GCN
GAT
GraphSAGE
GIN

\\color{#FE6100} h_v^{(0)}

=

x_v

Node

v

's initial embedding.

... is just node

v

's original features.

for all

v \in V.

and for

k = 1, 2, \ldots

upto

K

:

\\color{#FE6100} h_v^{(k)}

=

GCN
GAT
GraphSAGE
GIN

\\color{#FE6100} h_v^{(0)}

=

x_v

Node

v

's initial embedding.

... is just node

v

's original features.

for all

v \\in V.

and for

k = 1, 2, \\ldots

upto

K

:

\\color{#FE6100} h_v^{(k)}

=

{\\color{#4D9DB5} f^{(k)}}\\left({\\color{#4D9DB5} W^{(k)}} \\cdot \\frac{\\sum\\limits_{u \\in \\mathcal{N}(v)} {\\color{#A95AA1} h_u^{(k - 1)}}}{|\\mathcal{N}(v)|} + {\\color{#4D9DB5} B^{(k)}} \\cdot {\\color{#FE6100} h_v^{(k - 1)}}\\right)

Node

v

's embedding at step

k

.

Mean of

v

's neighbour's embeddings at step

k - 1

.

Node

v

's embedding at step

k - 1

.

for all

v \\in V.

Color Codes:

Embedding of node $v$ .
Embedding of a neighbour of node $v$ .
(Potentially) Learnable parameters.

\\color{#FE6100} h_v^{(0)}

=

x_v

Node

v

's initial embedding.

... is just node

v

's original features.

for all

v \\in V.

and for

k = 1, 2, \\ldots

upto

K

:

\\color{#FE6100} h_v^{(k)}

=

{\\color{#4D9DB5} f^{(k)}}\\left({\\color{#4D9DB5} W^{(k)}} \\cdot \\left[\\sum\\limits_{u \\in \\mathcal{N}(v)} \\alpha_{vu}^{(k - 1)} {\\color{#A95AA1} h_u^{(k - 1)}} + \\alpha_{vv}^{(k - 1)} {\\color{#FE6100} h_v^{(k - 1)}}\\right]\\right)

Node

v

's embedding at step

k

.

Weighted mean of

v

's neighbour's embeddings at step

k - 1

.

Node

v

's embedding at step

k - 1

.

for all

v \\in V.

where the attention weights

\\alpha^{(k)}

are generated by an attention mechanism

{\\color{#4D9DB5} A^{(k)}}

, normalized such that the sum over all neighbours of each node

v

is

1

:

\\alpha_{vu}^{(k)}

=

\\frac{{\\color{#4D9DB5} A^{(k)}}({\\color{#FE6100} h_v^{(k)}}, {\\color{#A95AA1} h_u^{(k)}})}{\\sum\\limits_{w \\in \\mathcal{N}(v)}{\\color{#4D9DB5} A^{(k)}}({\\color{#FE6100} h_v^{(k)}}, {\\color{#A95AA1} h_w^{(k)}})}

for all

(v, u) \\in E.

Color Codes:

Embedding of node $v$ .
Embedding of a neighbour of node $v$ .
(Potentially) Learnable parameters.

\\color{#FE6100} h_v^{(0)}

=

x_v

Node

v

's initial embedding.

... is just node

v

's original features.

for all

v \\in V.

and for

k = 1, 2, \\ldots

upto

K

:

\\color{#FE6100} h_v^{(k)}

=

{\\color{#4D9DB5} f^{(k)}}\\left(\\sum\\limits_{u \\in \\mathcal{N}(v)} {\\color{#A95AA1} h_u^{(k - 1)}} + (1 + {\\color{#4D9DB5} \\epsilon^{(k)}}) \\cdot {\\color{#FE6100} h_v^{(k - 1)}}\\right)

Node

v

's embedding at step

k

.

Sum of

v

's neighbour's embeddings at step

k - 1

.

Node

v

's embedding at step

k - 1

.

for all

v \\in V.

Color Codes:

Embedding of node $v$ .
Embedding of a neighbour of node $v$ .
(Potentially) Learnable parameters.

\\color{#FE6100} h_v^{(0)}

=

x_v

Node

v

's initial embedding.

... is just node

v

's original features.

for all

v \\in V.

and for

k = 1, 2, \\ldots

upto

K

: