Subgraph Counts Research Articles

Parameterized Counting of Partially Injective Homomorphisms

We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and #mathsf {W[1]}-complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.

A Survey on Subgraph Counting

Computing subgraph frequencies is a fundamental task that lies at the core of several network analysis methodologies, such as network motifs and graphlet-based metrics, which have been widely used to categorize and compare networks from multiple domains. Counting subgraphs is, however, computationally very expensive, and there has been a large body of work on efficient algorithms and strategies to make subgraph counting feasible for larger subgraphs and networks. This survey aims precisely to provide a comprehensive overview of the existing methods for subgraph counting. Our main contribution is a general and structured review of existing algorithms, classifying them on a set of key characteristics, highlighting their main similarities and differences. We identify and describe the main conceptual approaches, giving insight on their advantages and limitations, and we provide pointers to existing implementations. We initially focus on exact sequential algorithms, but we also do a thorough survey on approximate methodologies (with a trade-off between accuracy and execution time) and parallel strategies (that need to deal with an unbalanced search space).

Faster algorithms for counting subgraphs in sparse graphs

Given a k-node pattern graph H and an n-node host graph G, the subgraph counting problem asks to compute the number of copies of H in G. In this work we address the following question: can we count the copies of H faster if G is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs. This decomposition gives a dynamic program for counting the homomorphisms of H in G by exploiting the degeneracy of G, which allows us to beat the state-of-the-art subgraph counting algorithms when G is sparse enough. For example, we can count the induced copies of any k-node pattern H in time 2^{O(k^2)} O(n^{0.25k + 2} log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n^{0.625k + 2} log n) if G has bounded average degree. These bounds are instantiations of a more general result, parameterized by the degeneracy of G and the structure of H, which generalizes classic bounds on counting cliques and complete bipartite graphs. We also give lower bounds based on the Exponential Time Hypothesis, showing that our results are actually a characterization of the complexity of subgraph counting in bounded-degeneracy graphs.

A Harary-Sachs theorem for hypergraphs

We generalize the Harary-Sachs theorem to k-uniform hypergraphs: the codegree-d coefficient of the characteristic polynomial of a uniform hypergraph H can be expressed as a weighted sum of subgraph counts over certain multi-hypergraphs with d edges. We include a detailed description of the aforementioned multi-hypergraphs and a formula for their corresponding weights.

Higher-order fluctuations in dense random graph models

Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. We are interested in these statistics because they are key to understanding fluctuations of regular subgraph counts — a cornerstone of dense graph limit theory. We also identify the resulting limiting Gaussian stochastic measures by means of the theory of generalised U-statistics and Gaussian Hilbert spaces, which we think is a suitable framework to describe and understand higher-order fluctuations in dense random graph models. With this article, we believe we answer the question “What is the central limit theorem of dense graph limit theory?”. We complement the theory with some statistical applications to illustrate the use of centred subgraph counts in network modelling.

Applications of Differential Privacy in Social Network Analysis: A Survey

Differential privacy provides strong privacy preservation guarantee in information sharing. As social network analysis has been enjoying many applications, it opens a new arena for applications of differential privacy. This article presents a comprehensive survey connecting the basic principles of differential privacy and applications in social network analysis. We concisely review the foundations of differential privacy and the major variants. Then, we discuss how differential privacy is applied to social network analysis, including privacy attacks in social networks, models of differential privacy in social network analysis, and a series of popular tasks, such as analyzing degree distribution, counting subgraphs and assigning weights to edges. We also discuss a series of challenges for future work.

An Approximate Blow-up Lemma for Sparse Hypergraphs

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.

Stein’s method of exchangeable pairs in multivariate functional approximations

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.

Bootstrap With Cluster‐Dependence in Two or More Dimensions

We propose a bootstrap procedure for data that may exhibit cluster‐dependence in two or more dimensions. The asymptotic distribution of the sample mean or other statistics may be non‐Gaussian if observations are dependent but uncorrelated within clusters. We show that there exists no procedure for estimating the limiting distribution of the sample mean under two‐way clustering that achieves uniform consistency. However, we propose bootstrap procedures that achieve adaptivity with respect to different uniformity criteria. Important cases and extensions discussed in the paper include regression inference, U‐ and V‐statistics, subgraph counts for network data, and non‐exhaustive samples of matched data.

