Graph Decomposition Research Articles

Decompositions of quasirandom hypergraphs into hypergraphs of bounded degree

We prove that any quasirandom uniform hypergraph H H can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the sparse setting when the density of H H quickly tends to 0 0 in terms of the number of vertices of H H . Our results answer and address questions of Kim, Kühn, Osthus and Tyomkyn; and Glock, Kühn and Osthus as well as Keevash. The provided approximate decompositions exhibit strong quasirandom properties which are very useful for forthcoming applications. Our results also imply approximate solutions to natural hypergraph versions of long-standing graph decomposition problems, as well as several decomposition results for (quasi)random simplicial complexes into various more elementary simplicial complexes such as triangulations of spheres and other manifolds.

Multiscale Methods for Signal Selection in Single-Cell Data.

Analysis of single-cell transcriptomics often relies on clustering cells and then performing differential gene expression (DGE) to identify genes that vary between these clusters. These discrete analyses successfully determine cell types and markers; however, continuous variation within and between cell types may not be detected. We propose three topologically motivated mathematical methods for unsupervised feature selection that consider discrete and continuous transcriptional patterns on an equal footing across multiple scales simultaneously. Eigenscores () rank signals or genes based on their correspondence to low-frequency intrinsic patterning in the data using the spectral decomposition of the Laplacian graph. The multiscale Laplacian score (MLS) is an unsupervised method for locating relevant scales in data and selecting the genes that are coherently expressed at these respective scales. The persistent Rayleigh quotient (PRQ) takes data equipped with a filtration, allowing the separation of genes with different roles in a bifurcation process (e.g., pseudo-time). We demonstrate the utility of these techniques by applying them to published single-cell transcriptomics data sets. The methods validate previously identified genes and detect additional biologically meaningful genes with coherent expression patterns. By studying the interaction between gene signals and the geometry of the underlying space, the three methods give multidimensional rankings of the genes and visualisation of relationships between them.

ASURAT: functional annotation-driven unsupervised clustering of single-cell transcriptomes.

MotivationSingle-cell RNA sequencing (scRNA-seq) analysis reveals heterogeneity and dynamic cell transitions. However, conventional gene-based analyses require intensive manual curation to interpret biological implications of computational results. Hence, a theory for efficiently annotating individual cells remains warranted.ResultsWe present ASURAT, a computational tool for simultaneously performing unsupervised clustering and functional annotation of disease, cell type, biological process and signaling pathway activity for single-cell transcriptomic data, using a correlation graph decomposition for genes in database-derived functional terms. We validated the usability and clustering performance of ASURAT using scRNA-seq datasets for human peripheral blood mononuclear cells, which required fewer manual curations than existing methods. Moreover, we applied ASURAT to scRNA-seq and spatial transcriptome datasets for human small cell lung cancer and pancreatic ductal adenocarcinoma, respectively, identifying previously overlooked subpopulations and differentially expressed genes. ASURAT is a powerful tool for dissecting cell subpopulations and improving biological interpretability of complex and noisy transcriptomic data.Availability and implementationASURAT is published on Bioconductor (https://doi.org/10.18129/B9.bioc.ASURAT). The codes for analyzing data in this article are available at Github (https://github.com/keita-iida/ASURATBI) and figshare (https://doi.org/10.6084/m9.figshare.19200254.v4).Supplementary information Supplementary data are available at Bioinformatics online.

Even Cycle Decompositions of index 3 by a novel coloring technique

An even cycle decomposition of an Eulerian graph is a partition of the edge set into even cycles. We color the even cycles so as two cycles sharing at least one vertex receive distinct colors. If k is the minimum number of required colors in such a coloring, then the even cycle decomposition has index k . We prove that the line graph of every bridgeless cubic graph of oddness 2 has an even cycle decomposition of index 3. The same property holds for the line graphs of some infinite families of class 2 cubic graphs with arbitrary large oddness. The construction of even cycle decompositions of index 3 in the line graph of a class 2 cubic graph is alternative to the constructions that are known in the literature; that one for the line graph of a cubic graph with arbitrary large oddness is also a new contribution to the more general problem on the existence of even cycle decompositions in the line graph of a bridgeless cubic graph. The constructions are obtained by applying a novel coloring technique on the edges of the line graph.

