Maximum Subgraph Research Articles

Efficient algorithms for the periodic subgraphs mining problem

Given a sequence G = 〈 G 0 , … , G T − 1 〉 of simple graphs over uniquely labeled vertices from a set V , the periodic subgraph mining problem consists in discovering maximal subgraphs that recur at regular intervals in G . For a periodic subgraph to be maximal, it is intended here that it cannot be enriched by adding edges nor can its temporal span be expanded in any direction. We give algorithms that improve the theoretical complexity of solutions previously available for this problem. In particular, we show an optimal solution based on an implicit description of the output subgraphs that takes time O ( | V | + | E ˜ | × T 2 / σ ) —where | E ˜ | is the average number of edges over the entire sequence G —to publish all maximal periodic subgraphs that meet or exceed a minimum occurrence threshold σ .

NIBBS-Search for Fast and Accurate Prediction of Phenotype-Biased Metabolic Systems

Understanding of genotype-phenotype associations is important not only for furthering our knowledge on internal cellular processes, but also essential for providing the foundation necessary for genetic engineering of microorganisms for industrial use (e.g., production of bioenergy or biofuels). However, genotype-phenotype associations alone do not provide enough information to alter an organism's genome to either suppress or exhibit a phenotype. It is important to look at the phenotype-related genes in the context of the genome-scale network to understand how the genes interact with other genes in the organism. Identification of metabolic subsystems involved in the expression of the phenotype is one way of placing the phenotype-related genes in the context of the entire network. A metabolic system refers to a metabolic network subgraph; nodes are compounds and edges labels are the enzymes that catalyze the reaction. The metabolic subsystem could be part of a single metabolic pathway or span parts of multiple pathways. Arguably, comparative genome-scale metabolic network analysis is a promising strategy to identify these phenotype-related metabolic subsystems. Network Instance-Based Biased Subgraph Search (NIBBS) is a graph-theoretic method for genome-scale metabolic network comparative analysis that can identify metabolic systems that are statistically biased toward phenotype-expressing organismal networks. We set up experiments with target phenotypes like hydrogen production, TCA expression, and acid-tolerance. We show via extensive literature search that some of the resulting metabolic subsystems are indeed phenotype-related and formulate hypotheses for other systems in terms of their role in phenotype expression. NIBBS is also orders of magnitude faster than MULE, one of the most efficient maximal frequent subgraph mining algorithms that could be adjusted for this problem. Also, the set of phenotype-biased metabolic systems output by NIBBS comes very close to the set of phenotype-biased subgraphs output by an exact maximally-biased subgraph enumeration algorithm ( MBS-Enum ). The code (NIBBS and the module to visualize the identified subsystems) is available at http://freescience.org/cs/NIBBS.

Modeling Flexible Pharmacophores with Distance Geometry, Scoring, and Bound Stretching

The study of pharmacophores, i.e., of common features between different ligands, is important for the quantitative identification of "compatible" enzymes and binding species. A pharmacophore-based technique is developed that combines multiple conformations with a distance geometry method to create flexible pharmacophore representations. It uses a set of low-energy conformations combined with a new process we call bound stretching to create sets of distance bounds, which contain all or most of the low-energy conformations. The bounds can be obtained using the exact distances between pairs of atoms from the different low-energy conformations. To avoid missing conformations, we can take advantage of the triangle distance inequality between sets of three points to logically expand a set of upper and lower distance bounds (bound stretching). The flexible pharmacophore can be found using a 3-D maximal common subgraph method, which uses the overlap of distance bounds to determine the overlapping structure. A scoring routine is implemented to select the substructures with the largest overlap because there will typically be many overlaps with the maximum number of overlapping bounds. A case study is presented in which 3-D flexible pharmacophores are generated and used to eliminate potential binding species identified by a 2-D pharmacophore method. A second case study creates flexible pharmacophores from a set of thrombin ligands. These are used to compare the new method with existing pharmacophore identification software.

