What Is a Diagram For?

All six diagram typesPart I · Framing · Chapter 1 of 12 · the opening chapter

The spring newsletter of a market town’s traders opened with this paragraph:

Since the council raised parking charges, fewer people come into town. The council’s own footfall counter shows visitors down by a fifth over two years, and our own figures show takings per shop falling with them, month by month. Seven shops have closed, and every owner gave falling takings as the reason. Every empty shop gives people one less reason to come, so fewer still come. Some blame online shopping, but people shopped online before the charges went up.

A variable is a quantity that can rise or fall, named as a noun phrase: visitors to the high street, takings per shop. An action is not a variable: shoppers stay away names something that happens, and visitors to the high street names what rises or falls when it does. The test for a name is in chapter 4 (chapter 4, Variables).

An effect is a line with an arrowhead from one variable to another: a change in the first changes the second. Takings per shop → empty shops: as takings fall, shops close.

Every figure in the chapter is the start of a causal loop diagram, whose effects gain a polarity, + or −, for which way the change runs: by the upgrade (chapter 3, Causal claims) in chapter 3, where a direction has been measured, and on every effect in chapter 4 (chapter 4, Polarity).

Evidence coding shows the author’s judgment of how well supported each effect is as a causal mechanism: in a diagram with a dashed effect, a solid effect is one the author judges more supported, and a dashed one less supported, a hypothesis. A dashed line means nothing else in this book.

Support comes in kinds, some stronger than others: a comparison of places where the first variable changed with places where it didn’t; a count of the second changing alongside the first; an explanation of how the first changes the second; testimony; a hunch. Where the line between solid and dashed falls is the author’s decision: one author draws solid only what a comparison supports, another anything better than a hunch, and a reader can question either.

The traders draw theirs at a count. They counted takings falling with visitors, month by month, and draw visitors → takings solid; the owners’ reasons for closing are testimony, and takings → empty shops is dashed, though a trader who took the owners at their word could draw it solid.

A variable counted alone supports no effect. The footfall counter measures visitors to the high street, down by a fifth, but nobody counted visitors alongside a change in the charges, in online shopping or in empty shops, so the traders dash those three effects too.

A diagram with no dashed effect carries no coding: its solid effects are its author’s claims, their support unrecorded.

The legend is a box listing the codings a diagram uses; any diagram with a dashed effect must have one. Its evidence entry shows a solid line for more supported and a dashed line for less supported. It also lists any viewpoint tags (chapter 5, Viewpoint tags), chapter 5’s small boxes naming who stresses an effect.

The paragraph runs from the charges to the shops; the drawing shows four things its order hides:

• A circle. Fewer still come leads back to the first variable: visitors → takings → empty shops → visitors is a closed path, in chapter 4 a loop (chapter 4, Loops).
• Two causes at one variable. Parking charges and online shopping both end at visitors to the high street. The traders blame one and dismiss the other, on the same evidence: none.
• Causes with nothing into them. Nothing drawn changes the charges or online shopping.
• Where the traders draw the line. Of the circle’s three effects, only visitors → takings is solid.

In the paragraph, what bears on visitors is spread over three sentences; in the drawing, it is the three lines ending at visitors. A diagram puts at one place what is used together, and shows what a paragraph leaves to be worked out (Larkin and Simon, 1987).

Of each effect, solid or dashed, ask what changes; whether a change in the first changes the second, or only comes before it; and how anyone knows. Of the coding, ask where the author drew the line between solid and dashed, and whether you would draw it there: a solid effect is its author’s judgment, not proof that one change causes the other. Of the whole, ask what is missing: a cause nobody drew, or an effect that would close another circle.

The lab’s checks test structure, never whether an effect holds: a check counting dashed effects passes a diagram with the wrong ones dashed.

What’s wrong with this diagram, from a youth club’s bid to the council for another year’s funding?

Every effect is solid, and three rest on counts: the club’s sign-in sheets count attendance by evening, the council publishes its youth grant each year beside complaints to the police in the club’s area, and the club’s accounts show how many evenings the grant pays for. That attending keeps young people off the streets at night, that fewer on the streets means fewer complaints, and that more volunteers would open more evenings are the club’s beliefs. The club draws no line between counts and beliefs, so a reader can’t tell the sign-in sheets from a belief about the streets at night.

A count of young people on the streets near the club, on nights it opens and nights it doesn’t, would give the club grounds to draw young people attending → young people on the streets at night solid.

Which type to draw for which question is in chapter 12 (chapter 12, What each type answers). You draw each exercise as one type in the lab, with only the marks you need for it.

One claim, or several that don’t meet, go in a sentence: the charges keep shoppers away. Causes with no effects between them go in a list. The chapter’s figures show no direction, amount or timing; each is drawn from a later chapter on. Each type’s case for a sentence instead is in chapter 12 (chapter 12, When not to draw).

Read one way, a solid effect is proven everywhere. Its support was gathered somewhere, under some conditions: induced traffic, solid on chapter 5’s road (chapter 5, Evidence coding), was found across American cities (Duranton and Turner, 2011), and a town of another shape may differ.

Read the other way, a dashed effect is false, or minor. It is less supported, not less important: the high street’s dashed empty shops → visitors closes the circle.

Evidence comes in degrees, and the coding has two: an effect counted in one town and one counted in twenty are drawn with the same solid line. The IPCC’s authors grade evidence as limited, medium or robust, agreement as low, medium or high, and an outcome’s likelihood in seven steps, from virtually certain to exceptionally unlikely (Mastrandrea and others, 2010).

