Predict a molecule's property from its bonds

Same six atoms, same six starting features, every time — only the bonds change. Watch the predicted property move as you switch structures. Nothing about any single atom changed; only who it's connected to did.

This capstone chains the last two chapters' rules into a single two-layer pipeline, then reduces the result to one number:

• $h^{(0)}$ — the atoms' initial input features, before either layer runs.
• $h^{(1)}$ — atom features after layer 1, the GCN (Graph Convolutional Network) neighborhood-mean update.
• $h^{(2)}$ — atom features after layer 2, the GAT (Graph Attention Network) attention update.
• $\hat{y}$ — the model's final predicted whole-molecule property.
• $v$ — index ranging over every atom in the molecule, summed by the readout.

1. Layer 1: GCN's plain neighborhood mean

   Layer 1 is Chapter 2's plain neighborhood mean — every atom blends with its bonded neighbors, no weighting.

2. Layer 2: GAT's attention, on top of layer 1

   Layer 2 is last chapter's attention rule, applied on top of layer 1's output, not the raw features — so by the time attention runs, every value already carries a first hop of neighbor information.

3. Readout: sum atom embeddings into one number

   The final sum — a readout — is the simplest way to turn six per-atom numbers into one whole-molecule prediction: real molecular property predictors do exactly this, summing (or averaging) learned atom embeddings into a single graph-level output.

Three different numbers, from three different arrangements of the exact same six atoms. The ring's prediction is the highest of the three — every atom in the ring's 3-cycle (nodes 0, 1, 2) has at least two neighbors to draw on, where the chain's end atoms have only one.

Atom $0$ on the ring structure, through both layers:

1. Layer 1 (GCN): mean over atom 0's neighborhood {0, 1, 2}

   $h^{(1)}_0 = \frac{1+2+0}{3} = 1.0$ (self $1$, neighbors $\{1,2\}$ with features $2,0$). The same mean rule for the rest of the ring:

   • $h^{(1)}_1 = \frac{2+1+0+3}{4} = 1.5$ (self $2$, neighbors $\{0,2,3\}$ with features $1,0,3$)
   • $h^{(1)}_2 = \frac{0+1+2}{3} = 1.0$ (self $0$, neighbors $\{0,1\}$ with features $1,2$)
   • $h^{(1)}_3 = \frac{3+2+1+2}{4} = 2.0$ (self $3$, neighbors $\{1,4,5\}$ with features $2,1,2$)
   • $h^{(1)}_4 = \frac{1+3}{2} = 2.0$ (self $1$, neighbor $\{3\}$ with feature $3$)
   • $h^{(1)}_5 = \frac{2+3}{2} = 2.5$ (self $2$, neighbor $\{3\}$ with feature $3$)

   These are needed next, since layer 2 attends over these values, not the raw ones.

2. Layer 2 (GAT): attention over the same neighborhood, using layer 1's output

   Scores for atom $0$ (neighbors $\{1,2\}$, using layer 1's values):

   • $e_{0,0}=\text{LeakyReLU}(1.0-1.0)=0$
   • $e_{0,1}=\text{LeakyReLU}(1.5-1.0)=0.5$
   • $e_{0,2}=\text{LeakyReLU}(1.0-1.0)=0$

   Softmax (subtracting the max, $0.5$):

   • $e^{0-0.5}\approx0.607$ (self)
   • $e^{0.5-0.5}=1$ (atom $1$)
   • $e^{0-0.5}\approx0.607$ (atom $2$)

   Sum $\approx2.213$. Normalized weights:

   • $0.607/2.213\approx0.274$
   • $1/2.213\approx0.452$
   • $0.607/2.213\approx0.274$

   Atom $1$ dominates, since it's the only neighbor layer 1 pushed higher than atom $0$ itself.

3. Atom 0's final embedding

   $h^{(2)}_0 \approx 0.274(1.0) + 0.452(1.5) + 0.274(1.0) \approx 1.226$.

4. The same two layers, for the ring's other five atoms

   Running the identical GAT step (scores → softmax → weighted sum) over each atom's own layer-1 neighborhood:

   • Atom $1$ (neighbors $\{0,2,3\}$, self $h^{(1)}_1=1.5$): scores $(0,-0.1,-0.1,0.5)$ for (self, $0$, $2$, $3$); softmax weights $\approx(0.224,0.203,0.203,0.370)$; $h^{(2)}_1\approx0.224(1.5)+0.203(1.0)+0.203(1.0)+0.370(2.0)\approx1.482$
   • Atom $2$ (neighbors $\{0,1\}$, self $h^{(1)}_2=1.0$): the same self/neighbor values as atom $0$ (one neighbor at $1.0$, one at $1.5$), so $h^{(2)}_2\approx1.226$ — identical to atom $0$
   • Atom $3$ (neighbors $\{1,4,5\}$, self $h^{(1)}_3=2.0$): scores $(0,-0.1,0,0.5)$ for (self, $1$, $4$, $5$); softmax weights $\approx(0.220,0.199,0.220,0.362)$; $h^{(2)}_3\approx0.220(2.0)+0.199(1.5)+0.220(2.0)+0.362(2.5)\approx2.082$
   • Atom $4$ (neighbor $\{3\}$ only, self $h^{(1)}_4=2.0=h^{(1)}_3$): self and neighbor share the same value, so softmax gives equal weights $0.5/0.5$ and $h^{(2)}_4=2.0$ exactly
   • Atom $5$ (neighbor $\{3\}$ only, self $h^{(1)}_5=2.5$, neighbor $h^{(1)}_3=2.0$): scores $(0,-0.1)$; softmax weights $\approx(0.525,0.475)$; $h^{(2)}_5\approx0.525(2.5)+0.475(2.0)\approx2.263$

   Summing all six: $1.226+1.482+1.226+2.082+2.0+2.263\approx10.278$ — the ring's whole-molecule prediction.

5. Same atoms, different bonds: the chain's prediction

   Wiring the exact same six atoms into a chain ($0\text-1\text-2\text-3\text-4\text-5$) instead changes every neighborhood, so layer 1 gives different means:

   • $h^{(1)}_0=\frac{1+2}{2}=1.5$
   • $h^{(1)}_1=\frac{2+1+0}{3}\approx1.0$
   • $h^{(1)}_2=\frac{0+2+3}{3}\approx1.667$
   • $h^{(1)}_3=\frac{3+0+1}{3}\approx1.333$
   • $h^{(1)}_4=\frac{1+3+2}{3}=2.0$
   • $h^{(1)}_5=\frac{2+1}{2}=1.5$

   Running the same layer-2 attention step on these values gives $h^{(2)}\approx(1.263,\ 1.462,\ 1.348,\ 1.739,\ 1.627,\ 1.811)$, summing to $1.263+1.462+1.348+1.739+1.627+1.811\approx9.250$ — lower than the ring's $10.278$, purely because the chain's end atoms ($0$ and $5$) have only one neighbor to draw on instead of two.