Subgraph Counts Research Articles

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.

Nonlinear large deviations: Beyond the hypercube

By extending (Adv. Math. 299 (2016) 396–450), we present a framework to calculate large deviations for nonlinear functions of independent random variables supported on compact sets in Banach spaces. Previous research on nonlinear large deviations has only focused on random variables supported on $\{-1,+1\}^{n}$, and accordingly we build theory for random variables with general distributions, increasing flexibility in the applications. As examples, we compute the large deviation rate functions for monochromatic subgraph counts in edge-colored complete graphs, and for triangle counts in dense random graphs with continuous edge weights. Moreover, we verify the mean field approximation for a class of vector spin models.

Motif Discovery of Protein Protein Interaction using Minimum Spanning Tree

In protein Interaction Networks, counting subgraph is a tedious task. From the list of non induced occurrence of the subgraph, motif topology calculated by using Combi Motif and Slider techniques. But, this approach was taken more time to execute. To reduce the execution time, the minimum weight value between the nodes, the Minimum spanning tree concept proposed. Prim’s method implemented with the greedy technique (as Kruskal’s algorithm) to calculate the minimum path between the nodes in the Protein interaction network. This technique uses to compare the similarity of the minimum spanning tree approach. Initially, this algorithm has discovered the path then calculated the weight matrix and found the minimum weight value. From the computational experiments, the proposed approach of MST providing better results in terms of time consumption and accuracy to count the motif pattern in the network of the interacted proteins.

Degeneracy in sparse ERGMs with functions of degrees as sufficient statistics

A sufficient criterion for “non-degeneracy” is given for Exponential Random Graph Models on sparse graphs with sufficient statistics which are functions of the degree sequence. This criterion explains why statistics such as alternating $k$-star are non-degenerate, whereas subgraph counts are degenerate. It is further shown that this criterion is “almost” tight. Existence of consistent estimates is then proved for non-degenerate Exponential Random Graph Models.

Bounds on the Spectral Radius of Digraphs from Subgraph Counts

The spectral radius of a directed graph is a metric that can only be computed when the structure of the network is completely known. However, in many practical scenarios, it is not possible to exactly retrieve the whole structure of the network; hence, the exact value of the spectral radius is not computable. Even in these scenarios, it is typically possible to extract local structural properties of a network using, for example, graph crawlers. In this paper, we develop a novel measure-theoretic framework to upper and lower bound the spectral radius of a directed graph using local structural information, in particular, using the counts of a collection of small subgraphs or motifs. Our framework is based on recent results relating the multivariate moment problem with semidefinite programming. Using these results, we develop a hierarchy of (small) semidefinite programs whose solutions provide upper and lower bounds on the spectral radius of a directed graph using, solely, subgraph and motif counts. We numerically validate the quality of our bounds using both random and real-world directed graphs.

Normal approximation for sums of weighted $U$-statistics – application to Kolmogorov bounds in random subgraph counting

We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and weighted $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraph counts in the Erdős–Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering recent results obtained for triangles and improving other bounds in the Wasserstein distance.

Poisson approximation of induced subgraph counts in an inhomogeneous random intersection graph model

In this paper, we consider a class of inhomogeneous random intersection graphs by assigning random weight to each vertex and two vertices are adjacent if they choose some common elements. In the inhomogeneous random intersection graph model, vertices with larger weights are more likely to acquire many elements. We show the Poisson convergence of the number of induced copies of a fixed subgraph as the number of vertices n and the number of elements m, scaling as m=⌊βnα⌋ (α,β>0), tend to infinity.

Subgraphs in preferential attachment models

AbstractWe consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$ . For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

The connected partition lattice of a graph and the reconstruction conjecture

AbstractPreviously we showed that many invariants of a graph can be computed from its abstract induced subgraph poset, which is the isomorphism class of the induced subgraph poset, suitably weighted by subgraph counting numbers. In this paper, we study the abstract bond lattice of a graph, which is the isomorphism class of the lattice of distinct unlabelled connected partitions of a graph, suitably weighted by subgraph counting numbers. We show that these two abstract posets can be constructed from each other except in a few trivial cases. The constructions rely on certain generalisations of a lemma of Kocay in graph reconstruction theory to abstract induced subgraph posets. As a corollary, trees are reconstructible from their abstract bond lattice. We show that the chromatic symmetric function and the symmetric Tutte polynomial of a graph can be computed from its abstract induced subgraph poset. Stanley has asked if every tree is determined up to isomorphism by its chromatic symmetric function. We prove a counting lemma, and indicate future directions for a study of Stanley's question.