Parallelization of network motif discovery using star contraction

Network motifs are widely used to uncover structural design principles of complex networks. Current sequential network motif discovery algorithms become inefficient as motif size grows, thus parallelization methods have been proposed in the literature. In this study, we use star contraction algorithm to partition complex networks efficiently for parallel discovery of network motifs. We propose two new heuristics to make star contraction more suitable for partitioning of complex networks. The effectiveness of our partitioning strategies is verified using the ESU algorithm for subgraph counting. We also propose a ghost vertices detection algorithm to ensure that all the motifs located in multiple parts are exactly found. We implement our method using MPI libraries and tested on real-life complex networks of different domains. We compared speedups of star contraction algorithm with speedups of other graph partitioning algorithms. Our algorithm obtained better speedups than those of other partitioning algorithms for most cases. Our algorithm provides significant speedups when compared to sequential ESU algorithm allowing discovery of larger network motifs.

Subgraph counts for dense random graphs with specified degrees

AbstractWe prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d1, …, dn) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes dj= λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

Establish the expected number of induced motifs on unlabeled graphs through analytical models

Complex networks are usually characterized by the presence of small and recurrent patterns of interactions between nodes, called network motifs. These small modules can help to elucidate the structure and the functioning of complex systems. Assessing the statistical significance of a pattern as a motif in a network G is a time consuming task which entails the computation of the expected number of occurrences of the pattern in an ensemble of random graphs preserving some features of G, such as the degree distribution. Recently, few models have been devised to analytically compute expectations of the number of non-induced occurrences of a motif. Less attention has been payed to the harder analysis of induced motifs. Here, we illustrate an analytical model to derive the mean number of occurrences of an induced motif in an unlabeled network with respect to a random graph model. A comprehensive experimental analysis shows the effectiveness of our approach for the computation of the expected number of induced motifs up to 10 nodes. Finally, the proposed method is helpful when running subgraph counting algorithms to get the number of occurrences of a topology become unfeasible.

Distributed subgraph counting

In this paper, we study local subgraph counting, which is to count the occurrences of a user-given pattern graph p around every node v in a data graph G , when v matches to a given orbit o in p , where the orbit serves as a center to count p. In general, the orbit can be a node, an edge, or a set of nodes in p. Local subgraph counting has played an important role in characterizing high-order local structures that exhibit in a large graph, and has been widely used in denser and relevant communities mining, graphlet degree distribution, discriminative features selection for link prediction, relational classification and recommendation. In the literature, almost all the existing works support a k -node pattern graph, for k ≤ 5, with either 1 node orbit or 1 edge orbit. Their approaches are difficult to support larger k due to the fact that subgraph counting is to count by subgraph isomorphism. In this work, we develop a new general approach to count any k pattern graphs with any orbits selected. The key idea behind is that we do local subgraph counting by homomorphism counting, which can be solved by relational algebra using joins, group-by and aggregation. By homomorphism counting, we do local subgraph counting by eliminating counts for those that are not subgraph isomorphism matchings from the total count for any possible matchings. We have developed a distributed system named DISC on Spark. Our extensive experiments validate the efficiency of our approach by testing 114 local subgraph counting queries used in the existing work over real graphs, where no existing work can support all.

Large deviations of subgraph counts for sparse Erdős–Rényi graphs

For any fixed simple graph H=(V,E) and any fixed u>0, we establish the leading order of the exponential rate function for the probability that the number of copies of H in the Erdős–Rényi graph G(n,p) exceeds its expectation by a factor 1+u, assuming n−κ(H)≪p≪1, with κ(H)=1/(2Δ), where Δ≥1 is the maximum degree of H. This improves on a previous result of Chatterjee and the second author, who obtained κ(H)=c/(Δ|E|) for a constant c>0. Moreover, for the case of cycle counts we can take κ as large as 1/2. We additionally obtain the sharp upper tail for Schatten norms of the adjacency matrix, as well as the sharp lower tail for counts of graphs for which Sidorenko's conjecture holds. As a key step, we establish quantitative versions of Szemerédi's regularity lemma and the counting lemma, suitable for the analysis of random graphs in the large deviations regime.

