Subgraph Counts Research Articles

On sampling from massive graph streams

We propose Graph Priority Sampling ( gps ), a new paradigm for order-based reservoir sampling from massive graph streams. gps provides a general way to weight edge sampling according to auxiliary and/or size variables so as to accomplish various estimation goals of graph properties. In the context of subgraph counting, we show how edge sampling weights can be chosen so as to minimize the estimation variance of counts of specified sets of subgraphs. In distinction with many prior graph sampling schemes, gps separates the functions of edge sampling and subgraph estimation. We propose two estimation frameworks: (1) Post-Stream estimation, to allow gps to construct a reference sample of edges to support retrospective graph queries, and (2) In-Stream estimation, to allow gps to obtain lower variance estimates by incrementally updating the subgraph count estimates during stream processing. Unbiasedness of subgraph estimators is established through a new Martingale formulation of graph stream order sampling, in which subgraph estimators, written as a product of constituent edge estimators, are unbiased, even when computed at different points in the stream. The separation of estimation and sampling enables significant resource savings relative to previous work. We illustrate our framework with applications to triangle and wedge counting. We perform a large-scale experimental study on real-world graphs from various domains and types. gps achieves high accuracy with < 1% error for triangle and wedge counting, while storing a small fraction of the graph with average update times of a few microseconds per edge. Notably, for billion-scale graphs, gps accurately estimates triangle and wedge counts with < 1% error, while storing a small fraction of < 0.01% of the total edges in the graph.

On Regularity Lemmas and their Algorithmic Applications

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation

A new Berry–Esseen bound for nonlinear functionals of nonsymmetric and nonhomogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin–Stein method and an analysis of the discrete Ornstein–Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős–Renyi random graph. A further application deals with a percolation problem on trees.

Universal limit theorems in graph coloring problems with connections to extremal combinatorics

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (nonnormal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erdős and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

Multivariate central limit theorems for Rademacher functionals with applications

Quantitative multivariate central limit theorems for general functionals of independent, possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincaré inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erdős-Rényi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application we prove a quantitative multivariate central limit theorem for vectors of intrinsic volumes induced by random cubical complexes.

Functional central limit theorem for subgraph counting processes

The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.

The parameterised complexity of counting even and odd induced subgraphs

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.

On the Lower Tail Variational Problem for Random Graphs

We study the lower tail large deviation problem for subgraph counts in a random graph. LetXHdenote the number of copies ofHin an Erdős–Rényi random graph$\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability$\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least forp≥n−αH(and conjecturally for a larger range ofp). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for everyH, and 0 < δ < δHfor some δH> 0, asp→ 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartiteHand δ close to 1.

Comparison of large networks with sub-sampling strategies.

Networks are routinely used to represent large data sets, making the comparison of networks a tantalizing research question in many areas. Techniques for such analysis vary from simply comparing network summary statistics to sophisticated but computationally expensive alignment-based approaches. Most existing methods either do not generalize well to different types of networks or do not provide a quantitative similarity score between networks. In contrast, alignment-free topology based network similarity scores empower us to analyse large sets of networks containing different types and sizes of data. Netdis is such a score that defines network similarity through the counts of small sub-graphs in the local neighbourhood of all nodes. Here, we introduce a sub-sampling procedure based on neighbourhoods which links naturally with the framework of network comparisons through local neighbourhood comparisons. Our theoretical arguments justify basing the Netdis statistic on a sample of similar-sized neighbourhoods. Our tests on empirical and synthetic datasets indicate that often only 10% of the neighbourhoods of a network suffice for optimal performance, leading to a drastic reduction in computational requirements. The sampling procedure is applicable even when only a small sample of the network is known, and thus provides a novel tool for network comparison of very large and potentially incomplete datasets.

Graph summarization with quality guarantees

We study the problem of graph summarization. Given a large graph we aim at producing a concise lossy representation (a summary) that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation. In this work we study a very natural type of summary: the original set of vertices is partitioned into a small number of supernodes connected by superedges to form a complete weighted graph. The superedge weights are the edge densities between vertices in the corresponding supernodes. To quantify the dissimilarity between the original graph and a summary, we adopt the reconstruction error and the cut-norm error. By exposing a connection between graph summarization and geometric clustering problems (i.e., k-means and k-median), we develop the first polynomial-time approximation algorithms to compute the best possible summary of a certain size under both measures. We discuss how to use our summaries to store a (lossy or lossless) compressed graph representation and to approximately answer a large class of queries about the original graph, including adjacency, degree, eigenvector centrality, and triangle and subgraph counting. Using the summary to answer queries is very efficient as the running time to compute the answer depends on the number of supernodes in the summary, rather than the number of nodes in the original graph.

Nonlinear large deviations

We present a general technique for computing large deviations of nonlinear functions of independent Bernoulli random variables. The method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs. Previous technology, based on Szemerédi's regularity lemma, works only for dense graphs. Applications are also made to exponential random graphs and three-term arithmetic progressions in random sets of integers.

A characterization of functions with vanishing averages over products of disjoint sets

Given α1,…,αm∈(0,1), we characterize all integrable functions f:[0,1]m→C satisfying ∫A1×⋯×Amf=0 for any collection of disjoint measurable sets A1,…,Am⊆[0,1] of respective measures α1,…,αm. We use this characterization to settle some of the conjectures in Janson and Sós (2015) about the relation between subgraph counts and quasi-randomness.