A modified Grassmann algebra approach to theorems on permanents and determinants

Abdesselam [1] showed how the techniques of Grassmann-Berezin calculus can be used to prove the all-minors matrix-tree theorem. Starting with any matrix A we associate a directed graph G to A equipped with two different weight functions on the edges. We extend the approach of Abdesselam to give interpretations for the permanents and determinants of all square sub-matrices of A in terms of weighted counts of sub-graphs of G using each of the edge-weight functions. We introduce a modification of Grassmann-Berezin variables to obtain our results.

F98. Counts of Small Subgraphs Within the Resting Functional Connectome are Parsimonious, Stable and Individualized Features in Healthy as Well as Disordered Mood

F98. Counts of Small Subgraphs Within the Resting Functional Connectome are Parsimonious, Stable and Individualized Features in Healthy as Well as Disordered Mood

Wasserstein-2 bounds in normal approximation under local dependence

We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erd\H{o}s-R\'enyi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.

Edge Correlations in Random Regular Hypergraphs and Applications to Subgraph Testing

Compared to the classical binomial random (hyper)graph model, the study of random regular hypergraphs is made more challenging due to correlations between the occurrence of different edges. We develop an edge-switching technique for hypergraphs which allows us to show that these correlations are limited for a large range of densities. This extends some previous results of Kim, Sudakov, and Vu for graphs. From our results we deduce several corollaries on subgraph counts in random $d$-regular hypergraphs. We also prove a conjecture of Dudek, Frieze, Ruciński, and Šileikis on the threshold for the existence of an $\ell$-overlapping Hamilton cycle in a random $d$-regular $r$-graph. Moreover, we apply our results to prove bounds on the query complexity of testing subgraph-freeness. The problem of testing subgraph-freeness in the general graphs model was first studied by Alon, Kaufman, Krivelevich, and Ron, who obtained several bounds on the query complexity of testing triangle-freeness. We extend some of these previous results beyond the triangle setting and to the hypergraph setting.

Quantitative CLTs for symmetric $U$-statistics using contractions

We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of contraction operators. Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called ‘dense parameter regime’ for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).

Time Windowed Data Structures for Graphs

We present data structures that can answer time windowed queries for a set of timestamped events in a relational event graph. We study the relational event graph as input to solve (a) time windowed decision problems for monotone graph properties, such as disconnectedness and bipartiteness, and (b) time windowed reporting problems such as reporting the minimum spanning tree, the minimum time interval, and the graph edit distance for obtaining spanning forests. We also present results of window queries for counting subgraphs of a given pattern, such as paths of length 2 (in general graphs) and paths of length 3 (in bipartite graphs), quadrangles and complete subgraphs of a fixed order or of all orders $\ell \geq 3$ (i.e., cliques of size $\ell$). These query results can be used to compute graph parameters that are important for social network analysis, e.g., clustering coefficients, embeddedness and neighborhood overlapping.

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

Abstract We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).

Assessment of model fit via network comparison methods based on subgraph counts

Assessment of model fit via network comparison methods based on subgraph counts

Mining Graphlet Counts in Online Social Networks

Counting subgraphs is a fundamental analysis task for online social networks (OSNs). Given the sheer size and restricted access of OSN, efficient computation of subgraph counts is highly challenging. Although a number of algorithms have been proposed to estimate the relative counts of subgraphs in OSNs with restricted access, there are only few works which try to solve a more general problem, i.e., counting subgraph frequencies. In this article, we propose an efficient random walk-based framework to estimate the subgraph counts. Our framework generates samples by leveraging consecutive steps of the random walk as well as by observing neighbors of visited nodes. Using the importance sampling technique, we derive unbiased estimators of the subgraph counts. To make better use of the degree information of visited nodes, we also design improved estimators, which increases the accuracy of the estimation with no additional cost. We conduct extensive experimental evaluation on real-world OSNs to confirm our theoretical claims. The experiment results show that our estimators are unbiased, accurate, efficient, and better than the state-of-the-art algorithms. For the Weibo graph with more than 58 million nodes, our method produces estimate of triangle count with an error less than 5% using only 20,000 sampled nodes. Detailed comparison with the state-of-the-art methods demonstrates that our algorithm is 2--10 times more accurate.

A large deviation principle for the Erdős–Rényi uniform random graph

Starting with the large deviation principle (LDP) for the Erdős–Renyi binomial random graph ${\mathcal G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph ${\mathcal G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in ${\mathcal G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.