On “From Local Structures to Size Generalization in Graph Neural Netorks”
Local structure depends on graph size, e.g., in \(G(n, p)\), the expected degree is \(np\), hence fixing \(p\) and increasing \(n\) changes the local structure of the graph.
To characterize the local structure by graph d-pattern: def. 4.1. and fig. 2. d-pattern is defined in a recursive way and is motivated by WL test (and the analysis in GIN), e.g., 1-pattern represents the node’s degree, and a 2-pattern of a node is a pair: (its degree, the set of its neighbor’s degrees).
To connect GNN with d-patterns: On one side, GNN can be programed to output any value on any d-pattern independently. (more precise version is Thr. 4.3) Conversely, Thr. 4.2: d-layer GNN (with a node-level output additionally) will output the same results for nodes with the same d-patterns. Combining them, we can independently control the (output) values of d-layer GNNs on the set of d-patterns and these values completely determine the GNN’s output.
The relation between size generalization and d-pattern discrepancy: Thr. 5.1 \(P_1, P_2\) be finitely supported distributions of graphs \(P_1^d, P_2^d\) the distribution of d-patterns over \(P_1, P_2\) respectively assume that any graph in \(P_2\) contains a node with a d-pattern in \(P_2^d \setminus P_1^d\) then any regression task solvable by a d-depth GNN, there exists a (d+3)-depth GNN that
- perfectly solves the task on \(P_1\)
- predicts with an arbitrary error on \(P_2\) The main idea is to leverage the unseen d-patterns from \(P_2^d\) to change the output on graphs from \(P_2\) Then Thr.5.2 The main idea is pick a set of d-patterns that has small probability, i.e., \(\epsilon\) on \(P_1^d\) but large probability in \(P_2^d\).
To improve size generalization: consdier domain adaptation setting where we have access to labeled samples from the source domain but the target domain is unlabeled SSL: propose node-regression pretext task, that is, to regression the histogram of each feature field where the histogram is aggregated at the focused node from its d-depth tree (d-hop neighborhood)