Directed versus undirected relationships

The arrow changes everything.

Consider two statements: "The sun heats the earth" and "The earth heats the sun." Both contain the same three elements — sun, heating, earth. Both describe a thermal relationship. But they are not the same statement. One is how the solar system works. The other is nonsense. The difference is the direction of the arrow.

This is not a trivial observation. It is a structural principle that governs how you model every system you will ever encounter — from the causal chains in your decision-making to the dependency structures in your projects, from the flow of authority in your organization to the direction of influence in your relationships. Some relationships have direction. Getting the direction wrong doesn't just introduce a minor error. It inverts your model of reality.

In the previous lesson (L-0243), you learned that relationships come in types — causal, temporal, hierarchical, associative, and more. But type alone doesn't capture the full structure of a relationship. Two relationships can be the same type and point in opposite directions, and the reversal changes everything. "A manages B" and "B manages A" are both hierarchical relationships. They are not the same relationship. The arrow matters.

Directed and undirected: the fundamental distinction

Graph theory — the branch of mathematics that studies relationships as formal structures — draws a hard line between two kinds of connections.

In an undirected relationship, the connection between A and B is symmetric. If A is connected to B, then B is connected to A, and the relationship means the same thing in both directions. Think of two people shaking hands: if I shake your hand, you are also shaking mine. There is no arrow. The relationship is mutual by definition. In the mathematical notation of graph theory, an undirected edge is an unordered pair — {A, B} is the same as {B, A}.

In a directed relationship, the connection from A to B is not the same as the connection from B to A. The relationship has an arrow — it flows from a source to a target, from an origin to a destination. "A follows B on Twitter" does not imply "B follows A on Twitter." "A reports to B" does not imply "B reports to A." "A causes B" emphatically does not imply "B causes A." In mathematical notation, a directed edge is an ordered pair — (A, B) is distinct from (B, A).

This distinction, formalized by mathematicians building on Leonhard Euler's 1736 analysis of the Konigsberg bridge problem — the work that founded graph theory itself — is one of the most consequential structural choices you can make when modeling a system. Every time you draw a connection between two things, you are implicitly deciding whether that connection has a direction. If you make that decision unconsciously, you will systematically lose information about how the system actually works.

Here is a quick test for any relationship: can you swap the two elements and preserve the meaning? "Alice and Bob are siblings" — swap them, and the meaning is identical. Undirected. "Alice mentors Bob" — swap them, and the meaning inverts. Directed. "Alice and Bob are co-located" — undirected. "Alice depends on Bob's output" — directed. The swap test takes two seconds and it reveals the structural DNA of the relationship.

Why direction matters: four domains where the arrow is everything

The distinction between directed and undirected relationships is not an academic curiosity confined to graph theory textbooks. It shows up in every domain where relationships carry meaning — which is to say, every domain that matters.

Causation

Judea Pearl, the computer scientist and philosopher whose work on causal inference earned him the Turing Award in 2011, built an entire mathematical framework around the direction of causal arrows. In Pearl's structural causal models, every arrow in a causal diagram represents a claim about the direction of influence: this variable affects that one, not the reverse. An arrow from smoking to lung cancer is a causal claim. Reversing that arrow — claiming that lung cancer causes smoking — is a different claim, and a false one.

Pearl's framework makes explicit something that statistical analysis alone cannot: the difference between correlation and causation is fundamentally about direction. Two variables can be correlated without either one causing the other — they might share a common cause. But causation always has a direction. "A causes B" is a directed relationship. Remove the arrow, and you have correlation. Restore it, and you have a causal claim that can be tested, falsified, and acted upon.

The practical implications are enormous. If you believe that employee satisfaction causes productivity (satisfaction -> productivity), you invest in improving satisfaction. If you believe the arrow points the other way — that productivity causes satisfaction (productivity -> satisfaction) — you invest in removing barriers to output. Same two variables. Opposite arrows. Opposite strategies. Getting the direction wrong doesn't just waste resources; it solves the wrong problem.

Directed acyclic graphs (DAGs) — directed graphs with no cycles, meaning you can never follow the arrows in a loop back to where you started — are the backbone of modern causal reasoning. Epidemiologists use DAGs to identify confounding variables. Economists use them to design natural experiments. Data scientists use them to determine which variables to control for in an analysis. In every case, the arrows encode causal direction, and the entire analysis depends on getting those arrows right.