With the coding, each effect carries one answer, which side of its author’s line it falls on, readable at a glance. Where the line falls, and the degrees on either side of it, go in the sources and the prose beside the drawing, where the IPCC’s authors put theirs: a traceable account of each finding’s evidence, in the text.

Passing the checks means meeting the brief, not being right.

• Jill H. Larkin and Herbert A. Simon, “Why a Diagram Is (Sometimes) Worth Ten Thousand Words” doi.org/10.1111/j.1551-6708.1987.tb00863.x, Cognitive Science 11, no. 1 (1987): 65–99, on diagrams that hold at one location what an inference uses, and make explicit what sentences leave implicit.
• Michael D. Mastrandrea and others, Guidance Note for Lead Authors of the IPCC Fifth Assessment Report on Consistent Treatment of Uncertainties www.ipcc.ch/site/assets/uploads/2017/08/AR5_Uncertainty_Guidance_Note.pdf (IPCC, 2010), on grading evidence, agreement and likelihood in words, with a traceable account in the text.
• Gilles Duranton and Matthew A. Turner, “The Fundamental Law of Road Congestion: Evidence from US Cities” doi.org/10.1257/aer.101.6.2616, American Economic Review 101, no. 6 (2011), measuring how driving rises with new lanes.
• David Wembridge, Grace Johnson, Nida Al-Fulaij and Steve Langton, The State of Britain’s Hedgehogs 2022 www.hedgehogstreet.org/wp-content/uploads/2022/02/SoBH-2022-Final.pdf (People’s Trust for Endangered Species and British Hedgehog Preservation Society, 2022), for DF 03.
• ESA Space Debris Office, ESA’s Annual Space Environment Report www.sdo.esoc.esa.int/publications/Space_Environment_Report_I10R0_20260501.pdf, issue 10.0 (2026), for DF 04.
• NASA Orbital Debris Program Office, “Satellite Collision Leaves Significant Debris Clouds” orbitaldebris.jsc.nasa.gov/quarterly-news/pdfs/odqnv13i2.pdf, Orbital Debris Quarterly News 13, no. 2 (2009), for DF 04.
• Donald J. Kessler and Burton G. Cour-Palais, “Collision Frequency of Artificial Satellites: The Creation of a Debris Belt” doi.org/10.1029/JA083iA06p02637, Journal of Geophysical Research 83, no. A6 (1978), for DF 04.

Drawing the Boundary

System context mapsPart I · Framing · Chapter 2 of 12 · the one chapter on system context maps

• People disagree about whose job something is.
• Something arrives that nobody expected to deal with: a bill, a request, a complaint.
• A new job, project or household starts, and nobody has listed who it deals with.
• A party outside is spoken of as if it were run from inside.
• A problem passes from one part to another, each saying it belongs elsewhere.

Priya and Sam rent a house with their ten-year-old daughter. Sam’s nephew Dev rents their spare room, furnished, and buys and cooks his own food. Sam shops and cooks for the family; Priya pays the rent and the bills.

Each month Priya’s wages arrive from her employer, and the household pays the landlord rent, the supermarket for groceries and the energy supplier for electricity. Dev pays the household rent for his room.

In January the boiler fails. Dev asks Sam when his room will be warm again; Sam phones the landlord, who sends a plumber three days later. The household owes Dev a warm room, and depends on the landlord to provide one.

A boundary is a thin, solid, rounded outline drawn around the system, with a tab carrying the system’s name. It marks what the map explains. What sits outside, the map doesn’t explain: it shows what the landlord gives the household and receives from it, not how the landlord runs a business.

Every boundary has a name, since drawing one means saying which system it bounds: here, the household. A boundary is never dashed: in this book a dashed line means a hypothesis.

An external entity is a party outside the boundary that the system exchanges something with: a person, an organization or, at the scale of a planet, the Sun. It’s drawn as its name in an oval. A party appears because something crosses between it and the system; one with nothing crossing is left off.

The system runs what is inside its boundary and only exchanges with what is outside. Dev lives in the house, but he pays for his room and runs his own meals, so he’s an external entity; the daughter, whose room and meals the household provides, is inside.

An external entity isn’t a variable: nothing on a system context map rises or falls.

An exchange is one named thing crossing the boundary between the system and an external entity: wages, rent, electricity. It’s drawn as a solid line with one arrowhead, joining an oval to the boundary, with its name beside its middle. One exchange carries one thing: rent and repairs is two exchanges. A line without a name shows that something crosses, not what.

An exchange’s direction runs from the party that gives to the party that receives. The household gives the landlord rent and receives repairs: two exchanges, pointing opposite ways.

The exchanges leaving the boundary are what the system answers for to others, such as Dev’s furnished room; those arriving are what it depends on, such as the landlord’s repairs.

A nested boundary is a boundary inside another, naming a part of the system. Nesting is read from position: a boundary drawn wholly inside another is its part. Two boundaries either nest or stay apart; overlapping ones leave it unclear which part a thing belongs to.

A part is named for what it does, not for who does it: meals, not Sam. If Priya takes over the cooking, the part stays.

An exchange’s owner is the boundary it reaches. With parts drawn, each exchange reaches the part that handles it: groceries and grocery payments reach meals, wages reach earnings, and rent, repairs, electricity and bill payments reach housing. When the boiler fails, chasing the landlord’s repairs falls to housing. An exchange drawn to the outer boundary belongs to the system as a whole.

With one or two parties, a sentence does what the map would: the household pays the landlord rent and gets repairs. When a reader needs only who the system deals with, not what crosses, as on a list of emergency phone numbers, a list of parties does it.

