contract: Group the Chosen Players in a Network

Description

contract is a fundmental tool for analyzing the group-level centrality. It contracts the chosen nodes into one following a specified rule, and thus the node-level method can be applied. There are four grouping criteria in the contract function: minimum, maximum, union, and addition, each of them responses to a specific interpretation of the edge values of the adjacency matrix. See details.

Usage

contract(adj.matrix, nodes, method = c("min", "max", "union", "add"))

Arguments

adj.matrix

Matrix indicating the adjacency matrix of the network.

nodes

Integer indicating the column index of the chosen player in the adjacenncy matrix. Input c(index1,index2,...) for multiple players

method

Indication of which grouping criterion should be used. method="min" indicates the "minimum" criterion (edge values as distances). method="max" indicates the "maximum" criterion (edge values as non-cummulative strengths).

Value

A new adjacency matrix after contracting the chosen nodes (named set).

Details

When the edge values are interpreted as distances, the contracting follows a "minimum" rule that the distance (edge value) between an objective and the contracting set is the minimun distance of each individual node in the set. For example, suppose A to C has distance 2 and B to C has distance 1, then the distance between C and the merged set AB is 1. When the edge values are interpreted as accumulative strength, the contracting follows an "addition" rule that the strength (edge value) of tie between an objective node and the contracting set is simply the summation of the edge value of each node in the set. In the above example, when the strength interpretation is used, the strangth of tie between C and the merged AB is 1+2=3. If the strength is not accumulative, then the "maximum" rule should be applied. When the edge values are interpreted as probability, the contracting follows an "union" rule that the link to the set can be formed when at least one of the nodes in the set is connected. The rule assumes that the link to each node in the set is independent to each other and thus computes the probability using 1 minus the product of the probability of not being connected for each node. For 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.

References

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