Information flow

Information rarely flows symmetrically. In economics, the concept of asymmetric information — formalized by George Akerlof, Michael Spence, and Joseph Stiglitz, who shared the 2001 Nobel Prize for this work — describes situations where one party in a transaction has more or better information than the other. The seller of a used car knows things about the car that the buyer doesn't. The employee knows things about their work effort that the employer can't observe. The patient knows things about their symptoms that the insurer cannot verify.

These are directed relationships. Information flows from the informed party toward (or away from) the uninformed party. Akerlof's famous "market for lemons" paper showed that this directional asymmetry doesn't just create unfairness — it can collapse entire markets. When buyers can't distinguish good used cars from bad ones, they offer a price that reflects the average quality. Sellers of good cars, unwilling to accept the average price, withdraw from the market. The only cars left are the lemons. The directional flow of information — seller knows, buyer doesn't — drives the entire pathology.

In your own life, every relationship where one person knows something the other doesn't is a directed information relationship. Your doctor knows things about medicine that you don't. Your financial advisor knows things about markets that you don't. But you know things about your body and your risk tolerance that they don't. Mapping these information arrows reveals who has epistemic power in each relationship — and where you need to close the gap.

Social networks

Mark Granovetter's landmark 1973 paper "The Strength of Weak Ties" — one of the most cited works in the history of social science, with over 78,000 citations — studied how people find jobs through their social networks. He discovered that weak ties (acquaintances) were more valuable than strong ties (close friends) for accessing new opportunities, because weak ties bridged between different social clusters while strong ties stayed within them.

But the deeper structural insight is about direction. Social networks contain both directed and undirected relationships, and mixing them up produces flawed models. Friendship on Facebook is undirected — it requires mutual acceptance. Following on Twitter is directed — no reciprocity required. Granovetter's later work, and the explosion of social network analysis that followed, revealed that the direction of ties profoundly shapes network behavior.

In a directed social network, you can have hubs — nodes with many incoming connections but few outgoing ones (celebrities, thought leaders, broadcast nodes). In an undirected network, every connection is necessarily reciprocal, so hubs look different — they are densely connected to nodes that are also densely connected to them. The topology of influence, the speed of information diffusion, the vulnerability to node removal — all of these change depending on whether the network's relationships are directed or undirected.

A 2022 large-scale experiment conducted on LinkedIn — involving 20 million people over a five-year period — confirmed and extended Granovetter's findings. Weak ties on the platform were significantly more likely to lead to new employment than strong ties. But the mechanism was directional: information about job opportunities flowed from loose acquaintances who moved in different professional circles, through directed pathways of recommendation and introduction, to the job seeker.

Dependencies and prerequisites

Every project plan is a directed graph. Task B depends on Task A means there is an arrow from A to B — A must be completed before B can begin. Reversing that arrow (claiming B must be done before A) produces a different plan, possibly an impossible one.

The curriculum you are navigating right now is a directed graph. L-0243 (Types of relationships) is a prerequisite for this lesson. This lesson enables L-0245 (Strength of relationships varies). Those arrows have direction. You can't meaningfully understand directed versus undirected relationships without first understanding that relationships come in types. And you need the concept of directionality before you can reason about relationship strength — because the strength of A's influence on B might be completely different from the strength of B's influence on A, and you need the arrow to keep track.

Software dependency graphs, academic prerequisite chains, manufacturing assembly sequences, recipe instructions — all directed. In each case, reversing an arrow doesn't just change the description; it breaks the system.

When undirected relationships are actually directed

One of the most common modeling errors is treating a directed relationship as undirected because the direction is socially invisible or psychologically uncomfortable to acknowledge.

"We have a good relationship" — this is the undirected framing. It implies symmetry: both parties experience the relationship as good, both invest equally, both benefit equally. But is that true? Map the actual flows. Who initiates contact more often? Who makes more accommodations? Who benefits more from the connection? In many cases, what looks like a mutual, undirected relationship is actually two directed relationships of different strengths running in opposite directions. A mentors B (strong arrow from A to B). B provides status and social validation to A (weaker arrow from B to A). Collapsing these two directed relationships into a single undirected "connection" loses the information about who gives what and who gets what.

