Bacteriophages (viruses that infect bacteria) are the most abundant biological life-forms on Earth. However, very little is known regarding the structure of phage-bacteria infections. In a recent study we showed that phage-bacteria infection assay datasets are statistically nested in small scale communities while modularity is not statistically present. We predicted that at large macroevolutionary scales, phage-bacteria infection assay datasets should be typified by a modular structure, even if there is nested structure at smaller scales. We evaluate and confirm this hypothesis using the largest study of the kind to date. The study in question represents a phage-bacteria infection assay dataset in the Atlantic Ocean region between the European continental shelf and the Sargasso Sea. We present here a digitized version of this study that consist of a bipartite network with 286 bacteria and 215 phages including 1332 positive interactions, together with an exhaustive structural analysis of this network. We evaluated the modularity and nestedness of the network and its communities using a variety of algorithms including BRIM (Bipartite, Recursively Induced Modules), NTC (Nestedness Temperature Calculator) and NODF (Nestedness Metric based on Overlap and Decreasing Filling). We also developed extensions of these standard methods to identify multi-scale structure in large phage-bacteria interaction datasets. In addition, we performed an analysis of the degree of geographical diversity and specialization among all the hosts and phages. We find that the largest-scale ocean dataset study, as anticipated by Flores et al. 2013, is highly modular and not significantly nested (computed in comparison to null models). More importantly is the fact that some of the communities extracted from Moebus and Nattkemper dataset were found to be nested. We examine the role of geography in driving these modular patterns and find evidence that phage-bacteria interactions can exhibit strong similarity despite large distances between sites. We discuss how models can help determine how coevolutionary dynamics between strains, within a site and across sites, drives the emergence of nested, modular and other complex phage-bacteria interaction networks. Finally, we releases a computational library (BiMAT)to help to help the ecology research community to perform bipartite network analysis of the same nature I did during my PhD.

The one- and two-sided bipartite graph drawing problem alms to find a layout of a bipartite graph, with vertices of the two parts placed on parallel imaginary lines, that has the minimum number of edge-crossings. Vertices of one part are in fixed positions for the one-sided problem, whereas all vertices are free to move along their lines in the two-sided version. Many different heuristics exist for finding approximations to these problems, which are NP-hard. New sequential and parallel methods for producing drawings with low edgecrossings are investigated and compared to existing algorithms, notably Penalty Minimisation and Sifting, the current leaders. For the one-sided problem, new methods that include those based on simple stochastic hillclimbing, simulated annealing and genet.ic algorithms were tested. The new block-crossover genetic algorithm produced very good results with lower crossings than existing methods, although it tended to be slower. However, time was a secondary aim, the priority being to achieve low numbers of crossings. This algorithm can also be seeded with the output of an existing algorithm to improve results; combining with Penalty Minimisation in this way improved both the speed and number of crossings. Four parallel methods for the one-sided problem have been created, although two were abandoned because they gave bad results for even simple graphs. The other two methods, based on stochastic hill-climbing, produced acceptable results in faster times than similar sequential methods. PVM was used as the parallel communication system. Two new heuristics were studied for the two-sided problem, for which the only known existing method is to apply one-sided algorithms iteratively. The first is based on a heuristic for the linear arrangment problem; the second is a method of performing stochastic hill-climbing on two sides. A way of applying anyone-sided algorithm iteratively was also created. The linear arrangement method based on the Koren-Harel multi-scale algorithm achieved the best results, outperforming iterative Barycentre (previously the best method) and iterative Penalty Minimisation. Another area of this work created three new heuristics for the k-planar drawing problem where k > 1. These are the first known practical algorithms to solve this problem. A sequential genetic algorithm based on TimGA is devised to work on k-planar graphs. Two parallel algorithms, one island model and the other a 'mesh' model, are also given. Comparison of results for k = 2 indicate that the parallel island method is better than the other two methods. MPI was used for the parallel communication. Overall, 14 new methods are introduced, of which 10 were developed into working algorithms. For the one-sided bipartite graph drawing problem the new block-crossover genetic algorithm can produce drawings with lower crossings than the current best available algorithms. The parallel methods do not perform as well as the sequential ones, although they generally achieved the same results faster. All of the new two-sided methods worked well; the weighted two-sided swap stochastic hill-climbing method was comparable to the existing best method, iterative Barycentre, and generally produced drawings with lower crossings, although it suffered with needing a good termination condition. The new methods based on the linear arrangement problem consistently produced drawings with lower crossings than iterative Barycentre, although they were nearly always slower. A new parallel algorithm for the k-planar drawing problem, based on the island model, generally created drawings with the lowest edge-crossings, although no algorithms were known to exist to make comparisons.

