Graph-structured data is ubiquitous across diverse domains like social networks, search, question answering, and drug discovery. Effective retrieval of (sub-)graphs with relevant substructures has become critical to the success of these applications. This proposed tutorial will introduce attendees to state-of-the-art neural methods for graph retrieval, highlighting architectures that effectively model relevance through innovative combinations of early and late interaction mechanisms.

Subgraph matching is vital in knowledge graph (KG) question answering, molecule design, scene graph, code and circuit search, etc. Neural methods have shown promising results for subgraph matching. Our study of recent systems suggests refactoring them into a unified design space for graph matching networks. Existing methods occupy only a few isolated patches in this space, which remains largely uncharted. We undertake the first comprehensive exploration of this space, featuring such axes as attention-based vs. soft permutation-based interaction between query and corpus graphs, aligning nodes vs. edges, and the form of the final scoring network that integrates neural representations of the graphs. Our extensive experiments reveal that judicious and hitherto-unexplored combinations of choices in this space lead to large performance benefits. Beyond better performance, our study uncovers valuable insights and establishes general design principles for neural graph representation and interaction, which may be of wider interest.

Estimating the clique number in a graph is central to various applications, e.g., community detection, graph retrieval, etc. Existing estimators often rely on non-differentiable combinatorial components. Here, we propose a full differentiable estimator for clique number estimation, which can be trained from distant supervision of clique numbers, rather than demonstrating actual cliques. Our key insight is a formulation of the maximum clique problem (MCP) as a maximization of the size of fully dense square submatrix, within a suitably row-column-permuted adjacency matrix. We design a differentiable mechanism to search for permutations that lead to the discovery of such dense blocks. However, the optimal permutation is not unique, which leads to the learning of spurious permutations. To tackle this problem, we view the MCP problem as a sequence of subgraph matching tasks, each detecting progressively larger cliques in a nested manner. This allows effective navigation through suitable node permutations. These steps result in MxNet, an end-to-end differentiable model, which learns to predict clique number without explicit clique demonstrations, with the added benefit of interpretability. Experiments on eight datasets show the superior accuracy of our approach.

Retrieving graphs from a large corpus, that contain a subgraph isomorphic to a given query graph, is a core operation in many real-world applications. While recent multi-vector graph representations and scores based on set alignment and containment can provide accurate subgraph isomorphism tests, their use in retrieval remains limited by their need to score corpus graphs exhaustively. We introduce CoRGII (COntextual Representation of Graphs for Inverted Indexing), a graph indexing framework in which, starting with a contextual dense graph representation, a differentiable discretization module computes sparse binary codes over a learned latent vocabulary. This text document-like representation allows us to leverage classic, highly optimized inverted indexes, while supporting soft (vector) set containment scores. Improving on this paradigm further, we replace the classical impact score of a `word' on a graph (such as defined by TFIDF or BM25) with a data-driven, trainable impact score. Crucially, CoRGII is trained end-to-end using only binary relevance labels, without fine-grained supervision of query-to-document set alignments. Extensive experiments show that CoRGII provides better trade-offs between efficiency and accuracy, compared to several baselines.

Graph Edit Distance (GED) measures the (dis-)similarity between two given graphs in terms of the minimum-cost edit sequence, which transforms one graph to the other. GED is related to other notions of graph similarity, such as graph and subgraph isomorphism, maximum common subgraph, etc. However, the computation of exact GED is NP-Hard, which has recently motivated the design of neural models for GED estimation. However, they do not explicitly account for edit operations with different costs. In response, we propose $\texttt{GraphEdX}$, a neural GED estimator that can work with general costs specified for the four edit operations, viz., edge deletion, edge addition, node deletion, and node addition. We first present GED as a quadratic assignment problem (QAP) that incorporates these four costs. Then, we represent each graph as a set of node and edge embeddings and use them to design a family of neural set divergence surrogates. We replace the QAP terms corresponding to each operation with their surrogates. Computing such neural set divergence requires aligning nodes and edges of the two graphs. We learn these alignments using a Gumbel-Sinkhorn permutation generator, additionally ensuring that the node and edge alignments are consistent with each other. Moreover, these alignments are cognizant of both the presence and absence of edges between node pairs. Through extensive experiments on several datasets, along with a variety of edit cost settings, we show that $\texttt{GraphEdX}$ consistently outperforms state-of-the-art methods and heuristics in terms of prediction error. The code is available at https://github.com/structlearning/GraphEdX.

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 $\texttt{IsoNet++}$, 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, $\texttt{IsoNet++}$ 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. 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.

In many search applications related to passage retrieval, text entailment, and subgraph search, the query and each 'document' is a set of elements, with a document being relevant if it contains the query. These elements are not represented by atomic IDs, but by embedded representations, thereby extending set containment to *soft* set containment. Recent applications address soft set containment by encoding sets into fixed-size vectors and checking for elementwise *vector* *dominance*. This 0/1 property can be relaxed to an asymmetric *hinge* *distance* for scoring and ranking candidate documents. Here we focus on data-sensitive, trainable indices for fast retrieval of relevant documents. Existing LSH methods are designed for mostly symmetric or few simple asymmetric distance functions, which are not suitable for hinge distance. Instead, we transform hinge distance into a proposed *dominance* *similarity* measure, to which we then apply a Fourier transform, thereby expressing dominance similarity as an expectation of inner products of functions in the frequency domain. Next, we approximate the expectation with an importance-sampled estimate. The overall consequence is that now we can use a traditional LSH, but in the frequency domain. To ensure that the LSH uses hash bits efficiently, we learn hash functions that are sensitive to both corpus and query distributions, mapped to the frequency domain. Our experiments show that the proposed asymmetric dominance similarity is critical to the targeted applications, and that our LSH, which we call FourierHashNet, provides a better query time vs. retrieval quality trade-off, compared to several baselines. Both the Fourier transform and the trainable hash codes contribute to performance gains.

