# List the degree centrality for group of node 2 and 3
kpcent(W,c(2,3),type="degree")

# Transform the edge value to distance interpretaion
# Compute the fragmentation centrality for node 2
A <- W
A[W!=0] <- 1/W[W!=0]
kpcent(A,2,type="fragment")

# Replicate the group-level degree centrality (normalized) when the weights
# are given by the inverse distances and report the outgoing score only
kpcent(A,c(2,3),type="mreach.closeness",binary=TRUE,M=1,cmode="outdegree")

# Transform the edge value to probability interpretation
# Compute the diffusion centrality with number of iteration 20
P <- 0.1*W
kpcent(P,c(2,3),type="diffusion",T=20)

keyplayer documentation built on Nov. 8, 2023, 9:06 a.m.