Subgraph Counts Research Articles

More on quasi-random graphs, subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases.The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.

The importance sampling technique for understanding rare events in Erdős–Rényi random graphs

In dense Erdős–Renyi random graphs, we are interested in the events where large numbers of a given subgraphs occur. The mean behaviour of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, what is the probability of an Erdős–Renyi graph containing an excessively large number of a given subgraph? Using the large deviation principle, we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erdős–Renyi graphs. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by the special choice of exponential random graph that is indistinguishable from the Erdős–Renyi graph conditioned to have many triangles. We show how this choice can be made for the conditioned Erdős–Renyi graphs both in the replica symmetric phase and also in parts of the replica breaking phase. Equally interestingly, we also show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme.

Motif-role-fingerprints: the building-blocks of motifs, clustering-coefficients and transitivities in directed networks.

Complex networks are frequently characterized by metrics for which particular subgraphs are counted. One statistic from this category, which we refer to as motif-role fingerprints, differs from global subgraph counts in that the number of subgraphs in which each node participates is counted. As with global subgraph counts, it can be important to distinguish between motif-role fingerprints that are ‘structural’ (induced subgraphs) and ‘functional’ (partial subgraphs). Here we show mathematically that a vector of all functional motif-role fingerprints can readily be obtained from an arbitrary directed adjacency matrix, and then converted to structural motif-role fingerprints by multiplying that vector by a specific invertible conversion matrix. This result demonstrates that a unique structural motif-role fingerprint exists for any given functional motif-role fingerprint. We demonstrate a similar result for the cases of functional and structural motif-fingerprints without node roles, and global subgraph counts that form the basis of standard motif analysis. We also explicitly highlight that motif-role fingerprints are elemental to several popular metrics for quantifying the subgraph structure of directed complex networks, including motif distributions, directed clustering coefficient, and transitivity. The relationships between each of these metrics and motif-role fingerprints also suggest new subtypes of directed clustering coefficients and transitivities. Our results have potential utility in analyzing directed synaptic networks constructed from neuronal connectome data, such as in terms of centrality. Other potential applications include anomaly detection in networks, identification of similar networks and identification of similar nodes within networks. Matlab code for calculating all stated metrics following calculation of functional motif-role fingerprints is provided as S1 Matlab File.

CombiMotif: A new algorithm for network motifs discovery in protein–protein interaction networks

Discovering motifs in protein–protein interaction networks is becoming a current major challenge in computational biology, since the distribution of the number of network motifs can reveal significant systemic differences among species. However, this task can be computationally expensive because of the involvement of graph isomorphic detection. In this paper, we present a new algorithm (CombiMotif) that incorporates combinatorial techniques to count non-induced occurrences of subgraph topologies in the form of trees. The efficiency of our algorithm is demonstrated by comparing the obtained results with the current state-of-the art subgraph counting algorithms. We also show major differences between unicellular and multicellular organisms. The datasets and source code of CombiMotif are freely available upon request.

Acc-Motif: Accelerated Network Motif Detection.

Network motif algorithms have been a topic of research mainly after the 2002-seminal paper from Milo et al. [1], which provided motifs as a way to uncover the basic building blocks of most networks. Motifs have been mainly applied in Bioinformatics, regarding gene regulation networks. Motif detection is based on induced subgraph counting. This paper proposes an algorithm to count subgraphs of size k + 2 based on the set of induced subgraphs of size k. The general technique was applied to detect 3, 4 and 5-sized motifs in directed graphs. Such algorithms have time complexity O(a(G)m), O(m(2)) and O(nm(2)), respectively, where a(G) is the arboricity of G(V, E). The computational experiments in public data sets show that the proposed technique was one order of magnitude faster than Kavosh and FANMOD. When compared to NetMODE, acc-Motif had a slightly improved performance.

