Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4-vertex patterns is highly challenging, and there are few practical results known that can scale to massive sizes. Indeed, even a highly tuned enumeration code takes more than a day on a graph with millions of edges. Most previous work that runs for truly massive graphs employ clusters and massive parallelization. We provide a sampling algorithm that provably and accurately approximates the frequencies of all 4-vertex pattern subgraphs. Our algorithm is based on a novel technique of 3-path sampling and a special pruning scheme to decrease the variance in estimates. We provide theoretical proofs for the accuracy of our algorithm, and give formal bounds for the error and confidence of our estimates. We perform a detailed empirical study and show that our algorithm provides estimates within 1 relative error for all subpatterns (over a large class of test graphs), while being orders of magnitude faster than enumeration and other sampling based algorithms. Our algorithm takes less than a minute (on a single commodity machine) to process an Orkut social network with 300 million edges.
Motif counting for bioinformatics was arguably initiated by a seminal paper of Milo @cite_5 . This technique has been used for graph modeling @cite_9 @cite_35 , graph comparisons @cite_9 @cite_13 , and even decomposing a network @cite_22 . Refer to @cite_27 @cite_7 for more details.
@cite_35 @cite_22 @cite_7 @cite_9 @cite_27 @cite_5 @cite_13
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A major challenge in molecular biology is reverse-engineering the cis-regulatory logic that plays a major role in the control of gene expression. This program includes searching through DNA sequences to identify “motifs” that serve as the binding sites for transcription factors or, more generally, are predictive of gene expression across cellular conditions. Several approaches have been proposed for de novo motif discovery–searching sequences without prior knowledge of binding sites or nucleotide patterns. However, unbiased validation is not straightforward. We consider two approaches to unbiased validation of discovered motifs: testing the statistical significance of a motif using a DNA “background” sequence model to represent the null hypothesis and measuring performance in predicting membership in gene clusters. We demonstrate that the background models typically used are “too null,” resulting in overly optimistic assessments of significance, and argue that performance in predicting TF binding or expression patterns from DNA motifs should be assessed by held-out data, as in predictive learning. Applying this criterion to common motif discovery methods resulted in universally poor performance, although there is a marked improvement when motifs are statistically significant against real background sequences. Moreover, on synthetic data where “ground truth” is known, discriminative performance of all algorithms is far below the theoretical upper bound, with pronounced “over-fitting” in training. A key conclusion from this work is that the failure of de novo discovery approaches to accurately identify motifs is basically due to statistical intractability resulting from the fixed size of co-regulated gene clusters, and thus such failures do not necessarily provide evidence that unfound motifs are not active biologically. Consequently, the use of prior knowledge to enhance motif discovery is not just advantageous but necessary. An implementation of the LR and ALR algorithms is available at http: code.google.com p likelihood-ratio-motifs . Many algorithms for motif finding that are commonly used in bioinformatics start by sampling r potential motif occurrences from n input sequences. The motif is derived from these samples and evaluated on all sequences. This approach works extremely well in practice, and is implemented by several programs. Li, Ma and Wang have shown that a simple algorithm of this sort is a polynomial-time approximation scheme. However, in 2005, we showed specific instances of the motif finding problem for which the approximation ratio of a slight variation of this scheme converges to one very slowly as a function of the sample size r, which seemingly contradicts the high performance of sample-based algorithms. Here, we account for the difference by showing that, for a variety of different definitions of “strong” binary motifs, the approximation ratio of sample-based algorithms converges to one exponentially fast in r. We also describe “very strong” motifs, for which the simple sample-based approach always identifies the correct motif, even for modest values of r. Network motifs are statistically overrepresented sub-structures (sub-graphs) in a network, and have been recognized as ‘the simple building blocks of complex networks’. Study of biological network motifs may reveal answers to many important biological questions. The main difficulty in detecting larger network motifs in biological networks lies in the facts that the number of possible sub-graphs increases exponentially with the network or motif size (node counts, in general), and that no known polynomial-time algorithm exists in deciding if two graphs are topologically equivalent. This article discusses the biological significance of network motifs, the motivation behind solving the motif-finding problem, and strategies to solve the various aspects of this problem. A simple classification scheme is designed to analyze the strengths and weaknesses of several existing algorithms. Experimental results derived from a few comparative studies in the literature are discussed, with conclusions that lead to future research directions. Summary: Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns that occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biological regulation networks. Existing algorithms for detecting network motifs act by exhaustively enumerating all subgraphs with a given number of nodes in the network. The runtime of such algorithms increases strongly with network size. Here, we present a novel algorithm that allows estimation of subgraph concentrations and detection of network motifs at a runtime that is asymptotically independent of the network size. This algorithm is based on random sampling of subgraphs. Network motifs are detected with a surprisingly small number of samples in a wide variety of networks. Our method can be applied to estimate the concentrations of larger subgraphs in larger networks than was previously possible with exhaustive enumeration algorithms. We present results for high-order motifs in several biological networks and discuss their possible functions. Availability: A software tool for estimating subgraph concentrations and detecting network motifs (mfinder 1.1) and further information is available at http: www.weizmann.ac.il mcb UriAlon