Given a query graph and a database of corpus graphs, a graph retrieval system aims to deliver the most relevant corpus graphs. Graph retrieval based on subgraph matching has a wide variety of applications, e.g., molecular fingerprint detection, circuit design, software analysis, and question answering. In such applications, a corpus graph is relevant to a query graph, if the query graph is (perfectly or approximately) a subgraph of the corpus graph. Existing neural graph retrieval models compare the node or graph embeddings of the query-corpus pairs, to compute the relevance scores between them. However, such models may not provide edge consistency between the query and corpus graphs. Moreover, they predominantly use symmetric relevance scores, which are not appropriate in the context of subgraph matching, since the underlying relevance score in subgraph search should be measured using the partial order induced by subgraph-supergraph relationship. Consequently, they show poor retrieval performance in the context of subgraph matching. In response, we propose ISONET, a novel interpretable neural edge alignment formulation, which is better able to learn the edge-consistent mapping necessary for subgraph matching. ISONET incorporates a new scoring mechanism which enforces an asymmetric relevance score, specifically tailored to subgraph matching. ISONET’s design enables it to directly identify the underlying subgraph in a corpus graph, which is relevant to the given query graph. Our experiments on diverse datasets show that ISONET outperforms recent graph retrieval formulations and systems. Additionally, ISONET can provide interpretable alignments between query-corpus graph pairs during inference, despite being trained only using binary relevance labels of whole graphs during training, without any fine-grained ground truth information about node or edge alignments.

The graph retrieval problem is to search in a large corpus of graphs for ones that are most similar to a query graph. A common consideration for scoring similarity is the maximum common subgraph (MCS) between the query and corpus graphs, usually counting the number of common edges (i.e., MCES). In some applications, it is also desirable that the common subgraph be connected, i.e., the maximum common connected subgraph (MCCS). Finding exact MCES and MCCS is intractable, but may be unnecessary if ranking corpus graphs by relevance is the goal. We design fast and trainable neural functions that approximate MCES and MCCS well. Late interaction methods compute dense representations for the query and corpus graph separately, and compare these representations using simple similarity functions at the last stage, leading to highly scalable systems. Early interaction methods combine information from both graphs right from the input stages, are usually considerably more accurate, but slower. We propose both late and early interaction neural MCES and MCCS formulations. They are both based on a continuous relaxation of a node alignment matrix between query and corpus nodes. For MCCS, we propose a novel differentiable network for estimating the size of the largest connected common subgraph. Extensive experiments with seven data sets show that our proposals are superior among late interaction models in terms of both accuracy and speed. Our early interaction models provide accuracy competitive with the state of the art, at substantially greater speeds.

After observing a snapshot of a social network, a link prediction (LP) algorithm identifies node pairs between which new edges will likely materialize in future. Most LP algorithms estimate a score for currently non-neighboring node pairs, and rank them by this score. Recent LP systems compute this score by comparing dense, low dimensional vector representations of nodes. Graph neural networks (GNNs), in particular graph convolutional networks (GCNs), are popular examples. For two nodes to be meaningfully compared, their embeddings should be indifferent to reordering of their neighbors. GNNs typically use simple, symmetric set aggregators to ensure this property, but this design decision has been shown to produce representations with limited expressive power. Sequence encoders are more expressive, but are permutation sensitive by design. Recent efforts to overcome this dilemma turn out to be unsatisfactory for LP tasks. In response, we propose PermGNN, which aggregates neighbor features using a recurrent, order-sensitive aggregator and directly minimizes an LP loss while it is `attacked' by adversarial generator of neighbor permutations. PermGNN has superior expressive power compared to earlier GNNs. Next, we devise an optimization framework to map PermGNN's node embeddings to a suitable locality-sensitive hash, which speeds up reporting the top-K most likely edges for the LP task. Our experiments on diverse datasets show that PermGNN outperforms several state-of-the-art link predictors by a significant margin, and can predict the most likely edges fast.

Given the large number of new musical tracks released each year, automated approaches to plagiarism detection are essential to help us track potential violations of copyright.Most current approaches to plagiarism detection are based on musical similarity measures, which typically ignore the issue of polyphony in music.We present a novel feature space for audio derived from compositional modelling techniques, commonly used in signal separation, that provides a mechanism to account for polyphony without incurring an inordinate amount of computational overhead.We employ this feature representation in conjunction with traditional audio feature representations in a classification framework which uses an ensemble of distance features to characterize pairs of songs as being plagiarized or not.Our experiments on a database of about 3000 musical track pairs show that the new feature space characterization produces significant improvements over standard baselines.