Formation of reciprocal appreciation patterns in small groups: an agent-based model

Complex Adaptive Systems Modeling

Table 1 The number \(N_{ind}\) and intensities \(W_{ind}\) of peer-appreciation patterns (ind) based on peer-appreciation strengths \(\kappa _{i j}\)

Pattern	ind	Number \(N_{ind}\)	Intensity \(W_{ind}\)
triad	300	\(\sum (S^{3} )/6\)	\(N_{300}^{-1} \sum (S'^{3} )^{\frac{1}{6}}/6 \)
dyadic L	201	\(\sum (S^{2} \circ \tilde{E})/2 \)	\(N_{201}^{-1} \sum (S'^{2} \circ \tilde{E})^{\frac{1}{4}}/2 \)
dyad	102	\(\sum (\tilde{E}^2 \circ S )/2\)	\(N_{102}^{-1} \sum (\tilde{E}^2 \circ S' )^{\frac{1}{2}}/2 \)
triadic L	210	\(\sum (A A^{T} \circ S)/2\)	\(N_{210}^{-1} \sum (A' A'^{T} \circ S')^{\frac{1}{4}}/2 \)
C triad	120D	\(\sum (A^{T} A \circ S)/2\)	\(N_{120D}^{-1} \sum (A'^{T} A' \circ S')^{\frac{1}{4}}/2 \)
B triad	120U	\(\sum (A A^{T} \circ S)/2\)	\(N_{120U}^{-1} \sum (A' A'^{T} \circ S')^{\frac{1}{4}}/2 \)
C dyad	111U	\(\sum S A^{T} \circ K \circ \widetilde{K^T}\)	\(N_{111U}^{-1} \sum (S' A'^{T} \circ K \circ \widetilde{K^T})^{\frac{1}{3}} \)
B dyad	111D	\(\sum S A \circ K \circ \widetilde{K^T}\)	\(N_{111D}^{-1} \sum (S' A' \circ K \circ \widetilde{K^T})^{\frac{1}{3}} \)
endorsed L	021U	\(\sum (A A^{T} \circ \tilde{E})/2\)	\(N_{021U}^{-1} \sum (A' A'^{T} \circ \tilde{E})^{\frac{1}{2}}/2 \)

The following descriptive names of patterns in shorthand notations are used: L leadership, C collateral, B broker. The number \(N_{ind}\) of different patterns can be counted from the adjacency matrix \(\mathrm{K'}\), which has elements \([\mathrm{K'}]_{i j}= \kappa _{i j}\) and \([\mathrm{K'}]_{i i}=0\) (excluding self-appreciations) providing all the information to count the number \(N_{ind}\) and intensities \(W_{ind}\) of patterns of interest. For this, six other auxiliary matrices derived from \(\mathrm{K'}\) are introduced: the symmetric part of \(\mathrm{K'}\), denoted by \(\mathrm{S'}\), and the asymmetric part, \(\mathrm{A'}=\mathrm{K'}-\mathrm{S'}\); matrices \(\mathrm{K}\), \(\mathrm{S}\) and \(\mathrm{A}\) where entries are 1 for all non-zero entries in \(\mathrm{S'}\) and \(\mathrm{A'}\), respectively; and the symmetric matrix \(\mathrm{E}\) constructed so that if \(\mathrm{K}_ {i j} \ne 0\) or \(\mathrm{K}_ {j i} \ne 0\), then \(\mathrm{E}_{i j} = \mathrm{E}_{j i}=1\). Standard matrix operations are used so that T denotes transpose, Tr trace and \(\circ \) is the element-wise multiplication (Hadamard product). The logical inverse (complement) of the matrix is denoted by \(\sim \) e.g. \(\tilde{\mathrm{K}}\) as the complement of \(\mathrm{K}\)