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\bigoplus

1. \bigoplus operator

This operator looks like this \(\rightarrow\bigoplus\). In LaTeX, we use $\bigoplus$ for this, and hence, the title name.(since you cannot use LaTeX math mode in title ;-;)

2. Definition

\(\bigoplus\) is a direct sum operator. I will give an example using graphs to explain this.

  • Suppose only the nodes {\(v_1\), \(v_2\), \(v_3\)} \(\in \mathcal{V}\) are given and I want to construct a subgraph \(\mathcal{G}^D\) = (\(\mathcal{V}^D\), \(\mathcal{E}^D\), \(\mathcal{R}^D\)) around it by sampling its \(k\)-hop neighborhood from the graph \(\mathcal{G}\).
  • Then we find the \(k\)-hop neighbors of each node \(v_m \in \mathcal{V}\), denoted as (\(\mathcal{V}_{m}\), \(\mathcal{E}_{m}\), \(\mathcal{R}_{m}\)).
  • \((\mathcal{V}_{m}, \mathcal{E}_{m}, \mathcal{R}_{m}) =\) subgraph obtained by sampling \(k\)-hop neighborhood of the node \(v_m \in \mathcal{V} \;\;\; \forall m = 1, 2, 3\).
  • Finally, you have sets of vertices, sets of edges and sets of relations. We take union of these sets respectively over \(m = 1, 2, 3\) to obtain the desired subgraph \(\mathcal{G}^D\).
  • \(\mathcal{V}^D = \bigcup_{m=1}^{m=3} \mathcal{V}_{m}\),   \(\mathcal{E}^D = \bigcup_{m=1}^{m=3} \mathcal{E}_{m}\)   and   \(\mathcal{R}^D = \bigcup_{m=1}^{m=3} \mathcal{R}_{m}\)
  • This can be equivalently represented as \(\mathcal{G}^D = \bigoplus_{m=1}^{m=3} (\mathcal{V}_{m}, \mathcal{E}_{m}, \mathcal{R}_{m})\)