Graph Neural Networks (GNNs) are a predominant method for graph representation learning. However, beyond subgraph frequency estimation, their application to network motif significance-profile (SP) prediction remains under-explored, with no established benchmarks in the literature. We propose to address this problem, framing SP estimation as a task independent of subgraph frequency estimation. Our approach shifts from frequency counting to direct SP estimation and modulates the problem as multitarget regression. The reformulation is optimised for interpretability, stability and scalability on large graphs. We validate our method using a large synthetic dataset and further test it on real-world graphs. Our experiments reveal that 1-WL limited models struggle to make precise estimations of SPs. However, they can generalise to approximate the graph generation processes of networks by comparing their predicted SP with the ones originating from synthetic generators. This first study on GNN-based motif estimation also hints at how using direct SP estimation can help go past the theoretical limitations that motif estimation faces when performed through subgraph counting.
Recent advances in graph databases (GDBs) have been driving interest in large-scale analytics, yet current systems fail to support higher-order (HO) interactions beyond first-order (one-hop) relations, which are crucial for tasks such as subgraph counting, polyadic modeling, and HO graph learning. We address this by introducing a new class of systems, higher-order graph databases (HO-GDBs) that use lifting and lowering paradigms to seamlessly extend traditional GDBs with HO. We provide a theoretical analysis of OLTP and OLAP queries, ensuring correctness, scalability, and ACID compliance. We implement a lightweight, modular, and parallelizable HO-GDB prototype that offers native support for hypergraphs, node-tuples, subgraphs, and other HO structures under a unified API. The prototype scales to large HO OLTP & OLAP workloads and shows how HO improves analytical tasks, for example enhancing accuracy of graph neural networks within a GDB by 44%. Our work ensures low latency and high query throughput, and generalizes both ACID-compliant and eventually consistent systems.
Subgraph counting the task of determining the number of instances of a query pattern within a large graph lies at the heart of many critical applications, from analyzing financial networks and transportation systems to understanding biological interactions. Despite decades of work yielding efficient algorithmic (AL) solutions and, more recently, machine learning (ML) approaches, a clear comparative understanding is elusive. This gap stems from the absence of a unified evaluation framework, standardized datasets, and accessible ground truths, all of which hinder systematic analysis and fair benchmarking. To overcome these barriers, we introduce BEACON: a comprehensive benchmark designed to rigorously evaluate both AL and ML-based subgraph counting methods. BEACON provides a standardized dataset with verified ground truths, an integrated evaluation environment, and a public leaderboard, enabling reproducible and transparent comparisons across diverse approaches. Our extensive experiments reveal that while AL methods excel in efficiently counting subgraphs on very large graphs, they struggle with complex patterns (e.g., those exceeding six nodes). In contrast, ML methods are capable of handling larger patterns but demand massive graph data inputs and often yield suboptimal accuracy on small, dense graphs. These insights not only highlight the unique strengths and limitations of each approach but also pave the way for future advancements in subgraph counting techniques. Overall, BEACON represents a significant step towards unifying and accelerating research in subgraph counting, encouraging innovative solutions and fostering a deeper understanding of the trade-offs between algorithmic and machine learning paradigms.
Graph spectra are an important class of structural features on graphs that have shown promising results in enhancing Graph Neural Networks (GNNs). Despite their widespread practical use, the theoretical understanding of the power of spectral invariants -- particularly their contribution to GNNs -- remains incomplete. In this paper, we address this fundamental question through the lens of homomorphism expressivity, providing a comprehensive and quantitative analysis of the expressive power of spectral invariants. Specifically, we prove that spectral invariant GNNs can homomorphism-count exactly a class of specific tree-like graphs which we refer to as parallel trees. We highlight the significance of this result in various contexts, including establishing a quantitative expressiveness hierarchy across different architectural variants, offering insights into the impact of GNN depth, and understanding the subgraph counting capabilities of spectral invariant GNNs. In particular, our results significantly extend Arvind et al. (2024) and settle their open questions. Finally, we generalize our analysis to higher-order GNNs and answer an open question raised by Zhang et al. (2024).
Graphs are essential data structures for modeling complex interactions in domains such as social networks, molecular structures, and biological systems. Graph-level tasks, which predict properties or classes for the entire graph, are critical for applications, such as molecular property prediction and subgraph counting. Graph Neural Networks (GNNs) have shown promise in these tasks, but their evaluations are often limited to narrow datasets, tasks, and inconsistent experimental setups, restricting their generalizability. To address these limitations, we propose a unified evaluation framework for graph-level GNNs. This framework provides a standardized setting to evaluate GNNs across diverse datasets, various graph tasks (e.g., graph classification and regression), and challenging scenarios, including noisy, imbalanced, and few-shot graphs. Additionally, we propose a novel GNN model with enhanced expressivity and generalization capabilities. Specifically, we enhance the expressivity of GNNs through a $k$-path rooted subgraph approach, enabling the model to effectively count subgraphs (e.g., paths and cycles). Moreover, we introduce a unified graph contrastive learning algorithm for graphs across diverse domains, which adaptively removes unimportant edges to augment graphs, thereby significantly improving generalization performance. Extensive experiments demonstrate that our model achieves superior performance against fourteen effective baselines across twenty-seven graph datasets, establishing it as a robust and generalizable model for graph-level tasks.
