“Networks that fire together, wire together” (paraphrase: Hebb, 1949)
Introduction
This document outlines the concept of phase transitions within Hyper-Dimensional Multi-Level - Relational Frame Responding (HDML-RFR) and the use of this model to further understand and control the phase transitions that occur within HDML-RFR. The model arises from integration of concepts and models from complexity science, network theory, and cognitive development, all within the context of the root assumptions and conceptual units of Relational Frame Theory (RFT). The outline presents an evolutionary model of relational framing across levels of increasing dimensionality and complexity, beginning with isolated relational frames and evolving into multi-level, dynamic networks of relations. It describes the phase transition from discrete relational frames to networked relational behavior, highlighting key features such as cognitive load, flexibility, and emergence.
Frames to Networks: Phase Transitions
As water freezing into ice or evaporating into mist—a phase transition is a nonlinear change in system features, function and behavior, not just an accumulation or dispersion of parts or processes. Both elemental and emergent relational structures are states of cognitive functioning.
Initial State: Discrete Relational Frames
Discrete & Linear -- early in learning, relational behavior is limited to individual relational frames (e.g., coordination, distinction, opposition).
Low Integration -- these early frames are contextually bound, often cued directly by antecedents, and not yet forming cohesive structures with emergent or flexible functions.
Cognitive Energy -- processing is effortful, and each relation is learned in isolation.
Critical Mass: Combinatorial Density Increases.
Continuous & Non-linear
Combinatorial Entailment Emerges -- when a response class is controlled by multiple frames, entailments begin to multiply nonlinearly (e.g., A=B, B=C → A=C).
Transformation of Function spreads across more complex relational networks.
This creates “stress” or “variance of integrity” of relational frames and the Crels and Cfuncs that drive them — analogous to increasing temperature or density in physical systems.
Conceptualized Phase Transition
Like a percolation threshold (Albert, R., Jeong, H., & Barabási, A.-L., 2000) in network theory: enough nodes (relata) and links (relations and entailments) lead to a composite, cohesive combinatorally dense network of interconnected relational nodes (stimuli) and linking Crel entailments. Behaviorally, this is when relational responding becomes more automatic, fluid, and contextually flexible. Percolation Threshold refers to that transition point at which enough nodes and connections exist in a system that an evolved connected composite structure appears. Beyond this point, transfer of function and control by Crels can rapidly and reliably spread across the entire network.
RFT as a Percolating Network
Pre-Percolation phase exists when (1) learners have a limited set of trained frames and relations, (2) stimuli are related in isolation or within small, localized frames, and (3) most relational responding is directly taught, and contextually rigid Combinatorial Density is low; frames are "siloed".
A Percolation Threshold occurs when (1) training and multiple exemplar instruction, results in more stimuli within a Crel, (2) entailments begin to chain — A=B, B=C, C=D..., and (3) learners start to show derived relational responses across stimuli never explicitly paired. This relational state is analogous to a "Critical Mass" phase.
Percolation Point — Phase transition is a tipping point where the relational network becomes self-organizing and new nodes (stimuli) can instantly connect to existing relations. This results in automaticity (fast, untaught responses), fluidity (shift among frames easily), and contextual flexibility (responding differently depending on Crels and Cfuncs)
Post-Threshold: Semantic Percolation. The system now supports abstract thinking, analogical reasoning, and generalized transformation of function, novel stimuli trigger relational cascades, and networked cognition appears; that is, meaning is no longer tied to individual relations, but to the structure of the whole network.
Think of it like rain on dry soil. Initially, droplets sink without much effect, but at a certain point, the ground becomes saturated, and water easily spreads everywhere. [Though the analogy does not capture the existence of long-range entailment connections across disparate relational networks, which likely provide the foundation for hierarchical frames.]
Percolation Threshold in RFT – Visual Model
X-axis is the number of trained/linked relational nodes (stimuli/frames).
Y-axis shows the probability of derived relational responding (DRR) across the network.
That the curve suddenly spikes up shows nonlinear growth. This inflection point is the “percolation threshold”.