Poisson approximation of subgraph counts in stochastic block models and a graphon model

Small subgraph counts can be used as summary statistics for large random graphs. We use the Stein–Chen method to derive Poisson approximations for the distribution of the number of subgraphs in the stochastic block model which are isomorphic to some fixed graph. We also obtain Poisson approximations for subgraph counts in a graphon-type generalisation of the model in which the edge probabilities are (possibly dependent) random variables supported on a subset of [ 0,1 ]. Our results apply when the fixed graph is a member of the class of strictly balanced graphs.

A Network Formation Model Based on Subgraphs

We develop a new class of random-graph models for the statistical estimation of network formation that allow for substantial correlation in links. Various subgraphs (e.g., links, triangles, cliques, stars) are generated and their union results in a network. The challenge in estimating the frequencies with which subgraphs 'truly' form is that subgraphs can overlap and may also incidentally generate new subgraphs, and so the true rate of formation of the subgraphs cannot generally be inferred just by counting their presence in the resulting network. We provide estimation techniques for recovering the rates at which the underlying subgraphs were formed from the observation of a single (large) network. We provide results on identification of the true underlying rates of subgraph formation from various statistics, as well as a new Central Limit Theorem for correlated random variables that establishes asymptotic normality for our estimators. We also show that if the network is sparse enough then direct counts of subgraphs are consistent and asymptotically normal estimators. We illustrate the models with applications.

The lower tail: Poisson approximation revisited

The well-known “Janson's inequality” gives Poisson-like upper bounds for the lower tail probability when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ) of Janson, Łuczak and Ruciński. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 219–246, 2016

The challenges of unbounded treewidth in parameterised subgraph counting problems

Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area. Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of these subgraphs in some sense; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting problem. The paper also gives a thorough survey of known results on this subject and the methods used, as well as discussing the relationships both between multicolour and uncoloured versions of subgraph counting problems, and between exact counting, approximate counting and the corresponding decision problems.

Some Hard Families of Parameterized Counting Problems

We consider parameterized subgraph counting problems of the following form: given a graph G , how many k -tuples of its vertices induce a subgraph with a given property? A number of such problems are known to be #W[1]-complete; here, we substantially generalize some of these existing results by proving hardness for two large families of such problems. We demonstrate that it is #W[1]-hard to count the number of k -vertex subgraphs having any property where the number of distinct edge densities of labeled subgraphs that satisfy the property is o ( k 2 ). In the special case in which the property in question depends only on the number of edges in the subgraph, we give a strengthening of this result, which leads to our second family of hard problems.

Rand-FaSE: fast approximate subgraph census

Determining the frequency of small subgraphs is an important graph mining primitive. One major class of algorithms for this task is based upon the enumeration of all sets of $$k$$ connected nodes. These are known as network-centric algorithms. FAst Subgraph Enumeration (FaSE) is a exact algorithm for subgraph counting that contrasted with its past approaches by performing the isomorphism tests while doing the enumeration, encapsulating the topological information in a g-trie and thus largely reducing the number of required isomorphism tests. Our goal with this paper is to expand this approach by providing an approximate algorithm, which we called Rand-FaSE. It uses an unbiased sampling estimator for the number of subgraphs of each type, allowing an user to trade some accuracy for even faster execution times. We tested our algorithm on a set of representative complex networks, comparing it with the exact alternative, FaSE. We also do an extensive analysis by studying its accuracy and speed gains against previous sampling approaches. With all of this, we believe FaSE and Rand-FaSE pave the way for faster network-centric census algorithms.

Parallel color-coding

New shared- and distributed-memory parallelizations of the color-coding method.Bioinformatics applications: motif detection, signaling pathway discovery in protein interaction networks.Parallel performance analysis of optimized color-coding implementations. We present new parallelization and memory-reducing strategies for the graph-theoretic color-coding approximation technique, with applications to biological network analysis. Color-coding is a technique that gives fixed parameter tractable algorithms for several well-known NP-hard optimization problems. In this work, by efficiently parallelizing steps in color-coding, we create two new biological protein interaction network analysis tools: Fascia for subgraph counting and motif finding and FastPath for signaling pathway detection. We demonstrate considerable speedup over prior work, and the optimizations introduced in this paper can also be used for other problems where color-coding is applicable.

Stratified sampling for the Ising model: A graph-theoretic approach

We present a new approach to a classical problem in statistical physics: estimating the partition function and thermodynamic quantities of the ferromagnetic Ising model. The standard approach to this problem is to use Markov chain Monte Carlo methods that are based on the classic work of Metropolis et al. (1953). Although great improvements to these original ideas have been made, there remains scope for improvement. The first polynomial time algorithm for the estimation of the partition function was developed by Jerrum and Sinclair (1993), who reduced the problem to counting subgraphs via the high-temperature expansion. However, the polynomial bound achieved has large degree and so yields an algorithm that is too slow for practical use. Our approach, which also uses the high-temperature expansion, yields a broad class of Monte Carlo algorithms that are not based on the work of Metropolis et al., but instead use heuristic sampling techniques. In particular, we estimate coefficients of a polynomial that, once obtained, can be used to determine the quantities of interest at all temperatures simultaneously. This class of algorithms can be applied to any underlying graph, with or without an external field. These algorithms are also highly parallelizable, which, among other features, makes their implementation possible in practice.