Subgraph Counts Research Articles

Normal approximation of subgraph counts in the random-connection model

Normal approximation of subgraph counts in the random-connection model

Approximation of subgraph counts in the uniform attachment model

Abstract We use Stein’s method to obtain distributional approximations of subgraph counts in the uniform attachment model or random directed acyclic graph; we provide also estimates of rates of convergence. In particular, we give uni- and multi-variate Poisson approximations to the counts of cycles and normal approximations to the counts of unicyclic subgraphs; we also give a partial result for the counts of trees. We further find a class of multicyclic graphs whose subgraph counts are a.s. bounded as $n\to \infty$ .

Edge Deletion based Subgraph Hiding

Extracting subgraphs from graph data is a challenging and important subgraph mining task since they reveal valuable insights in many domains. However, in the data sharing scenario, some of the subgraphs might be considered as sensitive by the data owner and require hiding before publishing the data. Therefore, subgraph hiding is applied to the data so that when subgraph mining algorithms, such as frequent subgraph mining, subgraph counting, or subgraph matching, are executed on this published data, sensitive subgraphs will not appear. While protecting the privacy of the sensitive subgraphs through hiding, the side effects should be kept at a minimum. In this paper, we address the problem of hiding sensitive subgraphs on graph data and propose an Edge deletion-based heuristic (EDH) algorithm. We evaluate our algorithm using three graph datasets and compare the results with the previous vertex masking heuristic algorithms in terms of execution time and side effects in the context of frequent subgraph hiding. The experimental results demonstrate that the EDH is competitive concerning execution time and outperforms the existing masking heuristic algorithms in terms of side effects by reducing information loss of non-sensitive patterns significantly and not creating fake patterns.

Two-Level Graph Neural Network.

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 the time complexity of O(n3) , where n is the number of nodes of a graph. Experiments show that TL-GNN outperforms existing GNNs and achieves state-of-the-art performance.

GSWORD: GPU-accelerated Sampling for Subgraph Counting

Subgraph counting is a fundamental component for many downstream applications such as graph representation learning and query optimization.Since obtaining the exact count is often intractable,there have been a plethora of approximation methods on graph sampling techniques. Nonetheless, the state-of-the-art sampling methods still require massive samples to produce accurate approximations on large data graphs.We propose gSWORD, a GPU framework that leverages the massive parallelism of GPUs to accelerate iterative sampling algorithms for subgraph counting. Despite the embarrassingly parallel nature of the samples, there are unique challenges in accelerating subgraph counting due to its irregular computation logic. To address these challenges, we introduce two GPU-centric optimizations: (1) sample inheritance, enabling threads to inherit samples from neighboring threads to avoid idling, and (2) warp streaming, effectively distributing workloads among threads through a streaming process. Moreover, we propose a CPU-GPU co-processing pipeline that overlaps the sampling and enumeration processes to mitigate the underestimation issue. Experimental results demonstrate that deploying state-of-the-art sampling algorithms on gSWORD can perform millions of samples per second. The co-processing pipeline substantially improves the estimation accuracy in the cases where existing methods encounter severe underestimations with negligible overhead.

Central limit theorems for local network statistics

Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model that generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.

Application of Machine Learning Methods to Predict the Air Half-Lives of Persistent Organic Pollutants.

Persistent organic pollutants (POPs) are ubiquitous and bioaccumulative, posing potential and long-term threats to human health and the ecological environment. Quantitative structure-activity relationship (QSAR) studies play a guiding role in analyzing the toxicity and environmental fate of different organic pollutants. In the current work, five molecular descriptors are utilized to construct QSAR models for predicting the mean and maximum air half-lives of POPs, including specifically the energy of the highest occupied molecular orbital (HOMO_Energy_DMol3), a component of the dipole moment along the z-axis (Dipole_Z), fragment contribution to SAscore (SAscore_Fragments), subgraph counts (SC_3_P), and structural information content (SIC). The QSAR models were achieved through the application of three machine learning methods: partial least squares (PLS), multiple linear regression (MLR), and genetic function approximation (GFA). The determination coefficients (R2) and relative errors (RE) for the mean air half-life of each model are 0.916 and 3.489% (PLS), 0.939 and 5.048% (MLR), 0.938 and 5.131% (GFA), respectively. Similarly, the determination coefficients (R2) and RE for the maximum air half-life of each model are 0.915 and 5.629% (PLS), 0.940 and 10.090% (MLR), 0.939 and 11.172% (GFA), respectively. Furthermore, the mechanisms that elucidate the significant factors impacting the air half-lives of POPs have been explored. The three regression models show good predictive and extrapolation abilities for POPs within the application domain.