A Decomposition Algorithm for Learning Bayesian Networks Based on Scoring Function

Learning Bayesian network (BN) structure from data is a typical NP‐hard problem. But almost existing algorithms have the very high complexity when the number of variables is large. In order to solve this problem(s), we present an algorithm that integrates with a decomposition‐based approach and a scoring‐function‐based approach for learning BN structures. Firstly, the proposed algorithm decomposes the moral graph of BN into its maximal prime subgraphs. Then it orientates the local edges in each subgraph by the K2‐scoring greedy searching. The last step is combining directed subgraphs to obtain final BN structure. The theoretical and experimental results show that our algorithm can efficiently and accurately identify complex network structures from small data set.

FISH: Fast Instruction SyntHesis for Custom Processors

This paper presents Fast Instruction SyntHesis (FISH), a system that supports automatic generation of custom instruction processors from high-level application descriptions to enable fast design space exploration. FISH is based on novel methods for automatically adapting the instruction set to match an application in a high-level language such as C or C++. FISH identifies custom instruction candidates using two approaches: 1) by enumerating maximal convex subgraphs of application data flow graphs and 2) by integer linear programming (ILP). The experiments, involving ten multimedia and cryptography benchmarks, show that our contributed algorithms are the fastest among the state-of-the-art techniques. In most cases, enumeration takes only milliseconds to execute. The longest enumeration run-time observed is less than six seconds. ILP is usually slower than enumeration, but provides us with a complementary solution technique. Both enumeration and ILP allow the use of multiple different merit functions in the evaluation of data-flow subgraphs. The experiments demonstrate that, using only modest additional hardware resources, up to 30-fold performance improvement can be obtained with respect to a single-issue base processor.

KeyPathwayMiner: Detecting Case-Specific Biological Pathways Using Expression Data

Abstract Recent advances in systems biology have provided us with massive amounts of pathway data that describe the interplay of genes and their products. The resulting biological networks can be modeled as graphs. By means of “omics” technologies, such as microarrays, the activity of genes and proteins can be measured. Here, data from microarray experiments is integrated with the network data to gain deeper insights into gene expression. We introduce KeyPathwayMiner, a method that enables the extraction and visualization of interesting subpathways given the results of a series of gene expression studies. We aim to detect highly connected subnetworks in which most genes or proteins show similar patterns of expression. Specifically, given network and gene expression data, KeyPathwayMiner identifies those maximal subgraphs where all but k nodes of the subnetwork are expressed similarly in all but l cases in the gene expression data. Since identifying these subgraphs is computationally intensive, we develope...

Modeling Evolution of Weighted Clique Networks

We propose a weighted clique network evolution model, which expands continuously by the addition of a new clique (maximal complete sub-graph) at each time step. And the cliques in the network overlap with each other. The structural expansion of the weighted clique network is combined with the edges' weight and vertices' strengths dynamical evolution. The model is based on a weight-driven dynamics and a weights' enhancement mechanism combining with the network growth. We study the network properties, which include the distribution of vertices' strength and the distribution of edges' weight, and find that both the distributions follow the scale-free distribution. At the same time, we also find that the relationship between strength and degree of a vertex are linear correlation during the growth of the network. On the basis of mean-field theory, we study the weighted network model and prove that both vertices' strength and edges' weight of this model follow the scale-free distribution. And we exploit an algorithm to forecast the network dynamics, which can be used to reckon the distributions and the corresponding scaling exponents. Furthermore, we observe that mean-field based theoretic results are consistent with the statistical data of the model, which denotes the theoretical result in this paper is effective.

Encoding Custom Instruction Generation as Satisfiability Problem

The emergence of Application-specific Instruction-set Processor (ASIP) has encouraged the proliferation of tool-chains used to streamline its design flow. One of the features much sought-after in these tool-chains is notably the automatic generation of Application-specific Functional Units (AFUs) which, in turn, involves the custom instruction generation as a crucial step. Whereupon an additional step is assumed to pipeline the patterns identified for fulfilling the I/O constraint, custom instructions that correspond to maximal valid subgraphs are mostly beneficial to the speedup gain. Therefore, we present in this paper a propositional satisfiability approach to efficiently identify the custom instructions which contain a large number of valid nodes. Our approach is different substantially from the previous works where it uses an edge classification method to reduce the search space for convexity checking. The experiment results show that our method can, in a matter of few seconds, identify a set of custom instructions that speed up the application to a few times faster.