Motif discovery is a powerful and insightful method to quantify network structures and explore their function. As a case study, we present a comprehensive analysis of regulatory motifs in the connectome of the model organism Caenorhabditis elegans (C. elegans). Leveraging the Efficient Subgraph Counting Algorithmic PackagE (ESCAPE) algorithm, we identify network motifs in the multi-layer nervous system of C. elegans and link them to functional circuits. We further investigate motif enrichment within signal pathways and benchmark our findings with random networks of similar size and link density. Our findings provide valuable insights into the organization of the nerve net of this well documented organism and can be easily transferred to other species and disciplines alike.
Identifying frequent subgraphs, also called network motifs, is crucial in analyzing and predicting properties of real-world networks. However, finding large commonly-occurring motifs remains a challenging problem not only due to its NP-hard subroutine of subgraph counting, but also the exponential growth of the number of possible subgraphs patterns. Here we present Subgraph Pattern Miner (SPMiner), a novel neural approach for approximately finding frequent subgraphs in a large target graph. SPMiner combines graph neural networks, order embedding space, and an efficient search strategy to identify network subgraph patterns that appear most frequently in the target graph. SPMiner first decomposes the target graph into many overlapping subgraphs and then encodes each subgraph into an order embedding space. SPMiner then uses a monotonic walk in the order embedding space to identify frequent motifs. Compared to existing approaches and possible neural alternatives, SPMiner is more accurate, faster, and more scalable. For 5- and 6-node motifs, we show that SPMiner can almost perfectly identify the most frequent motifs while being 100x faster than exact enumeration methods. In addition, SPMiner can also reliably identify frequent 10-node motifs, which is well beyond the size limit of exact enumeration approaches. And last, we show that SPMiner can find large up to 20 node motifs with 10-100x higher frequency than those found by current approximate methods.
Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a unified framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric.
We suggest the use of hash functions to cut down the communication costs when counting subgraphs under edge local differential privacy. While various algorithms exist for computing graph statistics, including the count of subgraphs, under the edge local differential privacy, many suffer with high communication costs, making them less efficient for large graphs. Though data compression is a typical approach in differential privacy, its application in local differential privacy requires a form of compression that every node can reproduce. In our study, we introduce linear congruence hashing. With a sampling rate of $s$, our method can cut communication costs by a factor of $s^2$, albeit at the cost of increasing variance in the published graph statistic by a factor of $s$. The experimental results indicate that, when matched for communication costs, our method achieves a reduction in the $\ell_2$-error for triangle counts by up to 1000 times compared to the performance of leading algorithms.
Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the $k$-dimensional Weisfeiler-Leman ($k$WL) test, a widely-recognized method for verifying graph isomorphism. This connection has reignited interest in comprehending the specific graph properties effectively distinguishable by the $k$WL test. A central focus of research in this field revolves around determining the least dimensionality $k$, for which $k$WL can discern graphs with different number of occurrences of a pattern graph $P$. We refer to such a least $k$ as the WL-dimension of this pattern counting problem. This inquiry traditionally delves into two distinct counting problems related to patterns: subgraph counting and induced subgraph counting. Intriguingly, despite their initial appearance as separate challenges with seemingly divergent approaches, both of these problems are interconnected components of a more comprehensive problem: "graph motif parameters". In this paper, we provide a precise characterization of the WL-dimension of labeled graph motif parameters. As specific instances of this result, we obtain characterizations of the WL-dimension of the subgraph counting and induced subgraph counting problem for every labeled pattern $P$. We additionally demonstrate that in cases where the $k$WL test distinguishes between graphs with varying occurrences of a pattern $P$, the exact number of occurrences of $P$ can be computed uniformly using only local information of the last layer of a corresponding GNN. We finally delve into the challenge of recognizing the WL-dimension of various graph parameters. We give a polynomial time algorithm for determining the WL-dimension of the subgraph counting problem for given pattern $P$, answering an open question from previous work.
Graph neural networks (GNNs) are effective machine learning models for many graph-related applications. Despite their empirical success, many research efforts focus on the theoretical limitations of GNNs, i.e., the GNNs expressive power. Early works in this domain mainly focus on studying the graph isomorphism recognition ability of GNNs, and recent works try to leverage the properties such as subgraph counting and connectivity learning to characterize the expressive power of GNNs, which are more practical and closer to real-world. However, no survey papers and open-source repositories comprehensively summarize and discuss models in this important direction. To fill the gap, we conduct a first survey for models for enhancing expressive power under different forms of definition. Concretely, the models are reviewed based on three categories, i.e., Graph feature enhancement, Graph topology enhancement, and GNNs architecture enhancement.