The map shows no amounts and no timing: not how much the rent is, nor when it’s due. Nor does it show causes, such as why the electricity bill rose; influence maps and causal loop diagrams, from chapter 3, show those. What passes between parts, such as earnings paying for meals, stays in a sentence.

What’s wrong with this map of a volunteer-run festival?

Its boundary is the organizing committee, with nested boundaries for the chair, the treasurer and the volunteer coordinator: an organization chart, showing who sits on the committee, not what the festival runs. Its five exchanges have no names and each runs one way only, so the map shows that the town council, the sponsor, bands, food stallholders and visitors deal with someone, not what they give or receive. The lab’s check reports:

5 of 5 exchanges have no name yet: from the town council to the chair; from the treasurer to bands; from food stallholders to the treasurer; from visitors to the volunteer coordinator; from the sponsor to the chair. Name the one thing each carries.

The lab checks names, not what a boundary is drawn around: the committee’s map with every exchange named would pass.

Corrected, the boundary is the festival, with parts for what it runs: the stage, the food court and the gate. The same five parties exchange named things both ways, each with the part that handles it: bands give the stage performances and receive fees; visitors give the gate entry fees and receive wristbands. The council’s event permit and the sponsor’s money reach the festival as a whole.

A new treasurer leaves the corrected map unchanged; a second stage would add a part.

Passing the checks means meeting the brief, not being right.

• Tom DeMarco, Structured Analysis and System Specification (Yourdon Press, 1978), on structured analysis’s top-level diagram: the whole system drawn as one process among the parties outside it.
• Edward Yourdon, Modern Structured Analysis (Yourdon Press, 1989), on the environmental model, the part of structured analysis that draws a system’s surroundings.
• Simon Brown, The C4 model: System context diagram c4model.com/diagrams/system-context, on the C4 model’s top level: a software system drawn as one box among the people and other systems it deals with.
• Donella H. Meadows, Thinking in Systems: A Primer (Chelsea Green, 2008), chapter 4, on boundaries as choices made afresh for each problem.
• John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World (Irwin/McGraw-Hill, 2000), chapter 3, on the model boundary chart, which lists what a model treats as inside, as outside, and leaves out.
• Rebecca Lindsey, “Climate and Earth’s Energy Budget” science.nasa.gov/earth/earth-observatory/climate-and-earths-energy-budget (NASA Earth Observatory, 2009), on the sunlight Earth receives and the reflected sunlight and heat it sends to space: SC 02’s situation.

Surfacing Influences

Influence mapsPart I · Framing · Chapter 3 of 12 · the one chapter on influence maps

• Many things bear on one outcome, and nobody knows which way most of them act.
• A list of causes keeps growing, with no order to it.
• People argue over which way something pushes before anyone has counted it.
• Some effects have been measured and others only suspected, and a causal loop diagram would show them alike.

Nia is ten, and her parents would like her to read more for pleasure. Asked what bears on the time she spends reading, they list how well she reads, the length of the books she picks, how much she enjoys reading, the books at home, their visits to the library, her screen time, her after-school clubs, and the evenings one of them reads with her.

Her school’s reading log holds the minutes she reads each day and each book she finishes; her teacher’s assessments give her reading level each term. Over a year the two measured three effects: in terms when she logged more minutes, her level rose further; as her level rose, she picked longer books; and with a longer book on the go, she logged more minutes a day. Nothing records the rest of the list.

Her parents rank her enjoyment highest, her screen time and the length of her books next, and the books at home and her clubs lowest.

The central outcome is the one variable the map is drawn around: here, time Nia spends reading. Its name is double-underlined. A map has exactly one; a second outcome gets a map of its own.

Like every variable, it is a quantity that can rise or fall: time Nia spends reading, not Nia reads.

An influence claim says that one variable bears on another, without saying which way. More screen time might mean less reading or, with books on a tablet, more. It is drawn as a solid line with an arrowhead and no + or −, from the variable that bears to the one it bears on.

Two variables that bear on each other take two claims, one each way.

A direct influence is a claim ending at the central outcome: screen time → time Nia spends reading. An indirect influence ends at another variable and reaches the outcome through it: library visits per month reaches time Nia spends reading through books at home.

From every variable, a path of claims runs to the central outcome. A variable with no path there either gets the claim it lacks or comes off the map.

A cluster holds two or more variables that share something, under a label naming what they share: books to hand holds books at home and library visits per month. It is drawn as four corner brackets, the label inside the top-left one. A variable belongs to the cluster whose brackets it sits within.

Clusters neither nest nor overlap, and the central outcome sits in none. With no continuous outline, a cluster can’t be read as a boundary (chapter 2, The boundary), which marks what a model explains.

Weight is an influence’s relative importance, where influences are ranked: one, two or three dots beside the middle of the claim’s line. Nia’s parents rank her enjoyment highest, so its claim carries three dots, and her clubs’ claim one.

Weight says how much, not which way or how sure. Every line is drawn alike: a thicker one means nothing.

A causal claim says which way a change runs, by its polarity: + when the two variables move in the same direction, − when they move in opposite directions. Giving an influence claim a polarity is an upgrade; the claim keeps its ends and its weight. Taking the polarity off makes it an influence claim again.

Upgrade a claim where you can say how you know its direction. The reading log and the assessments measured three effects, each +, so those three claims are upgraded. Enjoyment, weighted highest, stays an influence claim: nothing measured it.

