SmallGroupNetwork: SmallGroupNetwork

In stephen-l-jones/SmallGroupNetwork: Fit Network Configurations to Small Group Networks

SmallGroupNetwork

Description

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Fit configurations to a small group network to determine the best-fitting configuration(s).

Details

Configurations are network structures of theoretical interest. They are fitted to an empirical small group network to see which configuration within a configuration_set best approximates the group network. The function fit_configuration_set determines the best-fitting configuration. Configurations and group networks are represented as square adjacency matrices. They can be binary or weighted, directed or undirected, and can include or exclude loops (i.e., self-references). A configuration must have the same dimensions as a group network to be fitted to it.

Binary configurations

When fitting binary configurations, a group network's negative values indicate the absence of an edge (i.e., tie) and positive values indicate the presence of an edge. More negative (or positive) values in the group network give stronger evidence of the absence (or presence) of a tie. The function will attempt to match a configuration's 0-valued elements to negative network values and match a configuration's 1-valued elements to positive values.

Given a binary configuration \mjeqnff in set \mjeqnFF and network \mjeqnx_gx_g for group \mjeqngg, a score for group \mjeqngg and configuration \mjeqnff is calculated as: \mjdeqnscore_g,f = \sum_i=1^N_g \sum_j=1^N_g x_g,ij \cdot b_ij, score_g,f = \sum_i \sum_j x_g,ij x b_ij, where \mjeqnx_g,ij x_g,ij is the (i, j)th element in \mjeqnx_gx_g and \mjeqnb_ij = +1b_ij = +1 when \mjeqnf_ij = 1f_ij = 1, \mjeqnb_ij = -1b_ij = -1 when \mjeqnf_ij = 0f_ij = 0, and \mjeqnb_ij = 0b_ij = 0 when \mjeqnf_ijf_ij is NA. The order of rows and columns in any configuration \mjeqnff is arbitrary. Thus, the function determines the ordering of rows and columns in \mjeqnff that maximizes \mjeqnscore_g,fscore_g,f.

All binary configurations in set \mjeqnFF that have the same dimensions (\mjeqnN_g \times N_gN_g x N_g) as \mjeqnx_gx_g are fit using fit_group_network, which returns the the best-fitting configuration \mjeqnf^\astf*. The fit matrix in the returned configuration_fit object gives the reordered rows and columns of configuration \mjeqnf^\astf* that maximizes the score.

Weighted configurations

When fitting weighted configurations, a group network's values indicate a level of the measured relationship. A configurations n-valued elements are matched to a network's edges with the closest values to n.

Given a weighted configuration \mjeqnff in set \mjeqnFF and network \mjeqnx_gx_g, a score for group \mjeqngg and configuration \mjeqnff is calculated as: \mjdeqnscore_g,f = \sum_i=1^N_g \sum_j=1^N_g \mathrmabs(x_g,ij - f_ij) \cdot d_ij,score_g,f = \sum_i \sum_j abs(x_g,ij - b_g,ij) x d_ij, where \mjeqn\mathrmabsabs is the absolute value function and \mjeqnd_ij = 0d_ij = 0 when \mjeqnf_ijf_ij is NA; otherwise, \mjeqnd_ij = 1d_ij = 1. All weighted configurations in set \mjeqnFF that have the same dimensions (\mjeqnN_g \times N_gN_g x N_g) as \mjeqnx_gx_g are fit, and the function returns the reordered weighted configuration \mjeqnf^\astf* that minimizes the score.

stephen-l-jones/SmallGroupNetwork documentation built on April 25, 2022, 11:15 p.m.