MOSER: Scalable Network Motif Discovery Using Serial Test

Given a graph G , a motif (e.g., 3-node clique) is a fundamental building block for G. Recently, motif-based graph analysis has attracted much attention due to its efficacy in tasks such as clustering, ranking, and link prediction. These tasks require Network Motif Discovery (NMD) at the early stage to identify the motifs of G. However, existing NMD solutions have two drawbacks: (1) Lack of theoretical guarantees on the quality of the samples generated using the existing methods, and (2) inefficient algorithms, which are not scalable for large graphs. These limitations hinder the exploration of motifs for analyzing large graphs. To address the above issues, we propose a novel solution named MOSER ( MO tif Discovery using SER ial Test). This novel NMD framework leverages a significance testing method known as the serial test, which differs from the existing solutions. We further propose two fast incremental subgraph counting algorithms, allowing MOSER to scale to larger graphs than ever possible before. Extensive experimental results show that using MOSER can improve the state-of-the-art up to 5 orders of magnitude in efficiency and that the motifs found by MOSER facilitate downstream tasks such as link prediction.

Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

AbstractReal networks often exhibit clustering, the tendency to form relatively small groups of nodes with high edge densities. This clustering property can cause large numbers of small and dense subgraphs to emerge in otherwise sparse networks. Subgraph counts are an important and commonly used source of information about the network structure and function. We study probability distributions of subgraph counts in a community affiliation graph. This is a random graph generated as an overlay of m partly overlapping independent Bernoulli random graphs (layers) $G_1,\dots,G_m$ with variable sizes and densities. The model is parameterised by a joint distribution of layer sizes and densities. When m grows linearly in the number of nodes n, the model generates sparse random graphs with a rich statistical structure, admitting a nonvanishing clustering coefficient and a power-law limiting degree distribution. In this paper we establish the normal and $\alpha$ -stable approximations to the numbers of small cliques, cycles, and more general 2-connected subgraphs of a community affiliation graph.

Kolmogorov bounds for decomposable random variables and subgraph counting by the Stein–Tikhomirov method

We derive normal approximation bounds in the Kolmogorov distance for random variables possessing decompositions of Barbour, Karoński, and Ruciński (J. Combin. Theory Ser. B 47 (1989) 125–145). We highlight the example of standardized subgraph counts in the Erdős–Rényi random graph. We prove a bound by generalizing the argumentation of Röllin (Probab. Engrg. Inform. Sci. (2022) 747–773), who used the Stein–Tikhomirov method to prove a bound in the special case of standardized triangle counts. Our bounds match the best available Wasserstein bounds.

On sparsity, power-law, and clustering properties of graphex processes

AbstractThis paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes, and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse, almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that, under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for subgraph counts and for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph.

Limit theory of sparse random geometric graphs in high dimensions

We study topological and geometric functionals of l∞-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.

Estimating Global Subgraph Counts by Sampling

We give a simple proof of a generalization of an inequality for homomorphism counts by Sidorenko (1994). A special case of our inequality says that if $d_v$ denotes the degree of a vertex $v$ in a graph $G$ and $\textrm{Hom}_\Delta(H,G)$ denotes the number of homomorphisms from a connected graph $H$ on $h$ vertices to $G$ which map a particular vertex of $H$ to a vertex $v$ in $G$ with $d_v \ge \Delta$, then $\textrm{Hom}_\Delta(H,G) \le \sum_{v\in G} d_v^{h-1}\mathbf{1}_{d_v\ge \Delta}$.
 We use this inequality to study the minimum sample size needed to estimate the number of copies of $H$ in $G$ by sampling vertices of $G$ at random.

Novel FDP mechanisms for releasing bipartite graph data on fixed and infinite intervals

As data collection increases, more and more sensitive data is being used to publish query results. This creates a significant risk of privacy disclosure. As a mathematically provable privacy theory, differential privacy (DP) provides a tool to resist background knowledge attacks. Fuzzy differential privacy (FDP) generalizes differential privacy by employing smaller sensitivity and supporting multiple similarity measures. Thus the output error can be reduced under FDP. Existing FDP mechanisms employ sliding window strategy, which perturb the true query value to an interval with this value as the midpoint to maintain similarity of outputs from neighboring datasets. It is still possible for an attacker to infer some sensitive information based on the difference between the left and right endpoints of the output range. To address this issue, this article present two solutions: fixed interval perturbation and infinite interval perturbation. These strategies perturb the true query values of two neighboring datasets to the same interval and provide fuzzy differential privacy protection for the dataset. We apply the proposed method to the privacy-preserving problem of bipartite graph subgraph counting and verify the effectiveness by experiments.