A conversion carries an influence map into a new causal loop diagram. The variables and the causal claims carry over; the influence claims, weights, clusters and the central outcome’s double underline are left behind. An influence claim isn’t carried over without a polarity: in a causal loop diagram, an effect without one is unfinished.

Nia’s three causal claims close a path: more minutes, a higher level, longer books, more minutes. Declared in the causal loop diagram, the path is a loop. It has no − effects, so by the parity rule (chapter 4, The parity rule) it is a reinforcing loop: a change in any of the three comes back around in the same direction.

With a few influences, each acting on the outcome and none on another, a list of variables does what the map would. With one influence in question, a sentence does: Nia’s parents think screen time cuts into her reading.

The map shows no amounts and no timing: not how many minutes screen time takes, nor when.

What’s wrong with this map of turnout in a town’s local election?

Every claim carries a + or −. The council’s ward records measured only two effects: more postal votes issued, higher turnout; a longer distance to a polling station, lower turnout. The other five polarities, on rainfall, candidates, news stories, trust in the council and newcomers, are guesses drawn as findings. Converted, the map would carry all seven into a causal loop diagram.

The lab’s checks test structure, not evidence: they report this map only under a brief requiring at least one influence claim.

Corrected, the two measured claims keep their polarities and the other five are influence claims: the same variables and the same claims, with five fewer polarities.

A count of turnout on wet and dry polling days, ward by ward, would let a third claim be upgraded.

Passing the checks means meeting the brief, not being right.

• The Open University, Diagramming for development 2: exploring interrelationships www.open.edu/openlearn/digital-computing/diagramming-development-2-exploring-interrelationships/content-section-3.1 (OpenLearn), §3.1, Influence diagram, on influence diagrams: a snapshot of what can influence a situation, two influences that run both ways drawn as two lines, and stronger influences drawn thicker.
• Geoff Coyle, Qualitative Modelling in System Dynamics, or What Are the Wise Limits of Quantification? proceedings.systemdynamics.org/1999/PAPERS/KEYNOTE1.PDF, keynote to the System Dynamics Society’s conference (Wellington, 1999), on influence diagrams drawn and studied without simulation, and what they are used for: a problem on one page, loops found in the diagram, and a basis for a quantified model.
• John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World (Irwin/McGraw-Hill, 2000), chapter 5, on polarity in causal loop diagrams, with §5.2.1, Causation versus Correlation.
• George P. Richardson, “Problems with Causal-Loop Diagrams” doi.org/10.1002/sdr.4260020207, System Dynamics Review 2, no. 2 (1986), on what polarity does and doesn’t mean.

Loops That Amplify, Loops That Correct

Causal loopsPart II · Feedback · Chapter 4 of 12 · the first of two chapters on causal loops

• A change seems to feed itself.
• Something keeps drifting back to where it was, however hard anyone pushes on it.
• A fix worked for a while, then stopped, or began working against you.
• An explanation’s becauses run in a circle.
• People disagree about what causes what, and arguing hasn’t settled it.

A deadline costs you two nights’ sleep. Foggy on Wednesday afternoon, you have a coffee at three, finish the job, and lie awake until one. On Thursday, foggier, you have a double at two. By Friday the caffeine has you sleeping five hours a night.

At the weekend, noticing how tired you are, you go to bed at nine two nights running, and by midweek you’re yourself again.

One change fed itself: less sleep, more coffee, less sleep. The other corrected itself: the further behind you fell, the harder you caught up, until there was nothing left to catch up on.

Every name on the diagram is a variable: a quantity that can rise or fall, such as sleep debt or hours slept.

That rules out actions and events. Test a name by putting more in front of it: more afternoon caffeine makes sense; more drink coffee doesn’t. Naming a variable as an action is the bad diagram’s mistake.

An effect means that a change in one variable causes a change in another; its polarity shows which way. A + effect means they move in the same direction: more sleep debt, more afternoon caffeine. A − effect means they move in opposite directions: more afternoon caffeine, fewer hours slept.

+ doesn’t mean good and − doesn’t mean bad, and neither gives size: caffeine might cost ten minutes of sleep or two hours.

In a causal loop diagram every effect carries a polarity; an influence claim (chapter 3, Influence claims), drawn in an influence map, carries none until it is upgraded to a causal claim.

Sleep debt raises afternoon caffeine, caffeine cuts hours slept, and fewer hours slept raise sleep debt. A closed path of effects like this is a loop. A chain shows how one thing affects another; a loop shows how a thing ends up affecting itself.

A reinforcing loop makes a change snowball, in whichever direction it starts: debt brings coffee and more debt, but a good night likewise brings less coffee and another good night. A balancing loop makes a change settle, moving a variable toward a target and opposing any push away from it: the further behind on sleep you fall, the harder you try to catch up.

Each loop’s label, drawn inside it with a curved mark showing which way it circulates, is its kind and a number: R1, R2 for reinforcing loops, B1, B2 for balancing ones. Its name describes what it does, not what it’s about: spiral, not coffee.

The parity rule: a loop with an even number of − effects is reinforcing, and one with an odd number is balancing. Zero counts as even.

Each − effect reverses a change passing through it. Two reversals cancel, so the change comes back pointing the way it set out, and grows; one or three leave it pointing against itself, and it shrinks.

For the spiral, the lab’s check gives this working:

R1 (spiral) has 2 − effects (afternoon caffeine → hours slept, hours slept → sleep debt): an even number, so it is reinforcing, as declared.

At planetary scale, less Arctic sea ice reflects less sunlight, so the ocean warms, and warmer water melts more ice: again two − effects, so reinforcing.

