How Information Theory Explains Complex Systems

Short answer: How they connect

Information theory gives us tools to count and track information. Claude Shannon showed how to measure uncertainty with entropy, and how much data a channel can carry with channel capacity.

Complex systems are groups of many parts that interact and change.

By treating interactions as flows of information, we can use Shannon s ideas to measure how organized or random a system is, how parts influence each other, and where change or learning happens. This link helps explain emergent behavior in things like ant colonies, ecosystems, and large language models.

What is information theory?

Information theory is a math way to think about messages and uncertainty. It began with Claude Shannon.

He asked: how much information is in a message? How much can a noisy channel send?

Two key ideas are:

Bit: a single yes/no choice. It s the basic unit of information.
Entropy: a measure of uncertainty or surprise. High entropy means more randomness; low entropy means more predictability.

Shannon s classic paper, A Mathematical Theory of Communication, lays this out. Modern summaries like Quanta Magazine s article and the MIT News explainer make the ideas easier to digest.

What is a complex system?

A complex system has many parts that interact. Examples include ant colonies, the weather, the stock market, and neural networks.

These systems show emergence, new behaviors that come from many simple rules. Complexity science studies these patterns.

See the Santa Fe Institute collection and a good survey at Complex Systems: A Survey.

How does information theory model complexity?

Short answer: it treats interactions as information flow. Here are the main ideas:

States and messages: Each part of a system can be in a state. Changes are like messages sent between parts.
Entropy as complexity: We can measure how much disorder or variability a system has. That helps compare systems or track change over time.
Mutual information: This measures how much one part tells you about another. High mutual information means strong coordination.
Channel capacity: Limits how much information can pass through a link. Bottlenecks shape behavior.

By using these tools, we can say things like "this subsystem carries a lot of information about the whole" or "this connection is a bottleneck for coordination." The New England Complex Systems Institute explains the difference between information and meaning in systems at NECSI s page.

Simple analogy

Think of a school hallway. Each student is a part, and notes passed are messages.

If everyone whispers, information moves slowly and is noisy. If a teacher uses a loudspeaker, messages travel fast and clearly.

Entropy tells you how mixed up the notes are. Mutual information tells you how well one student's note predicts what another student knows.

Three clear examples

1) Ant colony (biology)

Ants use pheromones and local rules to build paths and find food. Each ant has limited information.

The colony s behavior emerges from many local messages. We can measure:

Entropy of pheromone maps (how random are paths?).
Mutual information between scout ants and the nest (how much does a scout s signal change colony behavior?).

Researchers apply information theory to study how robust the colony is to noise and how it learns new routes.

2) Large language models (AI)

LLMs store patterns from text. Think of model layers as channels.

Information theory helps us ask:

How much information flows from input to final prediction?
Which layers compress or expand information (source coding theorem ideas)?
Where are the bottlenecks that limit understanding?

Recent work frames "understanding" as a statistical measure tied to Shannon s ideas. See a recent conference paper that links understanding to internal language models (ICNLP 2024).

3) Financial markets (economics)

Prices carry information about supply and demand. Entropy measures unpredictability in prices.

Mutual information can reveal how events in one market affect another. Channel capacity ideas show where trading rules and delays limit how fast information spreads.

Practical 5-question checklist to apply information theory

Use this to study any complex system. Answer each in simple terms.

What are the parts and their possible states?
What counts as a message between parts?
Where are the noisy links or bottlenecks?
What measures will you use (entropy, mutual information, transfer entropy)?
What change or prediction do you want to test?

This checklist is a compact version of the downloadable framework suggested by complexity researchers. Quick checkpoint: can you name one noisy link in a system you work with?

What tools and measures to use

Common, practical metrics:

Shannon entropy: measures uncertainty.
Mutual information: measures shared information.
Transfer entropy: directional information flow.
Compression tests: use source coding ideas to see if data are structured or random.

Many libraries compute these measures. Start small: compute entropy of a single signal, then mutual information between two signals.

Limitations and things to watch for

Information theory measures quantity, not meaning. It won t tell you the purpose of a message.
Estimates need good data. Small samples give biased numbers.
Complex systems can be nonstationary: rules change over time. That complicates measurement.

Shannon himself noted the distinction between information and meaning in designing communication systems; see the NECI discussion at NECSI and reflections on Shannon s role in complexity science at the Santa Fe Institute.

Why this matters to you

You can use these ideas to make smarter models and clearer experiments.

If you build AI, measure where information is lost. If you study ecosystems, track how signals travel. If you work in teams, think of meetings as noisy channels that need clearer protocols.

Tiny teaching tip: try a small experiment this week. Pick a short log or trace from a system, compute its entropy, and ask what a higher or lower value means for behavior.