Thesis (M.S.)--Rochester Institute of Technology, 2006. / Typescript. Includes bibliographical references (leaves 38-39).

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18621号 / 情博第545号 / 新制||情||96(附属図書館) / 31521 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 永持 仁, 教授 太田 快人, 教授 髙橋 豊 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Clustering is the process in which data is differentiated, classified according to some criteria. As a result of partitioning process, data is grouped into clusters for specific purpose. In a social network, clustering of people is one of the most popular problems. Therefore, we mainly concentrated on finding an efficient algorithm for this problem. In our study, data is made up of two types of entities (e.g., people, groups vs. political issues, religious beliefs) and distinct from most previous works, signed weighted bipartite graphs are used to model relations among them. For the partitioning criterion, we use the strength of the opinions between the entities. Our main intention is to partition the data into k-clusters so that entities within clusters represent strong relationship. One such example from a political domain is the opinion of people on issues. Using the signed weights on the edges, these bipartite graphs can be partitioned into two or more clusters. In political domain, a cluster represents strong relationship among a group of people and a group of issues. After partitioning, each cluster in the result set contains like-minded people and advocated issues. Our work introduces a general mechanism for k-way partitioning of signed bipartite graphs. One of the great advantages of our thesis is that it does not require any preliminary information about the structure of the input dataset. The idea has been illustrated on real and randomly generated data and promising results have been shown.