"These two variables are correlated" — correlation is the undirected version of a relationship between variables. It tells you that A and B move together, but not which one drives the other. As every statistics textbook warns, correlation does not imply causation — which is precisely the statement that an undirected relationship does not imply a directed one. To get from correlation to causation, you need to establish direction. Pearl's entire research program is essentially the mathematics of adding arrows to correlations.

"We depend on each other" — mutual dependence is real, but it is almost never symmetric. A depends on B for technical expertise. B depends on A for client relationships. These are two directed dependencies, not one undirected one. Collapsing them into "we're interdependent" obscures the specific vulnerabilities that each party faces if the relationship ends.

The discipline here is to resist the comfort of symmetry. Undirected relationships are simpler to model, easier to communicate, and less politically charged. But they are also less accurate. Every time you draw an undirected line between two things, ask: is this genuinely symmetric, or am I hiding an arrow?

Feedback loops: when direction creates circles

There is one critical case where directed relationships produce a structure that looks, at first glance, like it might be undirected: the feedback loop.

In a feedback loop, A affects B and B affects A. Stress causes poor sleep. Poor sleep causes stress. The relationship appears symmetric — both arrows exist. But it is not undirected. It is two directed relationships forming a cycle. And the cycle itself has emergent properties that neither arrow has alone.

Norbert Wiener, the mathematician who founded cybernetics in 1948, defined the field as the study of "circular causal and feedback mechanisms in biological and social systems." The Macy Conferences that formalized cybernetics (1946-1953) brought together mathematicians, biologists, psychologists, and anthropologists who all recognized the same pattern: directed causal arrows, when they form loops, produce self-reinforcing or self-correcting behavior that cannot be understood by looking at any single arrow in isolation.

A positive feedback loop amplifies: confidence leads to action, action leads to success, success leads to more confidence. Each arrow is directed. The loop as a whole is a growth engine (or, running in the opposite valence, a death spiral). A negative feedback loop stabilizes: a thermostat detects high temperature, triggers cooling, which reduces temperature, which turns off the cooling. Each arrow is directed. The loop as a whole maintains equilibrium.

The lesson for relationship mapping is precise: when you find a relationship that seems to "go both ways," don't collapse it into an undirected connection. Map both directed arrows separately. You will often discover that the two arrows have different strengths (A's effect on B is stronger than B's effect on A), different speeds (A affects B immediately but B affects A only over months), or different mechanisms (A affects B through information but B affects A through emotion). These asymmetries are invisible if you model the relationship as undirected. They become visible — and actionable — when you preserve the arrows.

This connects directly to L-0252, later in this phase, where you will study feedback loops as circular relationships. The concept of direction is the prerequisite: you cannot understand circular causation until you understand that each causal link in the circle has its own arrow.

Your Third Brain: how direction structures machine reasoning

Every time you ask a large language model to reason about causes and effects, you are asking it to navigate directed relationships. And the architecture underlying modern AI is built, at its deepest level, on the mathematics of directed graphs.

Bayesian networks — one of the foundational frameworks for probabilistic AI — are defined as directed acyclic graphs where each node represents a variable and each directed edge represents a conditional dependency. The direction of the arrow encodes which variable conditions which: an arrow from "rain" to "wet grass" means that the probability of wet grass depends on whether it rained, not the reverse. Bayesian networks can perform inference — given that the grass is wet, what is the probability that it rained? — precisely because they know which way the arrows point. Without direction, the network cannot distinguish cause from effect, and inference collapses into ambiguity.

Neural network computation itself flows in a directed manner. In a feedforward neural network, signals propagate from input layer to hidden layers to output layer — always forward, never backward. The computation is a directed acyclic graph. During training, gradients flow backward through the network (backpropagation), but this is a separate directed process running in the opposite direction. The entire learning algorithm depends on maintaining clear directional flow in both the forward pass and the backward pass.

When you use AI tools as part of your cognitive infrastructure — your "Third Brain" — understanding direction helps you collaborate with these systems more effectively. If you ask an AI to analyze a set of relationships, specify the direction. Don't say "these two things are related." Say "this thing causes that thing" or "this thing depends on that thing" or "information flows from here to there." The more directional precision you provide, the more accurately the AI can reason about the system you're describing. Directed inputs produce directed outputs. Undirected inputs produce ambiguous ones.