Moderate deviations of subgraph counts in the Erdős-Rényi random graphs 𝐺(𝑛,𝑚) and 𝐺(𝑛,𝑝)

The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erdős-Rényi random graph G ( n , m ) G(n,m) . Our approach is based on applying Freedman’s inequalities for the probability of deviations of martingales to a martingale representation of subgraph count deviations. In addition, we prove that subgraph count deviations of different subgraphs are all linked, via the deviations of two specific graphs, the path of length two and the triangle. We also deduce new bounds for the related G ( n , p ) G(n,p) model.

On Weisfeiler-Leman invariance: Subgraph counts and related graph properties

The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a fruitful approach to the Graph Isomorphism problem. 2-WL corresponds to the original algorithm suggested by Weisfeiler and Leman over 50 years ago. 1-WL is the classical color refinement routine. Indistinguishability by k-WL is an equivalence relation on graphs that is of fundamental importance for isomorphism testing, descriptive complexity theory, and graph similarity testing which is also of some relevance in artificial intelligence. Focusing on dimensions k=1,2, we investigate subgraph patterns whose counts are k-WL invariant, and whose occurrence is k-WL invariant. We achieve a complete description of all such patterns for dimension k=1 and considerably extend the previous results known for k=2.

A refined Cramér-type moderate deviation for sums of local statistics

We prove a refined Cramér-type moderate deviation result by taking into account of the skewness in normal approximation for sums of local statistics of independent random variables. We apply the main result to $k$-runs, U-statistics and subgraph counts in the Erdős–Rényi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order tail probability expansions via Stein’s method.

Logarithmic Sobolev inequalities for finite spin systems and applications

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdős–Rényi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

Testing for Equivalence of Network Distribution Using Subgraph Counts

We consider that a network is an observation, and a collection of observed networks forms a sample. In this setting, we provide methods to test whether all observations in a network sample are drawn from a specified model. We achieve this by deriving the joint asymptotic properties of average subgraph counts as the number of observed networks increases but the number of nodes in each network remains finite. In doing so, we do not require that each observed network contains the same number of nodes, or is drawn from the same distribution. Our results yield joint confidence regions for subgraph counts, and therefore methods for testing whether the observations in a network sample are drawn from: a specified distribution, a specified model, or from the same model as another network sample. We present simulation experiments and an illustrative example on a sample of brain networks where we find that highly creative individuals’ brains present significantly more short cycles than found in less creative people. Supplementary materials for this article are available online.

Estimating the number of connected components in a graph via subgraph sampling

Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman (Ann. Math. Stat. 20 (1949) 572–579) and Frank (Scand. J. Stat. 5 (1978) 177–188). We revisit a problem formulated by Frank (Scand. J. Stat. 5 (1978) 177–188) of estimating the number of connected components in a large graph based on the subgraph sampling model, in which we randomly sample a subset of the vertices and observe the induced subgraph. The key question is whether accurate estimation is achievable in the sublinear regime where only a vanishing fraction of the vertices are sampled. We show that it is impossible if the parent graph is allowed to contain high-degree vertices or long induced cycles. For the class of chordal graphs, where induced cycles of length four or above are forbidden, we characterize the optimal sample complexity within constant factors and construct linear-time estimators that provably achieve these bounds. This significantly expands the scope of previous results which have focused on unbiased estimators and special classes of graphs such as forests or cliques. Both the construction and the analysis of the proposed methodology rely on combinatorial properties of chordal graphs and identities of induced subgraph counts. They, in turn, also play a key role in proving minimax lower bounds based on construction of random instances of graphs with matching structures of small subgraphs.