A balancing loop closes a gap: the difference between the value someone wants and the value they have. Here it’s sleep debt, the sleep you need minus the sleep you get.

Draw it as its own variable, fed by the target from outside the loop. That shows the loop’s target, and raises two questions about any balancing loop: who set it, and could it move?

In a feedback control diagram, the words target, gap and measured value (the reading the target is compared with) are written above the names of the variables in those parts, and a person chooses the target: chapter 8’s target and gap (chapter 8, Target and gap).

A delay, two short strokes across an effect, means that the change arrives later than the cause: late enough to change what the loop does. Sleep debt builds for days before anyone acts on it, so the effect from sleep debt to early nights carries a delay, and the spiral, whose changes arrive within a day, gets a head start.

In a balancing loop with a delay, the correction continues after the gap closes; the variable overshoots its target and oscillates instead of settling.

In a stock & flow diagram, a delay on a flow holds material in transit and a delay on a signal holds a reading of the material: chapter 7’s material and signal delays (chapter 7, Material and signal delays).

A behavior note names, on the diagram, the behavior you expect the structure to produce: oscillation, overshoot, decline, then recovery. It makes the diagram a prediction: if what happens differs, the structure you drew is missing something.

Loops that share a variable are coupled: a change in one loop changes the other. In the full sleep diagram, the spiral and catching up share sleep debt, hours slept and the effect between them.

Which loop dominates depends on which is stronger at the time: early in the week, the spiral; once the debt is felt, catching up.

A leverage point is a variable proposed as the place to intervene, drawn as a box with a question mark on each side. In the sleep diagram it’s afternoon caffeine: cutting it weakens the spiral and leaves the slow balancing loop less to correct.

It’s a hypothesis, not a proven fix: someone else might propose early nights, arguing that noticing the debt sooner would help more. Either proposal can be tested against the diagram.

A causal loop diagram shows structure only, not size: when amounts decide the question you need numbers, which chapter 6’s stocks begin to supply.

Nor does it show sequence: for the order in which things happen, use a numbered list. A loop diagram of a sequence is the bad diagram’s mistake.

Nor everything: the sleep diagram leaves out stress, exercise and the deadline, keeping only the loops that explain the behavior in question.

When nothing feeds back, a sentence does the job.

What’s wrong with this diagram of keeping a household pantry stocked?

Its structure is sound: a closed loop, one − effect, labeled B1, a delay on the right effect. But its names, stock the shelves, order groceries and receive the delivery, are actions done in order and fail the more test, so the diagram is a to-do list bent into a circle, and its polarities mean nothing: what would more receive the delivery cause?

In the lab it passes every check, with a notice on each of its three names.

The correction keeps every effect, every polarity, the delay and the loop, and renames the three variables as quantities.

Now each effect is a checkable claim: more food in the pantry, fewer groceries ordered (−); more ordered, more delivered a few days later (+, delayed); more delivered, more food in the pantry (+). One − effect, so the loop is balancing: it holds the pantry near full. With the delay: reorder before the first delivery arrives and the pantry overfills.

Passing the checks means meeting the brief, not being right.

• Donella H. Meadows, Thinking in Systems: A Primer (Chelsea Green, 2008), chapter 1, on reinforcing and balancing loops, and chapter 2, on delays.
• Donella H. Meadows, Leverage Points: Places to Intervene in a System donellameadows.org/archives/leverage-points-places-to-intervene-in-a-system/ (Sustainability Institute, 1999), on leverage points.
• John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World (Irwin/McGraw-Hill, 2000), chapter 5, on causal loop diagrams: polarity, a loop’s kind, delays, and naming variables as nouns rather than actions.
• John D. Sterman, “System Dynamics Modeling: Tools for Learning in a Complex World” doi.org/10.2307/41166098, California Management Review 43, no. 4 (2001), a short introduction to the same ideas.
• George P. Richardson, “Problems with Causal-Loop Diagrams” doi.org/10.1002/sdr.4260020207, System Dynamics Review 2, no. 2 (1986), on what polarity does and doesn’t mean.
• National Snow and Ice Data Center, Why sea ice matters nsidc.org/learn/parts-cryosphere/sea-ice/why-sea-ice-matters, on why melting ice speeds further melting.

Delays, External Drivers, and Competing Loops

Causal loopsPart II · Feedback · Chapter 5 of 12 · the second of two chapters on causal loops

A town’s main road jams every morning, so the council widens it from two lanes to four. For a year the drive across town takes twelve minutes instead of twenty-five.

Then the road fills. People who had taken the bus, driven at other hours or stayed home drive at rush hour. Developers build estates on the outskirts, sold on the short drive into town, and their residents drive the main road too. Meanwhile the region’s population grows, as it did before the widening. By the fifth year the drive takes twenty-five minutes again, with twice as many cars, and the council proposes six lanes.

A boundary (chapter 2, The boundary), a named outline with a tab, marks what the model explains. The road diagram’s boundary is the town’s road network, around the lanes, travel time, car trips and homes on the outskirts.

An external driver is a variable outside every boundary that affects something inside and isn’t affected back. Its name is drawn in angle brackets: ‹regional population›. People move to the region for its jobs and housing, and nothing in the town’s road network changes how many come, so regional population sits outside the boundary with one effect leaving it, into car trips.

An external entity (chapter 2, External entities) also sits outside a boundary, but it is a party, not a quantity, and exchanges run to it as well as from it; it is drawn only in system context maps, which have no variables, so the two never share a diagram.

