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Computation Graphs

A computation graph breaks a calculation into nodes so values flow forward and sensitivities flow backward.

published · difficulty 2/5 · 13 min read

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Intuition

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Suppose one intermediate value gets used twice. If a=xya=xy, b=sin⁡(a)b=\sin(a), and c=a+bc=a+b, then aa affects cc directly and also through bb. How should the derivative remember both uses without differentiating one huge expanded formula by hand?

A computation graph makes that bookkeeping visible.

Instead of treating a function as one large expression, we split it into small operations: multiply here, apply a nonlinearity, add there, reuse an intermediate value. Each operation becomes a node, and each dependency becomes an edge.

A computation graph is not just an expression tree. The expanded expression xy+sin⁡(xy)xy+\sin(xy) repeats xyxy. The graph stores a=xya=xy once and records that two later operations depend on that same value.

This matters because the graph gives backpropagation a route. Values move forward through the graph to produce an output or loss. Derivatives move backward through the same graph, using local slopes on each edge. The chain rule stops feeling like one giant symbolic derivative and becomes bookkeeping over small local changes.

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Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

InspectTrack the same object through the notation and check each symbol.Leave with the invariant the equations preserve.
Equation 1Inspect equation objecta = xy,\qquad b = \sin(a),\qquad c = a + b.Equation 2Inspect equation object\bar v = (\partial c)/(\partial v).Equation 3Inspect equation object\bar a \mathrel{+}= \bar c(\partial c)/(\partial a)\Big|_{direct} = 1,\qquad \bar b \mathrel{...

In this first example, every node is scalar. Let x,y∈Rx,y\in\mathbb R, and draw edges from inputs to the values that depend on them. The graph for one forward pass is a directed acyclic graph: later nodes depend on earlier nodes, not the other way around.

Consider the scalar computation

a=xy,b=sin⁡(a),c=a+b.a = xy,\qquad b = \sin(a),\qquad c = a + b.

The graph has inputs x,yx,y, intermediate nodes a,ba,b, and output cc. The forward pass stores each node value. The important detail is reuse: aa is stored once, but two later uses depend on it. The backward pass asks how a small change in each node would affect the output.

For any node vv, write

vˉ=∂c∂v.\bar v = \frac{\partial c}{\partial v}.

The output seed is cˉ=1\bar c=1. The final add node c=a+bc=a+b sends one unit of sensitivity to both inputs:

aˉ+=cˉ∂c∂a∣direct=1,bˉ+=cˉ∂c∂b=1.\bar a \mathrel{+}= \bar c\frac{\partial c}{\partial a}\Big|_{\text{direct}} = 1,\qquad \bar b \mathrel{+}= \bar c\frac{\partial c}{\partial b} = 1.

The sine branch also sends a local contribution into the same stored node aa. The important invariant is local: a node's cotangent is the sum of contributions from each immediate downstream use. The demo hides the sine-branch contribution until you predict whether it will lower, preserve, or raise the direct baseline.

The backward pass does not expand every symbolic path; each downstream node has already summarized everything beyond it.

For vector or tensor nodes, the same idea uses Jacobians or local vector-Jacobian product rules. With column-vector cotangents, if vj=f(vi)v_j=f(v_i) and Jji=∂vj/∂viJ_{ji}=\partial v_j/\partial v_i, the reverse update is

vˉi+=JjiTvˉj.\bar v_i \mathrel{+}= J_{ji}^{\mathsf T}\bar v_j.

Reverse-mode autodiff is the automated version of this bookkeeping: save forward values, then run local backward rules in reverse topological order.

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Code

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TraceMatch variables to symbols before reading the implementation.Leave with a runnable witness for the math.
Code witness 1import math x, y = 2.0, 3.0 # Forward pass: store intermediate node values. a = x * y b = mat...python
import math

x, y = 2.0, 3.0

# Forward pass: store intermediate node values.
a = x * y
b = math.sin(a)
c = a + b

# Reverse pass bookkeeping: every downstream use appends one local contribution.
bar_c = 1.0
bar_a = 0.0
bar_b = 0.0
bar_x = 0.0
bar_y = 0.0

# c = a + b sends one unit to both inputs.
bar_a += bar_c * 1.0
bar_b += bar_c * 1.0

# b = sin(a) sends another contribution into reused node a.
bar_a += bar_b * math.cos(a)

# a = x * y sends the accumulated sensitivity to x and y.
bar_x += bar_a * y
bar_y += bar_a * x

print("c:", round(c, 4))
print("stored a once:", a)
print("downstream uses of a:", ["direct add edge", "sine edge"])
print("bar_a:", round(bar_a, 4))
print("dc/dx:", round(bar_x, 4))
print("dc/dy:", round(bar_y, 4))

The code is deliberately manual. The += updates are the graph mechanism: a reused node collects every downstream local contribution before sending its accumulated sensitivity to its own inputs.

