PB Space: A Mathematical Framework for Modeling Presence and Implication Balance in Psychological Change Through Fuzzy Cognitive Maps

Abstract

Understanding psychological change requires a quantitative framework capable of capturing the complex and dynamic relationships among personal constructs. Personal Construct Psychology emphasizes the hierarchical reorganization of bipolar constructs, yet existing qualitative methods inadequately model the reciprocal and graded influences involved in such change. This paper introduces the Presence–Balance (PB) space, a centrality measure for constructs represented within Fuzzy Cognitive Maps (FCMs). FCMs model cognitive systems as directed, weighted graphs, allowing for nuanced analysis of construct interactions. The PB space operationalizes two orthogonal dimensions: Presence, representing the overall connectivity and activation of a construct, and Implication Balance, quantifying the directional asymmetry between influences exerted and received. By formalizing Hinkle’s hierarchical theory within a rigorous mathematical framework, the PB space enables precise identification of constructs that drive or resist transformation. This dual-dimensional model provides a structured method for analyzing personal construct systems, supporting both theoretical exploration and clinically relevant interpretations in the study of psychological change.