QSAR classification model for diverse series of antimicrobial agents using classification tree configured by modified particle swarm optimization

In the present study, classification tree induced by modified particle swarm optimization (MPSOCT) was invoked to construct a quantitative structure–activity relationship (QSAR) classification model for diverse series of antimicrobial agents against Candida albicans (CA). This may enable the screening of the antimicrobial agents in silico. In MPSOCT, modified particle swarm optimization (MPSO) was used to induce a globally optimal CT via simultaneously searching the optimal splitting parameters (i.e., splitting variables and values) and appropriate structure in tree. Based on more than hundred descriptors calculated by Material Studio 4.0 software, the classification model by MPSOCT categorized the compounds into the actives or inactives, compared with those by classification tree (CT), radial basis function network (RBFN), and partial least-squares discriminant analysis (PLS-DA). Experimental results revealed that MPSOCT offered more satisfactory classification accuracy than the other three classification algorithms. In addition, descriptors HOMO eigenvalue, Dipole_Y, S_ssCH2, S_dsCH, S_dO, AlogP98, Molecular flexibility, Wiener index, Kappa-2, Subgraph counts (2): path, Subgraph counts (3): cluster, Chi (0), Bond information content, Principal moment of inertia X, Ellipsoidal volume, Shadow_YZ, Shadow_ZX, Total molecular mass, and Atom count were found to play the most predominant roles in the antimicrobial activities against CA.

Spectral moments of regular graphs in terms of subgraph counts

We give equations that relate the spectral moments of a regular graph G to the numbers of certain subgraphs that occur within G. The equations are derived by using generating functions to count certain closed walks in G. The subgraphs that determine the i-th spectral moment are precisely the connected graphs of minimum degree 2 that are induced by at least one closed walk of length i.

Counting relations for general Zagreb indices

The first and second general Zagreb indices of a graph G, with vertex set V and edge set E, are defined as Mk1=Σuєv d(u)k and Mk2=ΣuvєE [d(u)d(v)]k where d(v) is the degree of the vertex v of G. We present combinatorial identities, relating Mk1 and Mk2 with counts of various subgraphs contained in the graph G.

Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering

Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic independence of U-statistics following Gaussian and Poisson asymptotic regimes. Our results generalize and unify several previous findings in the field. We provide an application to joint sub-graph counting in random geometric graphs.

G-Tries: a data structure for storing and finding subgraphs

The ability to find and count subgraphs of a given network is an important non trivial task with multidisciplinary applicability. Discovering network motifs or computing graphlet signatures are two examples of methodologies that at their core rely precisely on the subgraph counting problem. Here we present the g-trie, a data-structure specifically designed for discovering subgraph frequencies. We produce a tree that encapsulates the structure of the entire graph set, taking advantage of common topologies in the same way a prefix tree takes advantage of common prefixes. This avoids redundancy in the representation of the graphs, thus allowing for both memory and computation time savings. We introduce a specialized canonical labeling designed to highlight common substructures and annotate the g-trie with a set of conditional rules that break symmetries, avoiding repetitions in the computation. We introduce a novel algorithm that takes as input a set of small graphs and is able to efficiently find and count them as induced subgraphs of a larger network. We perform an extensive empirical evaluation of our algorithms, focusing on efficiency and scalability on a set of diversified complex networks. Results show that g-tries are able to clearly outperform previously existing algorithms by at least one order of magnitude.

Model Selection for Social Networks Using Graphlets

Several network models have been proposed to explain the link structure observed in online social networks. This paper addresses the problem of choosing the model that best fits a given real-world network. We implement a model-selection method based on unsupervised learning. An alternating decision tree is trained using synthetic graphs generated according to each of the models under consideration. We use a broad array of features, with the aim of representing different structural aspects of the network. Features include the frequency counts of small subgraphs (graphlets) as well as features capturing the degree distribution and small-world property. Our method correctly classifies synthetic graphs, and is robust under perturbations of the graphs. We show that the graphlet counts alone are sufficient in separating the training data, indicating that graphlet counts are a good way of capturing network structure. We tested our approach on four Facebook graphs from various American universities. The models th...

Subgraph fluctuations in random graphs

The pattern of over- and under-representations of three-node subgraphs has become a standard method of characterizing complex networks and evaluating how this intermediate level of organization contributes to network function. Understanding statistical properties of subgraph counts in random graphs, their fluctuations, and their interdependences with other topological attributes is an important prerequisite for such investigations. Here we introduce a formalism for predicting subgraph fluctuations induced by perturbations of unidirectional and bidirectional edge densities. On this basis we predict the over- and under-representation of subgraphs arising from a density mismatch between a network and the corresponding pool of randomized graphs serving as a null model. Such mismatches occur, for example, in modular and hierarchical graphs.

