Graph Entropy: Measuring Structural Uncertainty

 

🔢 Graph Entropy: Measuring Structural Uncertainty

Graph entropy is a way to quantify how much information is embedded in the structure of a graph. It was introduced by János Körner in the 1970s and is rooted in Shannon’s information theory.

🧠 Intuition:

• Imagine each node in your graph represents a symbol.

• Some symbols can be confused (i.e., they’re connected), others are distinguishable.

• Graph entropy measures the maximum rate at which you can transmit information through this graph without confusion.

📐 Formal Idea:

• Given a graph $GG$, entropy $H(G)H(G)$ is computed over independent sets—groups of nodes with no edges between them.

• The more independent sets a graph has, the higher its entropy—meaning more distinguishable information can be encoded.

📘 You can explore this further in or the .

🔁 Information Flow: Tracing Influence Through the Graph

Information flow refers to how data, influence, or control propagates through a graph’s edges. In your BFS traversals, this is exactly what you’re modeling:

• Nodes with high visit frequency are central to the flow.

• Nodes with low frequency are peripheral or isolated.

• The directionality of edges determines how information moves.

🔥 In Traversal Heatmaps:

• You’re visualizing information flow intensity.

• Nodes that light up across many BFS runs are hot zones—they’re critical for connectivity and influence.

• This mirrors entropy: high-flow nodes contribute more to the graph’s informational complexity.

🧬 Why It Matters for Your Schema

In your ETL graph:

• Graph entropy helps you understand the encoding capacity of your schema—how much structural variation exists.

• Information flow helps you optimize execution order—which entities are central, which are leaf nodes, and how data dependencies ripple through the graph.

Together, they give you a semantic fingerprint of your schema: not just how it’s built, but how it behaves.