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Interactive Demo

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ManipulateChange one control and predict the visible response before reveal.Leave with the observed invariant or a repaired model.

Live Concept Demo

Explore Computation Graphs

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 2/5undergraduatecode-aligned
Demo inquiry checkpoint

Manipulate one control and predict the visible change.

01Choose lensTrace a quantity
02ObserveDemo state pending
03GroundName the equation, invariant, or control that explains it.
04CarryNext: Reverse-Mode Automatic Differentiation

Choose what to inspect in Computation Graphs. This shared fallback is an observation guide, not evidence of learning.

Loading interactive demo...

Use the sliders to change xx and yy. In Forward mode, watch the stored value a=xya=xy feed both b=sin⁡(a)b=\sin(a) and the final add node c=a+bc=a+b.

Before switching into the backward numbers, predict whether the hidden accumulated sensitivity at the reused node aa should land lower than, nearly equal to, or higher than the direct contribution from c=a+bc=a+b.

Use Case A and Case B as neutral graph states. Commit first, then reveal how the downstream paths combine before aa sends sensitivity back to xx and yy.

Section prompt

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4/4 sections ready

Concept: Computation Graphs

What is the smallest example that makes Computation Graphs click without losing the math?

BeforeChain RuleNow4/4 sections readyTryManipulate one control and predict the visible change.NextReverse-Mode Automatic Differentiation
Object contextCalculus
StateCalculus
1 source attachedChecking local snapshot
concept:calculus/computation-graphs
Math3 equation objectsa = xy,\qquad b = \sin(a),\qquad c = a + b.CodeComputation Graphs code witness 1print("downstream uses of a:", ["direct add edge", "sine edge"])DemoComputation Graphs predictionManipulate one control and predict the visible change.
ConceptLearner lens

Computation Graphs

What is the smallest example that makes Computation Graphs click without losing the math?

Mode questionCan I say the mechanism back in one sentence before I reveal anything?

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Take this move

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01PredictCommit before reveal.02ExplainName the invariant.03Ground1 source link nearby.04ExperimentChange one knob.05CarrySave the next question.

Route back through the notebook

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4/4 sections ready
Carry inChain Rule

Bring the mental model from Chain Rule; this page will reuse it instead of restarting from zero.

Work hereComputation Graphs

A computation graph breaks a calculation into nodes so values flow forward and sensitivities flow backward.

Carry outReverse-Mode Automatic Differentiation

The next edge should feel earned: use the demo prediction here before following Reverse-Mode Automatic Differentiation.

Test the linkManipulate one control and predict the visible change.Then continue to Reverse-Mode Automatic Differentiation

After The First Pass

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ConceptComputation GraphsCalculus
01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary1 source link attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

Mechanism Storyboard

See the idea move before the page explains it

A computation graph breaks a calculation into nodes so values flow forward and sensitivities flow backward.

Demo notes open01 / Intuition
Editorial autodiff illustration of computation graph nodes with forward value flow and backward sensitivity arrows.
Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Computation Graphs should make visible.

Visual Inquiry

Make the image answer a mathematical question

A computation graph breaks a calculation into nodes so values flow forward and sensitivities flow backward.

4/4 stages readyDemo notes connected
Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Computation Graphs easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

Object - ConceptComputation GraphsQuestion

What is the smallest example that makes Computation Graphs click without losing the math?

concept:calculus/computation-graphs
Boundary

sources: baydin-2018-ad-survey

Check

Open the closest source note before trusting the local explanation.

Evidence

1 selected-object source shown first; 1 reference total.

Next move

Audit the claim boundary, then ask from the same selected object.

selected object source · paper · 2018Automatic differentiation in machine learning: a surveyBaydin et al.
Located CF editorial boundary

Grounds computation graphs as the program structure used by automatic differentiation systems.

Used here as

Baydin et al. describe AD over evaluation traces of elementary operations, computational graphs for dependency relations, and reverse mode as a forward pass that populates intermediate va...

Caveat

Covers differentiable primitives in one executed DAG/program trace and local scalar/vector-VJP teaching model, not symbolic simplification, mutation/aliasing, checkpointing policy, nonsmo...

Open source

Claim Review

A computation graph breaks a calculation into nodes so values flow forward and sensitivities flow backward.

Object - ConceptComputation GraphsQuestion

What is the smallest example that makes Computation Graphs click without losing the math?

concept:calculus/computation-graphs
Boundary

sources: baydin-2018-ad-survey

Check

Treat every claim as provisional until source support and a local witness agree.