Quantitative Structure–Reactivity Study of Electrochemical Oxidation of Phenolic Compounds at the SnO2–Based Electrode

In the present study, the electrochemical oxidation of 22 phenolic compounds was systemically examined at the RuO(2)-SnO(2)-Sb(2)O(5) electrode to elucidate the inherent structure-reactivity correlation. The oxidation process was monitored in situ by UV-vis spectroscopy. A variety of substituents (e.g., -CH(3), -NH(2), -Cl, -OH, -COOH, -NO(2), -CHO) were employed in order to cover various possible electronic effects. Our experimental results revealed that the relationship between the Hammett constant and rate constant for the electrochemical oxidation of phenolic compounds at the RuO(2)-SnO(2)-Sb(2)O(5) electrode was different from the results obtained at a platinum electrode. The substituted phenols with electron-withdrawing groups were electrochemically oxidized more rapidly than those with electron-donating groups. To decipher the effects of physiochemical properties on the electrochemical reactivity of phenolic compounds, 140 molecular descriptors were calculated and assessed for each phenolic compound; a quantitative structure property relationship (QSPR) model was developed. Correlations between the rate constants and quantum properties of the phenolic compounds were achieved using partial least-squares (PLS) analysis. The most crucial quantum descriptors responsible for the electrochemical reactivity of phenolic compounds were determined to be E(HOMO), chemical potential, total dipole, quadrupoles, subgraph counts, relative positive charged surface area, and pK(a). The proposed QSPR model was based on the quantum descriptors derived from the whole molecule, providing lucid explanation and effective prediction of the electrochemical reactivity of various phenolic compounds.

Counting Subgraphs via Homomorphisms

We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well-known results in algorithms and combinatorics, including the recent algorithm of Björklund, Husfeldt, and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn et al. for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color-coding-based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions, and the color coding technique, we obtain the following new results. The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time $ 2^{n+o(n)} $, in particular, in time $\mathcal{O}(2^{n}n^3)$ for trees and in time $2^{n+\mathcal{O}(\sqrt{n})}$ for planar graphs. Counting all maximum planar subgraphs, subgraphs of bounded genus, or more generally subgraphs excluding a fixed graph M as a minor can be done in $2^{\mathcal{O}(n)}$ time. Counting all subtrees with a given maximum degree (a generalization of counting Hamiltonian paths) of a given graph can be done in time $2^{\mathcal{O}(n)}$. A generalization of Ryser's formula is, Let G be a graph with an independent set of size $\ell$. Then the number of perfect matchings in G can be found in time $\mathcal{O}(2^{n-\ell} n^3)$. Let ${\cal H}$ be a graph class excluding a fixed graph M as a minor. Then the maximum number of vertex disjoint subgraphs from ${\cal H}$ in a graph G on n vertices can be found in time $2^{\mathcal{O}(n)}$. In order to show this, we prove that there exists a constant $c_M$ depending only on M such that the number of nonisomorphic n-vertex graphs in ${\cal H}$ is at most $c_M^n$. Let F be a k-vertex graph of treewidth t and let G be an n-vertex graph. A subgraph of G isomorphic to F (if one exists) can be found in $\mathcal{O}(4.32^k \cdot k \cdot t \cdot n^{t+1})$ expected time using $\mathcal{O}(\log{k} \cdot n^{t+1})$ space.

Arboricity, [formula omitted]-index, and dynamic algorithms

We propose a new data structure for manipulating graphs, called h-graph, which is particularly suited for designing dynamic algorithms. The structure itself is simple, consisting basically of a triple of elements, for each vertex of the graph. The overall size of all triples is O(n+m), for a graph with n vertices and m edges. We describe algorithms for performing the basic operations related to dynamic applications, as insertions and deletions of vertices or edges, and adjacency queries. The data structure employs a technique first described by Chiba and Nishizeki [Chiba, Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1) (1985) 210–223], and relies on the arboricity of graphs. Using the proposed data structure, we describe several dynamic algorithms for solving problems as listing the cliques of a given size, recognizing diamond-free graphs, and finding simple, simplicial and dominated vertices. These algorithms are the first of their kind to be proposed in the literature. In fact, the dynamic algorithms for the above problems lead directly to new static algorithms, and using the data structure we also design new static algorithms for the problems of counting subgraphs of size 4, recognizing cop-win graphs and recognizing strongly chordal graphs. The complexities of all of the proposed static algorithms improve over the complexities of the so far existing algorithms, for graphs of low arboricity. In addition, for the problems of counting subgraphs of size 4 and recognizing diamond-free graphs, the improvement is general.

