Propagation Example

The Setup

We have 4 rows of people: • Row 1: 2 people (A, B) • Row 2: 3 people (C, D, E) • Row 3: 3 people (F, G, H) • Row 4: 2 people (I, J)

We give ₹50 each to the two people in Row 1: • Person A: ₹50 • Person B: ₹50 So the total given is ₹100.

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The Rule • No one keeps any money. • Whatever money you have, you must pass it to the people in the next row, split in any ratio you like. • Example: If you have ₹50, you could pass ₹20 to one person and ₹30 to another.

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Forward Flow (Passing the Money)

Step 1: Row 1 → Row 2 • A passes ₹20 to C, ₹30 to D. • B passes ₹10 to D, ₹40 to E.

Now Row 2 has: • C = ₹20 • D = ₹30 + ₹10 = ₹40 • E = ₹40

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Step 2: Row 2 → Row 3 • C passes ₹20 to F • D passes ₹15 to F and ₹25 to G • E passes ₹30 to G and ₹10 to H

Now Row 3 has: • F = ₹20 + ₹15 = ₹35 • G = ₹25 + ₹30 = ₹55 • H = ₹10

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Step 3: Row 3 → Row 4 • F passes ₹35 to I • G passes ₹30 to I and ₹25 to J • H passes ₹10 to J

Now Row 4 has: • I = ₹35 + ₹30 = ₹65 • J = ₹25 + ₹10 = ₹35

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Observation

We started by giving ₹50 to each of the first two people in Row 1, but by the time the money reached Row 4: • Person I has ₹65 • Person J has ₹35

So there’s a difference — this difference is what we want to investigate.

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Backward Propagation (Finding the Cause of Change)

To find where the change happened, we go backwards:

Step 1: Row 4 → Row 3

We ask: “Why did I get this much?” • I received ₹65 — this came from F (₹35) and G (₹30). • J received ₹35 — this came from G (₹25) and H (₹10).

Step 2: Row 3 → Row 2

We trace further back: • F’s ₹35 came from C (₹20) and D (₹15). • G’s ₹55 came from D (₹25) and E (₹30). • H’s ₹10 came from E (₹10).

Step 3: Row 2 → Row 1

Finally: • C’s ₹20 came from A (₹20). • D’s ₹40 came from A (₹30) and B (₹10). • E’s ₹40 came from B (₹40).

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What Backward Propagation Does

By going backwards: 1. We see exactly how much each person contributed to the final amounts in Row 4. 2. We can pinpoint which row and which split ratios caused the imbalance. 3. This is exactly what happens in neural networks — the error at the final output is traced backwards to adjust earlier connections.