A nonlinear effect is weak over part of its range and sudden over another, or strong, then flat. It’s drawn with a small S beside the effect’s middle.

On a half-empty road, more car trips barely change travel time; near the road’s capacity, each extra car adds minutes. The point where an effect turns from weak to sudden is its threshold: here, the road’s capacity. Widening raised the threshold, so for a while more trips cost almost no time, and that drew the trips back.

In a diagram with a dashed effect, solid marks an effect its author judges more supported, and dashed marks one less supported, a hypothesis; a diagram with no dashed effect carries no coding (chapter 1 (chapter 1, Evidence coding)). In the road diagram, travel time → car trips is solid: induced traffic has been counted across American cities. Travel time → homes on the outskirts is dashed: the estates may have come for the short drive or for cheap land.

A viewpoint tag is a small labeled box beside an effect, naming a stakeholder who stresses that effect. The tag marks emphasis, not a different structure.

Commuters stress lanes on the main road → travel time, the drive the widening shortened. Residents along the main road stress homes on the outskirts → car trips, the traffic past their doors. The council stresses travel time → homes on the outskirts, the new estates and their taxes.

The legend is a box listing a diagram’s codings; any diagram with a dashed effect or a viewpoint tag must have one. Its evidence entry shows a solid line for more supported and a dashed line for less supported. Its viewpoints entry lists every viewpoint tag the diagram uses.

Loop dominance is which loop drives behavior at a given time. It is a reading of the structure, not a mark; the road diagram’s behavior note names which loop leads first and which later.

In the first year B1, relief by widening, dominates: four lanes cut travel time. B2, induced traffic, changes travel time within months, as shorter drives draw trips back. B3, building on the outskirts, changes it over years, through the delay on new homes. By the fifth year B2 and B3 dominate.

All three loops are balancing, toward different levels: B1 moves travel time toward what the council will accept, B2 and B3 toward what motorists will put up with. A fix that a system’s own loops pull back is policy resistance. Six lanes would set off B2 and B3 again.

What’s wrong with this diagram of the widened road?

Regional population, marked external, sits inside the boundary with an effect running into it from homes on the outskirts, since new homes house new people, and B3 runs through it. The lab’s check reports:

An external driver affects the system from outside and is not affected back, but 1 effect runs into one: homes on the outskirts → regional population. Remove that effect, or bring the variable inside the system by unmarking it as an external driver.

Corrected, homes on the outskirts run straight into car trips, and regional population sits outside the boundary, with its one effect leaving it.

B3 now closes inside the town’s road network. If the estates do draw people to the region, regional population belongs inside the boundary, unmarked.

Read one way, policy resistance means no fix works. The road diagram shows which loops pull widening back, and so where a fix could act on them: a charge for driving at rush hour weakens B2, so shorter drives draw fewer trips back.

Read the other way, a causal loop diagram is a set of separate effects, each judged alone. Lanes on the main road → travel time is true, and alone it means a shorter drive; the drive lengthens again through the loops it sits in.

A causal loop diagram doesn’t show accumulation. Homes on the outskirts pile up while drives are short and stay when drives lengthen, but a − effect from travel time can’t show that, so a reader can’t work out from the diagram which loop dominates when (Richardson, 1986). In experiments, business school students misjudge how the water in a bathtub changes as it flows in and drains out (Booth Sweeney and Sterman, 2000).

A behavior note is a prediction, not a derivation; stock & flow diagrams, from chapter 6, show accumulation and give numbers. Without them, the loop diagram still shows which loops exist and which way each moves travel time: widening sets off two loops against it.

Passing the checks means meeting the brief, not being right. The answer to CL 08, a parent who finishes the homework, has the structure of shifting the burden (chapter 10, Shifting the burden), an archetype in chapter 10.

• John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World (Irwin/McGraw-Hill, 2000), chapter 1, on policy resistance, and chapter 5, on causal loop diagrams of road building and traffic.
• John D. Sterman, “System Dynamics Modeling: Tools for Learning in a Complex World” doi.org/10.2307/41166098, California Management Review 43, no. 4 (2001), on policy resistance, with road building among its examples.
• Gilles Duranton and Matthew A. Turner, “The Fundamental Law of Road Congestion: Evidence from US Cities” doi.org/10.1257/aer.101.6.2616, American Economic Review 101, no. 6 (2011), measuring how driving rises with new lanes.
• George P. Richardson, “Problems with Causal-Loop Diagrams” doi.org/10.1002/sdr.4260020207, System Dynamics Review 2, no. 2 (1986), on what loop diagrams can’t show about accumulation.
• Linda Booth Sweeney and John D. Sterman, “Bathtub Dynamics: Initial Results of a Systems Thinking Inventory” doi.org/10.1002/sdr.198, System Dynamics Review 16, no. 4 (2000), on how people misjudge accumulation.
• Donella H. Meadows, Thinking in Systems: A Primer (Chelsea Green, 2008), chapter 2, on shifting dominance; chapter 4, on nonlinearities; chapter 5, on policy resistance.

What Accumulates

Stock & flow diagramsPart III · Accumulation · Chapter 6 of 12 · the first of two chapters on stock & flow diagrams

• Something builds up or runs down: savings, a backlog, a population.
• A cut in what comes in hasn’t stopped the total rising.
• Something leaves one place and has to turn up in another.
• Amounts have units, and the units have to agree.

With the plug in and the tap on, the water in a bathtub rises by the same amount every minute.

With the plug out, it rises more slowly as it deepens, since deeper water drains faster, until it drains as fast as it fills and holds one depth.

Turn off the tap and put the plug in: filling and draining stop, and the water is still there.