Before Percolation | After Percolation |
|---|---|
| Requires direct teaching | Learner derives new relations independently |
| Fragile frame use | Flexible, generative relational skills |
| Local generalization | Global network generalization |
| Effortful | Automatic and fast |
The percolation threshold in RFT marks that point at which combinatorial entailment density produces a generative effect, allowing fluid, context-sensitive derived relational responding to appear spontaneously. It is a nonlinear leap from frame-based behavior to network-based responding.
Post-Transition: Networked Relational Behavior
Relational Networks -- frames now relate dynamically; any node (stimulus) can activate a web of relations.
Multi-Level Abstraction -- meaning is derived from the whole network, not isolated, component relations.
Hyper-dimensionality -- relations are not just linear but cross levels (e.g., a relation about relations, or abstract relational clusters).
Cognitive Evolution
Occurs when (1) transformation of function happens across complex frame families like hierarchical, deictic, or temporal networks, (2) a high frequency of arbitrary applicable relational responding occurs, and (3) spontaneous derived relational responding becomes the norm.
Neurocognitive and Computational Parallels
Like neural network formation, where enough connectivity leads to emergent learning and pattern recognition, in Assembly Theory terms, the system reaches a complexity threshold where new functions appear that are not present in individual components (frames) of composites of components (networks and networks of networks).
Relational Hubs and Ming’s (2020, 2021) Applied RFT Model
Ming (2020, 2021) introduces the concept of relational hubs—stimuli or nodes within a relational network that maintain high connectivity across multiple relational frames. These hubs function as integrative anchors, enabling rapid generalization, transformation of function, and efficient transfer across contexts. In her applied curriculum work with learners with autism and language delays, Ming emphasizes fluency-based instruction and multiple exemplar training to strengthen these hubs.
Relational hubs serve as accelerators of percolation, allowing networks to reach the phase transition point faster by acting as bridges between otherwise isolated relational clusters. Their presence increases combinatorial density and contributes to the structural integrity and flexibility of the learner’s relational repertoire. For example, if the word "bigger" is linked via coordination, comparison, and hierarchy across various stimuli, it becomes a hub from which novel inferences can radiate. This approach reflects a strategic instructional focus: rather than teaching all possible relations exhaustively, strengthening a few well-connected hubs achieves greater generativity with fewer training trials.
In this way, Ming’s work operationalizes the network principles within applied RFT, offering tools to identify, assess, and build toward percolation thresholds and beyond. It connects theoretical models to curriculum design and decision-making in clinical practice.
Key Features of the Transition
Feature | Pre-Transition (Frames) | Transition Zone | Post-Transition (Networks) |
| Structure | Isolated relations | Emerging clusters | Interconnected networks |
| Flexibility | Rigid, context-bound | Emerging generalization | Highly flexible |
| Cognitive Load | High | Moderate | Low (due to fluency) |
| Emergence | Linear accumulation | Nonlinear leaps | Emergent relational meaning |
| Intervention Focus | Teaching individual frames | Teaching recombination | Teaching abstraction/metacognition |
Implications for Learning & Intervention
Early-stage programming should focus on (1) explicit teaching of operant, discriminated operant, conditional discriminated operant and mutually entailed Derived Relational Responding (DRR), (2) engineering a “Relational Transition Zone (RTZ)” that promotes recombinative generalization (e.g., Matrix Training), and (3) post-transition programming that requires the learner to engage in abstract reasoning, self-awareness, and perspective taking using relational networks.
References
Albert, R., Jeong, H., & Barabási, A.-L. (2000). Error and attack tolerance of complex networks. Nature, 406(6794), 378–382. https://doi.org/10.1038/35019019
Hebb, D. O. (1949). The organization of behavior: A neuropsychological theory. New York: Wiley.
Ming, S. (2020). Applied Relational Frame Theory: Developing a curriculum to establish generative language. The Psychological Record, 70(2), 271–290.
Ming, S. (2021). Relational hubs in language instruction: Building generativity through networked learning. Journal of Contextual Behavioral Science, 19, 59–72.