Mining Maximal Dense Subgraphs in Uncertain PPI Network

Several studies have shown that the prediction of protein function using PPI data is promising. However, the PPI data generated from experiments are noisy, incomplete and inaccurate, which promotes to represent PPI dataset as an uncertain graph. In this paper, we proposed a novel algorithm to mine maximal dense subgraphs efficiently in uncertain PPI network. It adopted several techniques to achieve efficient mining. An extensive experimental evaluation on yeast PPI network demonstrated that our approach had good performance in terms of precision and efficiency.

Combinatorial properties and further facets of maximum edge subgraph polytopes

Abstract Given a graph G and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset of G such that the number of edges within the subset is maximum. This NP-hard problem arises in the analysis of cohesive subgroups in social networks. In this work we study the polytope P ( G , k ) associated with a straightforward integer programming formulation of the maximum edge subgraph problem. We characterize the graph generated by P ( G , k ) and give a tight bound on its diameter. We give a complete description of P ( K 1 n , k ) , where K 1 n is the star on n + 1 vertices, and we conjecture a complete description of P ( m K 2 , k ) , where m K 2 is the graph composed by m disjoint edges. Finally, we introduce three families of facet-inducing inequalities for P ( G , k ) , which generalize known families of valid inequalities for this polytope.

Bus transport network model with ideal [formula omitted]-depth clique network topology

We propose an ideal n-depth clique network model. In this model, the original network is composed of cliques (maximal complete subgraphs) that overlap with each other. The network expands continuously by the addition of new cliques. The final diameter of the network can be set in advance, namely, it is controllable. Assuming that the diameter of the network is n, the network exhibits a logistic structure with (n+1) layers. In this structure, the 0th layer represents the original network and each node of the (m)th layer (1≤m≤n) corresponds to a clique in the (m−1)th layer. In the growth process of the network, we ensure that any (m)th layer network is composed of overlapping cliques. Any node in an (m)th layer network corresponds to an m-depth community in the original network, and the diameter of an m-depth community is m. Therefore, the (n−1)th layer network will contain only one clique, the (n)th layer network will contain only one node, and the diameter of the corresponding original network is n. Then an ideal n-depth clique network will be obtained. Based on the ideal n-depth clique network model, we construct a bus transport network model with an ideal n-depth clique network topology (ICNBTN). Moreover, our study compares this model with the real bus transport network (RealBTN) of three major cities in China and a recently introduced bus transport network model (BTN) whose network properties correspond well with those of real BTNs. The network properties of the ICNBTN are much closer to those of the RealBTN than those of the BTN are. At the same time, the ICNBTN has higher clustering extent of bus routes, smaller network diameter, which corresponds to shorter maximum transfer times in a bus network, and lower average shortest path time coefficient than the BTN and the RealBTN. Therefore, the ICNBTN can achieve higher transfer efficiency for a bus transport system.

Coefficients of chromatic polynomials and tension polynomials

We evaluate coefficients of chromatic polynomial of a graph $G$ as sums of zero values of tension polynomials of certain maximal subgraphs of $G$.

Speedup for a periodic subgraph miner

Given a series G = { G 0 , G 1 , … , G T } of graphs encompassing V vertices and E edges, a periodic graph is a spatially as well as temporally maximal subgraph of a subsequence of G in the form G i p = { G i , G i + p , … , G i + n p } , where n is not smaller than some predetermined threshold value σ. An algorithm for finding all such subgraphs is proposed taking time O ( ( E + V ) T 2 ln ( T / σ ) ) , which is faster by a factor of T than the method previously available.

Top-Down Algorithm for Mining Maximal Frequent Subgraph

In order to solution the problem of mining maximal frequent subgraph is very hard we proposed new algorithm Top-Down. The process of this algorithm is using decision tree to count support then firstly judge the biggest graph whether frequent and gradually reduce the graph which used to judge until can not mining maximal frequent subgraph, at the same time this algorithm is proposed a theorem and two principles these are improved the mining efficiency.

