Calculating Relational Class Cohesion as Eigenvalues
The use of Eigenvalues in determining the stability of a stimulus class in evoking a functional response class bridges linear algebra with stimulus class analysis, likely in the context of Relational Frame Theory (RFT) or stimulus equivalence.
The Matrix as a Model of Stimulus Relations
A matrix where [A-->B-->C-->D-->E, with reflectivity for each member] is symmetric and tridiagonal, meaning each stimulus (say A–E) is most strongly related to its immediate neighbors, with weaker or no direct relation to others. This mimics a typical scenario where stimuli are not equally connected to every other stimulus (as in a fully connected class), but instead are part of a chain-like or gradient relational structure.
What the Eigenvalues and Eigenvectors Tell Us
Eigenvalues → Stability of Modes
Larger eigenvalues correspond to more stable, dominant modes of behavioral control.
Smaller eigenvalues represent modes of behavioral control that are more fragile or sensitive to perturbation
In the hypothetical matrix used in this illustration, each stimulus coordinates with each of the other members of the set in the following proportions: A = 1, 0.5, 0, 0, 0; B = 0.5, 1, 0.5, 0, 0; C = 0, 0.5, 1, 0.5, 0; D = 0, 0, 0.5, 1, 0.5; E = 0, 0, 0, 0.5. Here, all five stimuli show perfect reflectivity and are directly associated with its immediate neighbors only.
The largest eigenvalue of this set (~1.866) corresponds to a smooth, symmetric eigenvector, indicating a consistent pattern in which all stimuli produce similar responses.
The smaller eigenvalues of this set (e.g., ~0.134) correspond to oscillatory modes — response patterns where stimuli diverge or differentiate. These are less stable and more sensitive to disruption.
Eigenvectors → Structure of Class Relations
The first eigenvector (associated with the largest eigenvalue) shows all stimuli having positive, symmetric contributions — this suggests cohesive class membership.
Other eigenvectors show opposing or alternating signs, indicating subclass formations or differentiation within the class under perturbation (e.g., extinction, new reinforcement contingencies).
Interpretation for a 5-Member Stimulus Class (All Evoking Same Response)
High stability in the dominant mode (eigenvalue ~1.866) indicates that this class structure is robust — the shared response is strongly supported by the structure of inter-stimulus relations.
However, the presence of lower eigenvalues indicates that disturbances (like altered reinforcement, contextual shifts, or introducing new stimuli) could lead to splintering or reorganization of the class, especially along the patterns shown by those oscillatory eigenvectors.
Psychological Analogy
Think of the eigenvectors like latent response tendencies: while your class is unified at the surface (same response), hidden gradients or structures within the stimuli might predispose them to split apart under pressure.
The system is stable, but not invulnerable — the farther apart stimuli are in the chain, the weaker their connection.