Extended dynamic subgraph statistics using [formula omitted]-index parameterized data structures

We present techniques for maintaining subgraph frequencies in a dynamic graph, using data structures that are parameterized in terms of h, the h-index of the graph. Our methods extend previous results of Eppstein and Spiro for maintaining statistics for undirected subgraphs of size three to directed subgraphs and to subgraphs of size four. For the directed case, we provide a data structure to maintain counts for all 3-vertex induced subgraphs in O(h) amortized time per update. For the undirected case, we maintain the counts of size-four subgraphs in O(h2) amortized time per update. These extensions enable a number of new applications in Bioinformatics and Social Networking research.

Faster algorithms for finding and counting subgraphs

In the Subgraph Isomorphism problem we are given two graphs F and G on k and n vertices respectively as an input, and the question is whether there exists a subgraph of G isomorphic to F. We show that if the treewidth of F is at most t, then there is a randomized algorithm for the Subgraph Isomorphism problem running in time O⁎(2kn2t). Our proof is based on a novel construction of an arithmetic circuit of size at most nO(t) for a new multivariate polynomial, Homomorphism Polynomial, of degree at most k, which in turn is used to solve the Subgraph Isomorphism problem. For the counting version of the Subgraph Isomorphism problem, where the objective is to count the number of distinct subgraphs of G that are isomorphic to F, we give a deterministic algorithm running in time and space O⁎((nk/2)n2p) or (nk/2)nO(tlogk). We also give an algorithm running in time O⁎(2k(nk/2)n5p) and taking O⁎(np) space. Here p and t denote the pathwidth and the treewidth of F, respectively. Our work improves on the previous results on Subgraph Isomorphism, it also extends and unifies most of the known results on sub-path and sub-tree isomorphisms.

The missing log in large deviations for triangle counts

AbstractThis paper solves the problem of sharp large deviation estimates for the upper tail of the number of triangles in an Erdős‐Rényi random graph, by establishing a logarithmic factor in the exponent that was missing till now. It is possible that the method of proof may extend to general subgraph counts. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 437–451, 2012

Quasi-random graphs and graph limits

We use the theory of graph limits to study several quasi-random properties, mainly dealing with various versions of hereditary subgraph counts. The main idea is to transfer the properties of (sequences of) graphs to properties of graphons, and to show that the resulting graphon properties only can be satisfied by constant graphons. These quasi-random properties have been studied before by other authors, but our approach gives proofs that we find cleaner, and which avoid the error terms and ε in the traditional arguments using the Szemerédi regularity lemma. On the other hand, other technical problems sometimes arise in analysing the graphon properties; in particular, a measure-theoretic problem on elimination of null sets that arises in this way is treated in an appendix.

ISIDA Property-Labelled Fragment Descriptors.

ISIDA Property-Labelled Fragment Descriptors (IPLF) were introduced as a general framework to numerically encode molecular structures in chemoinformatics, as counts of specific subgraphs in which atom vertices are coloured with respect to some local property/feature. Combining various colouring strategies of the molecular graph - notably pH-dependent pharmacophore and electrostatic potential-based flagging - with several fragmentation schemes, the different subtypes of IPLFs may range from classical atom pair and sequence counts, to monitoring population levels of branched fragments or feature multiplets. The pH-dependent feature flagging, pursued at the level of each significantly populated microspecies involved in the proteolytic equilibrium, may furthermore add some competitive advantage over classical descriptors, even when the chosen fragmentation scheme is one of the state-of-the-art pattern extraction procedures (feature sequence or pair counts, etc.) in chemoinformatics. The implemented fragmentation schemes support counting (1) linear feature sequences, (2) feature pairs, (3) circular feature fragments a.k.a. "augmented atoms" or (4) feature trees. Fuzzy rendering - optionally allowing nonterminal fragment atoms to be counted as wildcards, ignoring their specific colours/features - ensures for a seamless transition between the "strict" counts (sequences or circular fragments) and the "fuzzy" multiplet counts (pairs or trees). Also, bond information may be represented or ignored, thus leaving the user a vast choice in terms of the level of resolution at which chemical information should be extracted into the descriptors. Selected IPLF subsets were - tree descriptors, in particular - successfully tested in both neighbourhood behaviour and QSAR modelling challenges, with very promising results. They showed excellent results in similarity-based virtual screening for analogue protease inhibitors, and generated highly predictive octanol-water partition coefficient and hERG channel inhibition models.