Knowledge graphs — the structured databases that power search engines, recommendation systems, and AI assistants — are fundamentally directed. Google's Knowledge Graph, Wikidata, and similar systems represent facts as directed triples: (subject, predicate, object). "Einstein" -> "born in" -> "Ulm." "Einstein" -> "developed" -> "General Relativity." Every fact has a subject and an object, connected by a directed predicate. The direction is not optional. "Ulm was born in Einstein" is not a fact. The arrow is load-bearing.

Protocol: the direction audit

Here is the operational practice for integrating directionality into your relationship mapping. Use this whenever you are building a new map or reviewing an existing one.

1. Start with assumed direction. For every relationship you identify, draw an arrow rather than a line. Force yourself to specify: which element is the source and which is the target? Who acts on whom? What flows from where to where? If you cannot determine the direction, write a question mark on the arrow — but draw the arrow anyway.

2. Apply the swap test. For each arrow, mentally reverse it. Does the meaning change? If "A manages B" reversed to "B manages A" changes the meaning, the direction is real and must be preserved. If "A collaborates with B" reversed to "B collaborates with A" preserves the meaning, the relationship is genuinely undirected and can be drawn as a line.

3. Look for hidden asymmetries. For every relationship you marked as undirected, ask three questions: Is the investment symmetric? (Do both parties put in equal effort?) Is the benefit symmetric? (Do both parties gain equally?) Is the power symmetric? (Could either party end the relationship with equal consequences?) If any answer is no, consider mapping it as two directed arrows of different strengths.

4. Check for feedback loops. Where you find A -> B and B -> A, do not collapse them into A -- B. Keep both arrows. Note the strength, speed, and mechanism of each direction separately. These dual-arrow structures are feedback loops, and they will behave very differently from undirected connections.

5. Validate with evidence. For your most important directed relationships — especially causal claims — ask: what evidence supports this direction? Could the arrow actually point the other way? Could both variables be driven by a third cause? Directed relationships are claims about reality. They deserve the same scrutiny you would give any other claim.

6. Document your direction choices. When you finalize your map, annotate any relationship where the direction was ambiguous or contested. Future-you will want to know which arrows were established with confidence and which were provisional. This is the beginning of treating your relationship maps as living documents that evolve as your understanding deepens.

The bridge to strength

You now understand that relationships have types (L-0243) and that they have direction. But there is a third structural property that you have been encountering implicitly throughout this lesson and that deserves its own treatment: strength.

When we noted that the two arrows in a feedback loop might have different strengths — A's effect on B being stronger than B's effect on A — we were acknowledging that not all connections carry equal weight. When Granovetter distinguished weak ties from strong ties, he was measuring relationship strength. When you intuitively know that some dependencies are critical and others are nice-to-have, you are sensing strength differences.

Direction tells you which way the arrow points. Strength tells you how much flows along it. A weak causal arrow and a strong causal arrow point in the same direction but have radically different implications for how the system behaves.

That is where you go next. L-0245 — Strength of relationships varies — will give you the tools to quantify what you have so far only described structurally. Not all connections are equally strong, and the ability to distinguish a load-bearing relationship from a decorative one is what separates a useful map from a merely accurate one.

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Sources

• Pearl, Judea. Causality: Models, Reasoning and Inference. 2nd ed. Cambridge University Press, 2009.
• Granovetter, Mark S. "The Strength of Weak Ties." American Journal of Sociology 78, no. 6 (1973): 1360-1380.
• Akerlof, George A. "The Market for 'Lemons': Quality Uncertainty and the Market Mechanism." Quarterly Journal of Economics 84, no. 3 (1970): 488-500.
• Wiener, Norbert. Cybernetics: Or Control and Communication in the Animal and the Machine. MIT Press, 1948.
• Euler, Leonhard. "Solutio problematis ad geometriam situs pertinentis." Commentarii academiae scientiarum Petropolitanae 8 (1741): 128-140.
• Barabasi, Albert-Laszlo. Network Science. Cambridge University Press, 2016.
• Rajpurkar, Pranav et al. "Large-Scale Experiment on LinkedIn Confirms Granovetter's Weak Ties Theory." Science 377, no. 6612 (2022): 1304-1310.