Subgraph Counting

Given a pattern $P,$ that is more complicated than the patterns, we fragment $P$ into simpler patterns such that their exact count is known. In the subgraph GNN proposed earlier, look into the subgraph of the host graph. We have seen that this technique is scalable on large graphs. Also, we have seen that subgraph GNN is more expressive and efficient than traditional GNN. So, we tried to explore the expressibility when the pattern is fragmented into smaller subpatterns.

\emph{Canonical partition} $\mathcal{P}$ crops the index-restricted d-hop neighborhood around the center node from the target graph. $\mathcal{D}(G_t,v_i,v_c)$ means the shortest distance between $v_i$ and $v_c$ on $G_t$. \begin{equation} \mathcal{P}(G_t, v_c, d) = G_c, \operatorname{ s.t. } G_c \subseteq G_t, V_c = { v_i \in V_t|\mathcal{D}(G_t,v_i,v_c) \leq d , v_i \leq v_c} \end{equation} The graph $G_c$ obtained by canonical partition is called the \emph{canonical neighborhood}. Canonical neighborhoods can correctly substitute the target graph in canonical count. The subgraph count of query in target equals the summation of the canonical count of query in canonical neighborhoods for all target nodes. Canonical neighborhoods are acquired with canonical partition $\mathcal{P}$, given any $d$ greater than the diameter of the query. \begin{equation} \mathcal{C}(G_q,G_t) = \sum_{v_c \in V_t} \mathcal{C}c(G_q, \mathcal{P}(G_t, v_c, d),v_c), d \geq \max{v_i, v_j \in V_q} \mathcal{D}(G_q, v_i, v_j) \end{equation}