Let F be a family of sets. A graph G is an intersection graph of sets from the family F if there exists a mapping f : V (G)→ F such that, An interval graph is an intersection graph of a family of closed intervals on the real line. Interval graphs find application in diverse fields ranging from DNA analysis to VLSI design. An interval on the real line can be generalized to a k dimensional box or k-box. A k-box B = (R1.R2….Rk) is defined to be the Cartesian product R1 × R2 × …× Rk, where each Ri is a closed interval on the real line. If each Ri is a unit length interval, we call B a k-cube. Thus, an interval is a 1-box and a unit length interval is a 1-cube. A graph G has a k-box representation, if G is an intersection graph of a family of k-boxes in Rk. Similarly, G has a k-cube representation, if G is an intersection graph of a family of k-cubes in Rk. The boxicity of G, denoted by box(G), is the minimum positive integer k such that G has a k-box representation. Similarly, the cubicity of G, denoted by cub(G), is the minimum positive integer k such that G has a k-cube representation. Thus, interval graphs are the graphs with boxicity equal to 1 and unit interval graphs are the graphs with cubicity equal to 1. The concepts of boxicity and cubicity were introduced by F.S. Roberts in 1969. Deciding whether the boxicity (or cubicity) of a graph is at most k is NP-complete even for a small positive integer k. Box representation of graphs finds application in niche overlap (competition) in ecology and to problems of fleet maintenance in operations research. Given a low dimensional box representation, some well known NP-hard problems become polynomial time solvable. Attempts to find efficient box and cube representations for special classes of graphs can be seen in the literature. Scheinerman [6] showed that the boxicity of outerplanar graphs is at most 2. Thomassen [7] proved that the boxicity of planar graphs is bounded from above by 3. Cube representations of special classes of graphs like hypercubes and complete multipartite graphs were investigated in [5, 3, 4]. In this thesis, we present several bounds for boxicity and cubicity of special classes of graphs in terms of other graph parameters. The following are the main results shown in this work. 1. It was shown in [2] that, for a graph G with maximum degree Δ, cub(G) ≤ [4(Δ+ 1) log n]. We show that, for a k-degenerate graph G, cub(G) ≤ (k + 2)[2e log n]. Since k is at most Δ and can be much lower, this clearly is a stronger result. This bound is tight up to a constant factor. 2. For a k-degenerate graph G, we give an efficient deterministic algorithm that runs in O(n2k) time to output an O(k log n) dimensional cube representation. 3. Crossing number of a graph G is the minimum number of crossing pairs of edges, over all drawings of G in the plane. We show that if crossing number of G is t, then box(G) is O(t1/4 log3/4 t). This bound is tight up to a factor of O((log t)1/4 ). 4. We prove that almost all graphs have cubicity O(dav log n), where dav denotes the average degree. 5. Boxicity of a k-leaf power is at most k -1. For every k, there exist k-leaf powers whose boxicity is exactly k - 1. Since leaf powers are a subclass of strongly chordal graphs, this result implies that there exist strongly chordal graphs with arbitrarily high boxicity 6. Otachi et al. [8] conjectured that chordal bipartite graphs (CBGs) have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of CBGs that have unbounded boxicity. We first prove that the bipartite power of a tree (which is a CBG) is a CBG and then show that there exist trees whose bipartite powers have high boxicity. Later in Chapter ??, we prove a more generic result in bipartite powering. We prove that, for every k ≥ 3, the bipartite power of a bipartite, k-chordal graph is bipartite and k-chordal thus implying that CBGs are closed under bipartite powering. 7. Boxicity of a line graph with maximum degree Δ is O(Δ log2 log2 Δ). This is a log2 Δ log log Δ factor improvement over the best known upper bound for boxicity of any graph [1]. We also prove a non-trivial lower bound for the boxicity of a d-dimensional hypercube.

Microorganisms are all around us. Some of them even live in our body and are essential for our healthy being. Study of microbial communities based on their genetic content has become very popular with the development of new technologies, which enable easy reading of DNA or RNA. The key role of these studies is usually to characterize significant microbial patterns of an environment. However, currently used visualization tools have many drawbacks for such analyses. The subject of this thesis is to design a R/Bioconductor package for simple creation of bipartite graphs from microbial data. This type of visualization brings many advantages for microbiome analysis. Benefits of bipartite graphs are further demonstrated by analysis of main parameters affecting computer processing of microbial data.

Orientador: Prof. Dr. Daniel Morgato Martin / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Ciências da Computação, 2014. / Neste trabalho estudamos o Problema do Isomorfismo de Grafos e a sua complexidade para resolvê-lo. Nossa principal contribuição é a proposta de um algoritmo para o caso geral do Problema, baseado no particionamento do conjunto de vértices e em emparelhamentos perfeitos de grafos bipartidos. Estudamos também o algoritmo de Brendan McKay, que é o mais rápido algoritmo para o Problema do Isomorfismo de Grafos conhecido. Ao final, implementamos o algoritmo proposto nesta dissertação e o algoritmo de McKay. Após a comparação dos dois algoritmos, verificamos que os resultados obtidos pelo algoritmo proposto não foram satisfatórios, porém apresentamos possíveis melhorias de como deixá-lo mais eficiente. / In this work we study the Graph Isomorphism Problem and their complexity to solve it. Our main contribution is to propose an algorithm for the general case of the Problem, based on partitioning the set vertex and perfect matchings of bipartite graphs. We also studied the Brendan McKay¿s algorithm, who is the fastest algorithm for the Graph Isomorphism Problem known. At the end, we implemented the algorithm proposed in this dissertation and McKay¿s algorithm. After comparison of the two algorithms, we found that the results obtained by the proposed algorithm were not satisfactory, but improvements are possible as to make it more efficient.