4.2 Model

$$h_v = f(x_v, x_{co[v]}, h_{ne[v]}, x_{ne[v]}) $$

$$o_v = g(h_v, x_v)$$

• Functions
  • $f$: local transition function which is shared among all nodes
  • $g$ : local output function
• Symbols
  • x: the input feature
  • h: hidden state
  • $co[v]$: the set of edges connected to node $v$
  • $ne[v]$: the set of neighbors of node $v$
  • $x_v$: the features of $v$
  • $x_{co[v]}$: the features of its edges
  • $h_{ne[v]}$: the states of nodes in the neighborhood of v
  • $h_{co[v]}$: the states of features in the neighborhood of v

An example of the graph

Example for node $l_1$

• $x_{l_1}$: the input feature
• $co[l_1]$: $l_{(1, 4)}, l_{(1, 6)}, l_{(3, 1)}, l_{(1, 2)}$
• $ne[l_1]$: $l_2, l_3, l_4, l_6$

전체 노드에 대하여 아래의 식으로 나타낼 수 있음

$$H=F(H, X)$$

$$O=G(H, X_N)$$

• H: the matrices constructed by stacking all the states 중간 산물로 나온 모든 결과물
• O: the matrices constructed by all the outputs 마지막 output들
• X: the matrices constructed by all the features
• $X_n$: the matrices constructed by all the node features
• F: the global transition function
• G: the global output function

여러개의 layer을 거치게 될 경우, $$H^{t+1} = F(H^t, X)$$

Loss Function

$$loss = \sum_{i=1}^p (t_i - o_i)$$

여기서 $p$는 supervised nodes의 개수를 의미

Learning algorithm

based on a gradient descent strategy and is composed of the following steps

• The states $h_v^t$ are iteratively updated by $h_v = f(x_v, x_{co[v]}, h_{ne[v]}, x_{ne[v]}) $ until a time step $T$. Then obtain an approximate fixed point solution of $H=F(H, X)$ : $H(T) \approx H$
• The gradient of weights W is computed from the loss
• The weights W are updated according to the gradient computed in the last step.