Iteratively Refined Early Interaction Alignment for Subgraph Matching based Graph Retrieval

TL;DR

This work tackles subgraph isomorphism-based graph retrieval and introduces IsoNet++, an early-interaction GNN that iteratively refines an injective alignment between query and corpus graphs. It builds on a GW/QAP formulation, relaxing the alignment to a doubly stochastic matrix and updating it across multiple rounds while conducting cross-graph message passing. The three core innovations are node-pair partner interaction, multi-round lazy alignment refinement, and the option to perform eager vs lazy updates, with an edge-alignment variant providing further gains. Empirical results across six datasets show IsoNet++ achieving state-of-the-art performance in MAP and HITS@20, with clear advantages from node-pair interactions and lazy refinement, along with a transparent analysis of computation-time trade-offs and alignment quality. The approach offers practical implications for scalable, attack-surface-aware graph retrieval and provides avenues for extensions to other similarity measures and constrained alignments.

Abstract

Graph retrieval based on subgraph isomorphism has several real-world applications such as scene graph retrieval, molecular fingerprint detection and circuit design. Roy et al. [35] proposed IsoNet, a late interaction model for subgraph matching, which first computes the node and edge embeddings of each graph independently of paired graph and then computes a trainable alignment map. Here, we present IsoNet++, an early interaction graph neural network (GNN), based on several technical innovations. First, we compute embeddings of all nodes by passing messages within and across the two input graphs, guided by an injective alignment between their nodes. Second, we update this alignment in a lazy fashion over multiple rounds. Within each round, we run a layerwise GNN from scratch, based on the current state of the alignment. After the completion of one round of GNN, we use the last-layer embeddings to update the alignments, and proceed to the next round. Third, IsoNet++ incorporates a novel notion of node-pair partner interaction. Traditional early interaction computes attention between a node and its potential partners in the other graph, the attention then controlling messages passed across graphs. In contrast, we consider node pairs (not single nodes) as potential partners. Existence of an edge between the nodes in one graph and non-existence in the other provide vital signals for refining the alignment. Our experiments on several datasets show that the alignments get progressively refined with successive rounds, resulting in significantly better retrieval performance than existing methods. We demonstrate that all three innovations contribute to the enhanced accuracy. Our code and datasets are publicly available at https://github.com/structlearning/isonetpp.

Figures (6)

• Figure 1: Overview of IsoNet++. Panel (a) shows the pipeline of IsoNet++. Given a graph pair $(G_q,G_c)$, we execute $T$rounds, each consisting of $K$ GNN layer propagations. After a round $t$, we use the node embeddings to update the node alignment ${\bm{P}}={\bm{P}}_{t}$ from its previous estimate ${\bm{P}}={\bm{P}}_{t-1}$. Within each round $t\in[T]$, we compute the node embeddings of $G_q$ by gathering signals from $G_c$ and vice-versa, using GNN embeddings in the previous round and the node-alignment map ${\bm{P}}_{t}$. The alignment ${\bm{P}}_{t}$ remains consistent across all propagation layers $k\in[K]$ and is updated at the end of round $t$. Panel (b) shows our proposed node pair partner interaction in IsoNet++ (Node). When computing the message value of the node pair $(u,v)$, we also feed the node embeddings of the partners $u'$ and $v'$ in addition to the embeddings of the pairs $(u,v)$, where $u'$ and $v'$ is approximately aligned with $u$ and $v$, respectively (when converted to soft alignment, $u',v'$ need not be neighbors). Panel (c) shows the node pair partner interaction in IsoNet++ (Edge). In contrast to IsoNet++ (Node), here we feed the information from the message value of the partner pair $(u',v')$ instead of their node embeddings into the message passing network $\mathop{\mathrm{msg}}\nolimits_{\theta}$.
• Figure 2: Illustration of the three interaction modes. IsoNet has no/late interaction between $\bm{h} ^{(q)}$ and $\bm{h} ^{(c)}$. IsoNet++ and GMN allow interaction between the representations of the query and corpus nodes. Under node pair interaction, the individual node embeddings $\bm{h} ^{(q)}$ are used for message passing directly, thereby exposing them only to their neighbors. In the corresponding $\mathop{\mathrm{comb}}\nolimits_{\theta}$ step, nodes interact only with their respective partners, therefore missing out on information from the partners of its neighbors. However, under node pair partner interaction, the representation of a node is combined with that of its partner(s) first, using the $\mathop{\mathrm{inter}}\nolimits_\theta$ block to obtain $\bm{z} ^{(q)}$\ref{['eq:inter-1']}, which is used for message passing. Thus, when interacting with its neighbors, a node also gets information from the partners of its neighbors.
• Figure 3: Empirical probability density of similarity between the estimated alignments and the true alignments ${\bm{P}}^*,\bm{S}^*$ for both multi-round and multi-layer update strategies across different stages of updates ($t$ for multi-round and $k$ for multi-layer), for AIDS. Similarity is measured using $p(\text{Tr}({\bm{P}}_t ^{\top}{\bm{P}}^*)), p(\text{Tr}(\bm{S}_t ^{\top}\bm{S}^*))$ for multi-round lazy updates and $p(\text{Tr}({\bm{P}}_k ^{\top}{\bm{P}}^*)), p(\text{Tr}(\bm{S}_k ^{\top}\bm{S}^*))$ for multi-layer eager updates.
• Figure 4: Trade-off between MAP and inference time (batch size=128).
• Figure 5: Similar to Figure \ref{['fig:hist']}, we plot empirical probability density of $p(\text{Tr}({\bm{P}}_t ^{\top}{\bm{P}}^*))$ and $p(\text{Tr}(\bm{S}_t ^{\top}\bm{S}^*))$ for different values of $t$ lazy multi round updates and $p(\text{Tr}({\bm{P}}_k ^{\top}{\bm{P}}^*))$ and $p(\text{Tr}(\bm{S}_k ^{\top}\bm{S}^*))$ for different values of $k$ for eager multi layer updates. The first (last) two plots in the left (right) of each row are for multi-round IsoNet++ (Node) (multi-round IsoNet++ (Edge)).