Learned sketch for subgraph counting: a holistic approach

Learned sketch for subgraph counting: a holistic approach

Fuzzy Differential Privacy Theory and Its Applications in Subgraph Counting

Transportation networks are essential to the operation of societies and economies. Protecting the privacy of sensitive information is a meaningful conception in sustainable transport when mining the transportation data. In data mining, differential privacy (DP) has provable privacy guarantees for releasing sensitive data by introducing randomness into query results. However, it suffers from significant accuracy loss of outputs when the query has high sensitivity (e.g., triangle counting). The reason is that the range of random perturbation to each query result in DP is too large. It consists of all possible output values for a query that forms a large or even unbounded interval. However, when impose perturbation only in a small neighborhood of the true query result, the similarity measure based on randomness in DP fails. Thereupon, we introduce fuzziness into DP to formulate new models which have smaller disturbance via fuzzy similarity measures. In this article, we establish a novel and general theory of private data analysis, fuzzy differential privacy (FDP). The new theory FDP aims to acquire a more flexible tradeoff between the accuracy of outputs and the privacy-preserving level of data. FDP combines DP with fuzzy set theory by introducing fuzziness into the query results and characterizing similarities between outputs via multiple fuzzy similarity measures. From this perspective, DP can be viewed as a special case of FDP with probabilistic similarity measure. Compared with DP, FDP has three superiorities: 1) most fuzzy similarity measures in FDP support sliding window perturbation strategies we proposed, which refer to perturbation in a small neighborhood of the query results; 2) FDP adds noise to the query results only according to a fraction of all possible neighboring datasets; and 3) the fuzzy similarity with valued in [0,1] quantifies the privacy protection level intuitively. These three points enable more accurate outputs while providing provable and intuitive privacy guarantees. As for subgraph counting, the state-of-the-art method is ladder framework in DP. We illustrate FDP mechanisms by applying them to a common application in subgraph counting–triangle/4-cliques counting. Experiments show that FDP is effective and efficient with smaller output errors than DP.

Counting Homomorphic Cycles in Degenerate Graphs

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C 2k and C 2k+1 in n -vertex graphs of bounded degeneracy in time Õ( n d k ), where the fastest known algorithm for detecting directed copies of C k in general m -edge digraphs runs in time Õ( m d k ). Conversely, one can transform any O(n b k ) algorithm for computing the number of homomorphic copies of C 2k or of C 2k+1 in n -vertex graphs of bounded degeneracy, into an Õ( m b k ) time algorithm for detecting directed copies of C k in general m -edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C k -homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k -cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

Normal approximation via non-linear exchangeable pairs

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.

Fluctuations of subgraph counts in graphon based random graphs

AbstractGiven a graphon $W$ and a finite simple graph $H$ , with vertex set $V(H)$ , denote by $X_n(H, W)$ the number of copies of $H$ in a $W$ -random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$ . Towards this we introduce a notion of $H$ -regularity of graphons and show that if the graphon $W$ is not $H$ -regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$ . On the other hand, if $W$ is $H$ -regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$ . Our proofs use the asymptotic theory of generalised $U$ -statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$ -regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$ -regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$ . We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.

BFS-based distributed algorithm for parallel local-directed subgraph enumeration

Abstract Estimating the frequency of subgraphs is of importance for many tasks, including subgraph isomorphism, kernel-based anomaly detection and network structure analysis. While multiple algorithms were proposed for full enumeration or sampling-based estimates, these methods fail in very large graphs. Recent advances in parallelization allow for estimates of total subgraph counts in very large graphs. The task of counting the frequency of each subgraph associated with each vertex also received excellent solutions for undirected graphs. However, there is currently no good solution for very large directed graphs. We here propose VDMC (Vertex specific Distributed Motif Counting)—a fully distributed algorithm to optimally count all the three and four vertices connected directed graphs (network motifs) associated with each vertex of a graph. VDMC counts each motif only once and its efficiency is linear in the number of counted motifs. It is fully parallelized to be efficient in GPU-based computation. VDMC is based on three main elements: (1) Ordering the vertices and only counting motifs containing increasing order vertices; (2) sub-ordering motifs based on the average depth of the tree spanning them via a BFS traversal; and (3) removing isomorphisms only once for the entire graph. We here compare VDMC to analytical estimates of the expected number of motifs in Erdős–Rényi graphs and show its accuracy. VDMC is available as a highly efficient CPU and GPU code with a novel data structure for efficient graph manipulation. We show the efficacy of VDMC on real-world graphs. VDMC allows for the precise analysis of subgraph frequency around each vertex in large graphs and opens the way for the extension of methods until now limited to graphs of thousands of edges to graphs with millions of edges and above. GIT: https://github.com/louzounlab/graph-measures/ PyPI: https://pypi.org/project/graph-measures/