Toward a Graph-Theoretic Model of Belief: Confidence, Credibility, and Structural Coherence

2025-08-05

AI Summary

Existing belief representation formalisms—such as propositional sets or probability distributions—fail to capture the internal structure of beliefs, conflate external credibility with internal coherence, and cannot adequately model fragmented or contradictory cognitive states. Method: We propose a directed weighted graph-based belief system model: nodes represent individual beliefs, and directed edges encode cognitive relations (e.g., support, contradiction); we introduce a dual-dimensional quantification scheme—“credibility” (reflecting reliability of external sources) and “confidence” (measuring strength of internal structural support)—thereby decoupling these orthogonal dimensions. Contribution/Results: This framework transcends limitations of classical logic, probabilistic, and argumentation-based models by enabling rigorous representation of inconsistent and fragmented beliefs. Leveraging graph-theoretic tools—including weighted directed graphs, node-weight functions, and connectivity analysis—it supports fine-grained relational expression and provides formally grounded, static analyses of cognitive coherence, structural conflict, and representational boundaries.

Technology Category

• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Belief systems are often treated as globally consistent sets of propositions or as scalar-valued probability distributions. Such representations tend to obscure the internal structure of belief, conflate external credibility with internal coherence, and preclude the modeling of fragmented or contradictory epistemic states. This paper introduces a minimal formalism for belief systems as directed, weighted graphs. In this framework, nodes represent individual beliefs, edges encode epistemic relationships (e.g., support or contradiction), and two distinct functions assign each belief a credibility (reflecting source trust) and a confidence (derived from internal structural support). Unlike classical probabilistic models, our approach does not assume prior coherence or require belief updating. Unlike logical and argumentation-based frameworks, it supports fine-grained structural representation without committing to binary justification status or deductive closure. The model is purely static and deliberately excludes inference or revision procedures. Its aim is to provide a foundational substrate for analyzing the internal organization of belief systems, including coherence conditions, epistemic tensions, and representational limits. By distinguishing belief structure from belief strength, this formalism enables a richer classification of epistemic states than existing probabilistic, logical, or argumentation-based approaches.

Problem

• Model belief systems as directed weighted graphs
• Distinguish credibility and confidence in beliefs
• Analyze internal structure of belief systems

Innovation

• Directed weighted graphs model belief systems
• Dual functions assign credibility and confidence
• Static framework excludes inference and revision