MFC: Mining Maximal Frequent Dense Subgraphs without Candidate Maintenance in Imbalanced PPI Networks

The prediction of protein function is one of the most challenging problems in bioinformatics. Several studies have shown that the prediction using PPI is promising. However, the PPI data generated from high-throughput experiments are very noisy, which renders great challenges to the existing methods. In this paper, we propose an algorithm, MFC , to efficiently mine maximal frequent dense subgraphs without candidate maintenance in PPI networks. Instead of using summary graph, MFC produces frequent dense patterns by extending vertices. It adopts several techniques to achieve efficient mining. Due to the imbalance character of PPI network, we also propose to generate frequent patterns using relative support. We evaluate our approach on four PPI data sets. The experimental results show that our approach has good performance in terms of efficiency. With the help of relative support, more frequent dense functional interaction patterns in the PPI networks can be identified.

Maximum common subgraph isomorphism algorithms and their applications in molecular science: a review

AbstractThe intuitive description of small and large molecules using graphs has led to an increasing interest in the application of graph concepts for describing, analyzing, and comparing small molecules as well as proteins. Graph theory is a well‐studied field and many applications are present in various scientific disciplines. Recent literature describes a number of successful applications to biological problems. One of the most applied concepts aims at finding a maximal common subgraph (MCS) isomorphism between two graphs. We review exact MCS algorithms, especially designed for graphs obtained from small and large molecules, and give an overview of their successful applications. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 68–79 DOI: 10.1002/wcms.5This article is categorized under: Structure and Mechanism > Molecular Structures Computer and Information Science > Chemoinformatics Computer and Information Science > Computer Algorithms and Programming

A Parallel Graph Sampling Algorithm for Analyzing Gene Correlation Networks

Abstract Effcient analysis of complex networks is often a challenging task due to its large size and the noise inherent in the system. One popular method of overcoming this problem is through graph sampling, that is extracting a representative subgraph from the larger network. The accuracy of the sample is validated by comparing the combinatorial properties of the subgraph and the original network. However, there has been little study in comparing networks based on the applications that they represent. Furthermore, sampling methods are generally applied agnostically, without mapping to the requirements of the underlying analysis. In this paper,we introduce a parallel graph sampling algorithm focusing on gene correlation networks. Densely connected subgraphs indicate important functional units of gene products. In our sampling algorithm, we emphasize maintaining highly connected regions of the network through parallel sampling based on extracting the maximal chordal subgraph of the network. We validate our methods by comparing both combinatorial properties and functional units of the subgraphs and larger networks. Our results show that even with significant reduction of the network (on average 20% to 40%), we obtain reliable samplings and many of the relevant combinatorial and functional properties are retained in the subgraphs.

Computing maximum upward planar subgraphs of single-source embedded digraphs

We show how to compute a maximum upward planar single-source subgraph of a single-source embedded DAG G ? . We first show that finding a maximum upward planar subgraph of a single-source embedded digraph is NP-complete. We then give a new characterization of upward planar single-source digraphs. We use this characterization to present an algorithm that computes a maximum upward planar single-source subgraph of a single-source embedded DAG. This algorithm takes O(n 4) time in the worst case and O(n 3) time on average.

MARGIN

The exponential number of possible subgraphs makes the problem of frequent subgraph mining a challenge. The set of maximal frequent subgraphs is much smaller to that of the set of frequent subgraphs providing ample scope for pruning. MARGIN is a maximal subgraph mining algorithm that moves among promising nodes of the search space along the “border” of the infrequent and frequent subgraphs. This drastically reduces the number of candidate patterns in the search space. The proof of correctness of the algorithm is presented. Experimental results validate the efficiency and utility of the technique proposed.

On the Maximal Interval Subgraph of a Tree

We study the problem of finding the maximum interval subgraph in a tree. This problem is related to the Double Digestion Problem of DNA physical mapping. We show that the complexity of an algorithm of Wang is O(n). We also present a linear algorithm of our own. We study the case when the edges of the tree is weighted as well. An algorithm with complexity O(n³) is presented.