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Graph Distance as Surprise: Free Energy Minimization in Knowledge Graph Reasoning

Gaganpreet Jhajj, Fuhua Lin

TL;DR

Problem: the paper seeks to extend surprise minimization from tree-structured syntax to general knowledge graphs, using the Free Energy Principle to guide KG reasoning. Approach: define geometric surprise S_geo via shortest-path distance from context and pair it with a Kolmogorov-approximation term to form a free-energy objective F(e|C) that a KG-grounding must minimize; compute distances with BFS and compress paths with Lempel-Ziv. Contributions: a principled, graph-generalizable framework that recovers tree-based results, handles cycles, and links to GNNs and model-based RL; and an actionable path to grounding, embeddings, and architectural choices. Significance: provides a scalable, theory-driven method for ranking plausible groundings in KG-enabled agents, with implications for reasoning, memory, and planning in AI systems.

Abstract

In this work, we propose that reasoning in knowledge graph (KG) networks can be guided by surprise minimization. Entities that are close in graph distance will have lower surprise than those farther apart. This connects the Free Energy Principle (FEP) from neuroscience to KG systems, where the KG serves as the agent's generative model. We formalize surprise using the shortest-path distance in directed graphs and provide a framework for KG-based agents. Graph distance appears in graph neural networks as message passing depth and in model-based reinforcement learning as world model trajectories. This work-in-progress study explores whether distance-based surprise can extend recent work showing that syntax minimizes surprise and free energy via tree structures.

Graph Distance as Surprise: Free Energy Minimization in Knowledge Graph Reasoning

TL;DR

Problem: the paper seeks to extend surprise minimization from tree-structured syntax to general knowledge graphs, using the Free Energy Principle to guide KG reasoning. Approach: define geometric surprise S_geo via shortest-path distance from context and pair it with a Kolmogorov-approximation term to form a free-energy objective F(e|C) that a KG-grounding must minimize; compute distances with BFS and compress paths with Lempel-Ziv. Contributions: a principled, graph-generalizable framework that recovers tree-based results, handles cycles, and links to GNNs and model-based RL; and an actionable path to grounding, embeddings, and architectural choices. Significance: provides a scalable, theory-driven method for ranking plausible groundings in KG-enabled agents, with implications for reasoning, memory, and planning in AI systems.

Abstract

In this work, we propose that reasoning in knowledge graph (KG) networks can be guided by surprise minimization. Entities that are close in graph distance will have lower surprise than those farther apart. This connects the Free Energy Principle (FEP) from neuroscience to KG systems, where the KG serves as the agent's generative model. We formalize surprise using the shortest-path distance in directed graphs and provide a framework for KG-based agents. Graph distance appears in graph neural networks as message passing depth and in model-based reinforcement learning as world model trajectories. This work-in-progress study explores whether distance-based surprise can extend recent work showing that syntax minimizes surprise and free energy via tree structures.

Paper Structure

This paper contains 13 sections, 3 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. From Syntax to Semantics
  3. Theoretical Justification
  4. Implications and Future Work
  5. Worked Example: Free Energy Calculations
  6. Scenario and Knowledge Graph
  7. Computing Geometric Surprise
  8. Computing Algorithmic Complexity
  9. Free Energy Results
  10. Mathematical Details
  11. Breadth-First Search Algorithm
  12. Kolmogorov Complexity Approximation
  13. Connection to Active Inference

Figures (1)

  • Figure 1: Extending surprise from trees to knowledge graphs. Following standard KG design (e.g., Wikidata), we model "Prime Minister" as a position node. Given context "Canada", leaders (Trudeau, Harper) are at distance 1, the position node at distance 2, while disconnected entities (Biden) have distance $\infty$. The successor relation demonstrates cycle handling.