A stock is a quantity that accumulates: what would still be there if every flow stopped. The water in the tub can be measured at an instant; filling and draining exist only over time. A stock changes only through its flows.

It is drawn as a box around its name, its units beneath in smaller type. The world’s population is drawn the same way, filled by births and drained by deaths.

A flow moves material into or out of a stock. It is drawn as a double line, its material path, with an arrowhead where the material goes and, at its middle, a valve: a small circle crossed by an X, the flow’s name beside it.

A flow into a stock is one of its inflows, a flow out of it one of its outflows: filling is the tub’s inflow, draining its outflow. A flow’s value at an instant is its rate, such as 12 liters a minute.

A flow is named for what moves, filling or births, not for what sets it moving, such as tap.

A cloud, a small unnamed scalloped outline, is where a flow’s material comes from or goes to outside the model: the flow’s source where it starts, its sink where it ends. The bathtub’s model leaves out the water main and the sewer, so filling starts at a cloud and draining ends at one.

Each cloud ends one flow; whatever you name or count is a stock instead.

Every stock has units: liters for the water in the tub, people for the world’s population.

A diagram has one time unit, the unit every rate is per: the minute for the bathtub, the year for the population. A flow’s units are its stock’s units per time unit, so filling and draining run in liters per minute.

A transfer is a flow from one stock straight into another: an outflow of the first and an inflow of the second, such as pouring from water in bucket into water in tub.

Material that leaves a stock arrives somewhere, in another stock or at a cloud: conservation. So both stocks of a transfer count it in the same units, liters for the bucket and the tub.

An auxiliary is a variable that neither accumulates nor flows, such as a setting, a capacity or a ratio, drawn as its name alone: tap setting, drain opening, or the population’s births per person, the year’s births divided by the people alive. In Stella, a modeling program, an auxiliary is a converter.

A signal is a single thin line with an arrowhead, carrying a reading of one quantity to an auxiliary or to a flow’s valve; in the systems literature, an information link.

Tap setting → filling carries the tap’s setting, not water, and has a polarity as chapter 4’s effects (chapter 4, Polarity) do: +, since the further the tap is open, the faster the tub fills.

+ signals run from water in tub to draining and from world population to births and deaths. A signal from a stock to one of its own flows closes a loop through that stock (chapter 7 (chapter 7, Loops through stocks)).

A stock equation, one line beneath a stock’s box, adds its inflows and subtracts its outflows:

change in water in tub = filling − draining

Its right-hand side is the stock’s net flow. With filling at 12 liters a minute and draining at 8, the net flow is 4 liters a minute. In 2023 about 132 million people were born worldwide and 62 million died: a net flow of about 70 million people a year.

Auxiliaries and signals never appear in it; they change a stock only through a flow’s rate. A stock with no flows reads change in … = 0.

A delay on a flow, two short strokes across its double line, means its material arrives well after it leaves, as a change arrives late along a delayed effect (chapter 4, Delays). Grain shipped from a port reaches a mill weeks later: the flow from grain at the port to grain at the mill carries a delay, and the grain at sea is in neither stock.

A behavior-over-time graph shows one stock, flow or auxiliary over time: the quantity’s name (and a stock’s units) on the vertical axis, time on the horizontal, and one line of a named shape, rising or falling. The shapes are linear, exponential, approach (flattening toward a value), oscillation (swinging around a value), S-shaped and overshoot and collapse. A graph is a prediction from the structure, made before anything is measured.

A stock’s slope at any instant is its net flow: it rises while inflows exceed outflows, falls while outflows exceed inflows, and stays flat when they are equal.

With the plug in, the net flow is filling’s steady rate, so the water rises in a straight line.

With the plug out, draining speeds up as the water deepens, the net flow shrinks, and the water approaches the depth at which draining equals filling.

Asked to draw the water in a bathtub from graphs of its inflow and outflow, many graduate students at MIT’s business school drew a path its flows couldn’t produce (Booth Sweeney and Sterman, 2000); chapter 7’s Common misreading (chapter 7, Common misreading) gives a later study’s numbers.

When nothing accumulates, or the question is only which way one quantity moves another, a sentence does: opening the drain wider empties the tub faster. With several such effects running in a circle, a causal loop diagram does. When the question is how much there is now, not how it changes, a list of amounts does.

The diagram holds no numbers: rates and amounts go in a sentence or a table beside it.

What’s wrong with this diagram of a bathtub filling from the tap and emptying down the drain?

The tap, the water in the tub and the drain are three stocks in a row, joined by filling and draining. The tap is counted in liters per minute and the tub in liters, so filling’s units come out as liters per minute per minute. The tap holds no water: its setting fixes how fast filling runs, as the drain’s opening fixes draining. The lab’s check reports:

2 transfers join stocks in different units: filling, from tap (liters per minute) to water in tub (liters); draining, from water in tub (liters) to drain (liters per minute). What leaves one stock arrives in the other, so both count it in the same units.

The tap and the drain become auxiliaries, tap setting and drain opening, each with a + signal to its flow’s valve. Filling starts at a cloud and draining ends at one, leaving water in tub, in liters, the one stock, and both flows in liters per minute.

Passing the checks means meeting the brief, not being right.