Evidence

1 structured claim check on this concept.

Next move

Run the prediction or practice transfer before asking for a grounded review.

1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 1 reference and 3 local witnesses are available for inspection.

For one executed differentiable program, a computation graph uses elementary-operation nodes and dependency edges, stores each reused intermediate node value once in the forward pass, and makes reverse-mode/backprop local: downstream sensitivities from every use accumulate at a node before being sent to its inputs.
Registered CF editorial boundary
Automatic differentiation in machine learning: a surveyGrounds computation graphs as the program structure used by automatic differentiation systems.
Used here as

Baydin et al. describe AD over evaluation traces of elementary operations, computational graphs for dependency relations, and reverse mode as a forward pass that populates intermediate variables/records grap...

Local witness
source-span-baydin-2018-ad-surveymath-object-1math-object-2code-witness-1interactive-demo
Equation 1
a=xy,b=sin⁡(a),c=a+b.a = xy,\qquad b = \sin(a),\qquad c = a + b.
Equation 2
vˉ=∂c∂v.\bar v = \frac{\partial c}{\partial v}.
Caveat

Covers differentiable primitives in one executed DAG/program trace and local scalar/vector-VJP teaching model, not symbolic simplification, mutation/aliasing, checkpointing policy, nonsmooth primitives, or e...

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. Check caveats and exact source scope.

Checked Baydin et al. for one executed AD trace: programs become elementary-operation traces and graphs of intermediate-variable dependencies; reverse mode runs code forward, populates intermediates, records dependencies, then propagates adjoints backward; a reused variable's adjoint sums downstream contributions before input derivatives are obtained. Local math/code/demo instantiate a=xy,b=sin(a),c=a+b with one stored a, direct + sine-path cotangent accumulation, and propagation to x,y.

Reviewer: codex+oracle; reviewed 2026-05-07

Practice · Computation Graphs

Try the idea in your own words

A computation graph breaks a calculation into nodes so values flow forward and sensitivities flow backward.

Concept · Current object

Computation Graphs

Source boundary: sources: baydin-2018-ad-survey

Object context and links

Calculus

concept:calculus/computation-graphs
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Explain the mechanism

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    Grounded object roomClose
    Selected object routeAsk from this object; carry one invariant back.sources: baydin-2018-ad-survey
    1. ObjectConceptComputation Graphs
    2. PredictBefore revealComputation Graphs prediction
    3. WitnessCompare codeComputation Graphs code witness 1
    4. RoomAsk groundedChecking local snapshot
    ConceptComputation GraphsCalculus
    Code witness comparisonComputation Graphs code witness 1print("downstream uses of a:", ["direct add edge", "sine edge"])Prediction before revealComputation Graphs predictionManipulate one control and predict the visible change.
    Grounded room questionWhat is the smallest example that makes Computation Graphs click without losing the math?Checking local snapshot

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    conceptCalculus

    Computation Graphs

    Anchored question

    What is the smallest example that makes Computation Graphs click without losing the math?

    Source boundaryInspect source ids: baydin-2018-ad-surveyStable content-object key attached
    Role lenses for this object

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    Learner evidence requestAsk what would make "Computation Graphs" feel predictable rather than familiar.
    Assumption

    Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic.

    Source-checking summary

    Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion.

    Proposed experiment

    Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo.

    Next action

    The learner can state the mechanism in their own words

    Evidence4 checks
    PredictionChecking carried observation
    ActionReady for one action
    AILearner handoff ready
    Open source object
    01PredictionChecking browser-local route memory
    02EvidenceChecking for a carried observation
    03BoundaryInspect source ids: baydin-2018-ad-survey
    04Next moveSave one next action
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    Evidence to inspect
    • Source ids to inspect: baydin-2018-ad-survey
    • Definition, prerequisite, and contrast concept links
    • The equation or code witness that makes the concept operational
    • One demo state that shows the invariant instead of a slogan
    What would resolve this
    • The learner can state the mechanism in their own words
    • The learner can name the prerequisite that would repair confusion
    • The learner can predict how the mechanism changes under one perturbation
    Object-attached AI handoff

    I am working in Continuous Function's research reading room. Object: concept - Computation Graphs Object key: concept:calculus/computation-graphs Context: Calculus Anchor id: concept/concept-notebook/calculus/computation-graphs Open question: What is the smallest example that makes Computation Graphs click without losing the math? Evidence to inspect: - Source ids to inspect: baydin-2018-ad-survey - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Computation Graphs" feel predictable rather than familiar." | assumption: Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: baydin-2018-ad-survey" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Computation Graphs" feel predictable rather than familiar. - Assumption to keep visible: Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

    concept/concept-notebook/calculus/computation-graphs concept:calculus/computation-graphs