We introduce DeSCo, a scalable neural deep subgraph counting pipeline, designed to accurately predict both the count and occurrence position of queries on target graphs post single training. Firstly, DeSCo uses a novel canonical partition and divides the large target graph into small neighborhood graphs, greatly reducing the count variation while guaranteeing no missing or double-counting. Secondly, neighborhood counting uses an expressive subgraph-based heterogeneous graph neural network to accurately count in each neighborhood. Finally, gossip propagation propagates neighborhood counts with learnable gates to harness the inductive biases of motif counts. DeSCo is evaluated on eight real-world datasets from various domains. It outperforms state-of-the-art neural methods with 137x improvement in the mean squared error of count prediction, while maintaining the polynomial runtime complexity. Our open source project is at https://github.com/fuvty/DeSCo.
We perform the first adversarial robustness study into Graph Neural Networks (GNNs) that are provably more powerful than traditional Message Passing Neural Networks (MPNNs). In particular, we use adversarial robustness as a tool to uncover a significant gap between their theoretically possible and empirically achieved expressive power. To do so, we focus on the ability of GNNs to count specific subgraph patterns, which is an established measure of expressivity, and extend the concept of adversarial robustness to this task. Based on this, we develop efficient adversarial attacks for subgraph counting and show that more powerful GNNs fail to generalize even to small perturbations to the graph's structure. Expanding on this, we show that such architectures also fail to count substructures on out-of-distribution graphs.
In this paper, we propose localized versions of Weisfeiler-Leman (WL) algorithms in an effort to both increase the expressivity, as well as decrease the computational overhead. We focus on the specific problem of subgraph counting and give localized versions of $k-$WL for any $k$. We analyze the power of Local $k-$WL and prove that it is more expressive than $k-$WL and at most as expressive as $(k+1)-$WL. We give a characterization of patterns whose count as a subgraph and induced subgraph are invariant if two graphs are Local $k-$WL equivalent. We also introduce two variants of $k-$WL: Layer $k-$WL and recursive $k-$WL. These methods are more time and space efficient than applying $k-$WL on the whole graph. We also propose a fragmentation technique that guarantees the exact count of all induced subgraphs of size at most 4 using just $1-$WL. The same idea can be extended further for larger patterns using $k>1$. We also compare the expressive power of Local $k-$WL with other GNN hierarchies and show that given a bound on the time-complexity, our methods are more expressive than the ones mentioned in Papp and Wattenhofer[2022a].
As the popularity of graph data increases, there is a growing need to count the occurrences of subgraph patterns of interest, for a variety of applications. Many graphs are massive in scale and also fully dynamic (with insertions and deletions of edges), rendering exact computation of these counts to be infeasible. Common practice is, instead, to use a small set of edges as a sample to estimate the counts. Existing sampling algorithms for fully dynamic graphs sample the edges with uniform probability. In this paper, we show that we can do much better if we sample edges based on their individual properties. Specifically, we propose a weighted sampling algorithm called WSD for estimating the subgraph count in a fully dynamic graph stream, which samples the edges based on their weights that indicate their importance and reflect their properties. We determine the weights of edges in a data-driven fashion, using a novel method based on reinforcement learning. We conduct extensive experiments to verify that our technique can produce estimates with smaller errors while often running faster compared with existing algorithms.
A wide range of models have been proposed for Graph Generative Models, necessitating effective methods to evaluate their quality. So far, most techniques use either traditional metrics based on subgraph counting, or the representations of randomly initialized Graph Neural Networks (GNNs). We propose using representations from contrastively trained GNNs, rather than random GNNs, and show this gives more reliable evaluation metrics. Neither traditional approaches nor GNN-based approaches dominate the other, however: we give examples of graphs that each approach is unable to distinguish. We demonstrate that Graph Substructure Networks (GSNs), which in a way combine both approaches, are better at distinguishing the distances between graph datasets.
Graph Neural Networks (GNNs) are recently proposed neural network structures for the processing of graph-structured data. Due to their employed neighbor aggregation strategy, existing GNNs focus on capturing node-level information and neglect high-level information. Existing GNNs therefore suffer from representational limitations caused by the Local Permutation Invariance (LPI) problem. To overcome these limitations and enrich the features captured by GNNs, we propose a novel GNN framework, referred to as the Two-level GNN (TL-GNN). This merges subgraph-level information with node-level information. Moreover, we provide a mathematical analysis of the LPI problem which demonstrates that subgraph-level information is beneficial to overcoming the problems associated with LPI. A subgraph counting method based on the dynamic programming algorithm is also proposed, and this has time complexity is O(n^3), n is the number of nodes of a graph. Experiments show that TL-GNN outperforms existing GNNs and achieves state-of-the-art performance.