• Donella H. Meadows, Thinking in Systems: A Primer (Chelsea Green, 2008), chapter 1, The Basics, on stocks, flows, clouds as sources and sinks, and the bathtub.
• John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World (Irwin/McGraw-Hill, 2000), chapter 4, Structure and Behavior of Dynamic Systems, on the shapes behavior over time takes; chapter 6, Stocks and Flows, on the notation and on telling a stock from a flow by freezing a system in time; and chapter 7, Dynamics of Stocks and Flows, on reading a stock’s path from its net flow.
• Linda Booth Sweeney and John D. Sterman, “Bathtub Dynamics: Initial Results of a Systems Thinking Inventory” doi.org/10.1002/sdr.198, System Dynamics Review 16, no. 4 (2000), on how well students with no training in system dynamics read stocks and flows.
• isee systems, Working with Building Blocks iseesystems.com/resources/help/v3/Content/03-BuildingModels/Working_with_building_blocks.htm (Stella help), on the parts of a model in Stella, a modeling program.
• United Nations, World Population Prospects 2024, through Our World in Data, Number of births ourworldindata.org/grapher/number-of-births-per-year and Number of deaths ourworldindata.org/grapher/number-of-deaths-per-year, for the world’s births and deaths in 2023.

Stocks Inside Feedback

Stock & flow diagramsPart III · Accumulation · Chapter 7 of 12 · the second of two chapters on stock & flow diagrams

A new strain of flu reaches a region of a million people, none of whom has had it. Each day people who are ill meet people who have never had it, and some of those catch it. Each person who falls ill is ill for about a week, then recovers and doesn’t catch this strain again that season.

Reported cases reach the news a week after people fall ill. As the reports rise, people stay home and keep apart, and fewer fall ill.

The number of people ill rises for weeks, peaks and falls away. When the season ends, some people have never caught it.

A loop through a stock is a loop (chapter 4, Loops) whose path runs along one of the stock’s flows: a signal from the stock reaches the flow’s valve, directly or through auxiliaries, and the flow changes the stock. The parity rule (chapter 4, The parity rule) counts the flow as one step: + from the flow into the stock it fills, − from the flow out of the stock it drains.

Infectious contacts are meetings between someone ill and someone susceptible, so they rise with both stocks. Three loops run through falling ill:

• R1, contagion: people ill → infectious contacts → falling ill, which fills people ill. No − step: reinforcing.
• B1, depletion: people susceptible → infectious contacts → falling ill, which drains people susceptible. One − step: balancing.
• B2, caution: people ill → reported cases → caution → infectious contacts (−) → falling ill, which fills people ill. One − step: balancing.

Falling ill counts + in R1 and − in B1, since it fills one stock and drains the other.

Loop dominance (chapter 5, Loop dominance) shifts as stocks fill and empty. Early in the season nearly everyone is susceptible, R1 dominates, and people ill grows exponentially. As people susceptible empties, each person ill meets fewer people who can catch flu; B1 dominates, falling ill drops below recovering, and people ill falls.

Falling ill and recovering are transfers (chapter 6, Transfers), and the diagram has no cloud, so everyone who leaves one stock arrives in the next. The three stock equations,

change in people susceptible = − falling ill

change in people ill = falling ill − recovering

change in people recovered = recovering

add up to zero: the three stocks always total the region’s million people.

Births, deaths and moves are left out. Drawn in, each would run from a source or to a sink, and people would cross the region’s boundary (chapter 5, Boundaries) only along those flows. An external driver (chapter 5, External drivers) reaches a stock through a signal to an auxiliary or a valve, never to the stock: if school holidays cut contacts, ‹school holidays› → infectious contacts is −.

Evidence coding (chapter 5, Evidence coding) marks signals only. A flow is never dashed: that people move from susceptible to ill isn’t in question, only how fast, which the signals into its valve set.

An aging chain is a row of stocks joined by transfers, through which material passes in order: people susceptible, then ill, then recovered.

The time material spends in a stock is a delay. Someone who falls ill on a Monday is counted in people ill until about the next Monday, so a rise in falling ill shows in recovering about a week later, on average. People ill → recovering is +: with each person ill for about a week, about a seventh of those ill recover each day. Recovering carries no delay: the stock holds the week, and a delay on the flow would count it twice.

A delay in a stock & flow diagram, the two strokes of chapter 4’s delays (chapter 4, Delays), sits on a flow or on a signal. A material delay, on a flow, holds material in transit, in neither stock: grain at sea between a port and a mill. A signal delay, on a signal, holds news: the people are already ill, and the reading of how many arrives late. People ill → reported cases carries a signal delay of a week.

Caution is set by reported cases, a reading of people ill a week old, so after people ill peaks, reported cases and caution go on rising for about a week, and caution is highest once fewer people are ill.

A stock’s capacity is the most it can hold. As the stock nears it, the flow into it slows sharply: a nonlinear effect (chapter 5, Nonlinear effects) on a signal from the stock to its own inflow, its threshold an amount the stock holds. If the flu filled the region’s hospital beds, people in hospital → admissions would be − and nonlinear: weak while beds are free, strong as the last ones fill.

What’s wrong with this diagram of a town library’s books, lent from the shelves, returned to a trolley and reshelved?

The books on the shelves, on loan and in the returns trolley are three stocks, but lending runs from the shelves to a cloud, borrowing from a cloud into the books on loan, returning to a cloud and dropping off from a cloud. Between each stage and the next the same books leave the model and come back into it, so the three stocks’ total can rise or fall with no book bought or lost. The lab’s check for at least three transfers reports:

No transfers. Brief: at least 3. A transfer runs from one stock straight into another, with no cloud between them.

Corrected, lending, returning and reshelving are transfers around the three stocks; purchases arrive from a source and lost books leave to a sink, so each book is counted in one stock from the day it is bought until it is lost.