Non-centralised coordinated optimisation for maximising offshore wind farm power via a sparse communication architecture

The motivation for establishing sparse communication architecture is to reduce the complexity and size of the power optimisation problem being solved by centralised controllers of a large-scale offshore wind farm. In this regard, this article introduced the sub-graph extraction technique to construct sparse communication architectures while ignoring the weakly coupled wake effects between turbines to develop two non-centralised coordination optimisation strategies for maximising power output. The non-centralised optimisation framework mainly uses the idea of graph decomposition that creates an original wake digraph by quantifying wake intensity, defining the spectral norm of the wake weight matrix as a constraint to obtain pruned wake sub-digraph. Then, for determining communication neighbours, we used a depth-first tree search arithmetic to cluster subsets of turbines by which the impact of neighbouring turbines was considered. On this basis, designing a suitable optimisation-based distributed or decentralised strategy, defining shared variables and consensus stages among the local controllers, yaw angles from such controllers are optimally coordinated to maximise the total power of the whole wind farm. Finally, a case study of a wind farm with 36 turbine simulation results showed that the scale of the non-convex optimisation problem could be efficiently reduced without affecting the power output while reducing execution time under various turbulence intensities.

INDUCED \(nK_{2}\) DECOMPOSITION OF INFINITE SQUARE GRIDS AND INFINITE HEXAGONAL GRIDS

The induced \(nK_2\) decomposition of infinite square grids and hexagonal grids are described here. We use the multi-level distance edge labeling as an effective technique in the decomposition of square grids. If the edges are adjacent, then their color difference is at least 2 and if they are separated by exactly a single edge, then their colors must be distinct. Only non-negative integers are used for labeling. The proposed partitioning technique per the edge labels to get the induced \(nK_2\) decomposition of the ladder graph is the square grid and the hexagonal grid.

Graph decomposition for constructing blackstart restoration strategies in benchmark cases

This paper describes methods for constructing benchmark cases and solution strategies related to restoring an electric grid model from a blackstart condition. Existing public cases for grid restoration problems are limited in size and scope—this paper delineates the features required to create new synthetic, detailed datasets for this purpose. As part of validating these datasets, benchmark results are included for an electric grid restoration strategy based on a formulation of the restoration problem as a time sequence of ac power flow solutions. The algorithm produces a benchmark restoration sequence for an example 200-bus case, showing how critical loads can be restored within 45 min, the majority of the grid within 6 h, and all loads within 36 h. At the core of the solution strategy is a directed graph decomposition heuristic, as the algorithm builds a bus energization spanning forest. Because these datasets are built synthetically, they can have a high degree of realism in features such as geographic coordinates and modeling complexities, while avoiding concerns of data confidentiality, meaning that the data produced in this work can be made publicly available for the benefit of the research community.

GraceOutZip: Graceful labeled Graph summarization approach with Percolated anomalous discovery

Voluminous graph data management is a daunting problem in every real world application of a kind, besides the advancements in computation and storage technology. Efficient graph summarization techniques were contributed to achieve the substantial need for preserving novelty in social graphs. A GraceOutZip compression friendly graph reordering technique using graceful labeling strategy is adopted. User defined probabilistic selection method that provides unique labels for every identified outlier for potential use. Proposed method exploits unicycle-star based community representation rendering assignment of both node and edge labels based on graceful property. A novel mathematical programming model GraceOutZip is proposed to perform lossless compression with graph decomposition and unique label arrangement with intention for futuristic graph reconstruction. The experimental study on different real world network datasets demonstrates that GraceOutZip shows better scalable performance in perspective of interactive large-scale visual analytics and query optimization with 4 times better compression than the state-of-the-art representative method.

Graph decomposition methods for variance balanced block designs with correlated errors

Block designs have an important historical footing in the design of experiments. Let θ be the treatment mean vector in the statistical model with fixed block effects. It is desirable to have the covariance of the least squares estimator of the treatment effects, Cov(θ̂), be a completely symmetric matrix. When this occurs, we have a variance balanced design (VBD). It is known that a pairwise balanced design can be converted into a VBD by choosing blocks with a multiplicity inversely proportional to the block size. This assumes that experimental errors are uncorrelated. However, this need not be the case, especially in ecological experiments where blocks have spatial correlation or laboratory settings with temporal correlation. This paper proposes the use of graph decompositions as a generalization of balanced incomplete block designs for the construction of VBDs in the presence of correlated errors. A general result is obtained to construct VBDs for various error correlation structures. Several applications are given to illustrate the relationship between graph decompositions and VBDs.

Correction to my paper: "The quasi-series decomposition of two-terminal graphs"

Correction to my paper: "The quasi-series decomposition of two-terminal graphs"

The quasi-series decomposition of two-terminal graphs

The quasi-series decomposition of two-terminal graphs

Theory of collective topologically protected Majorana fermion excitations of networks of localized Majorana modes

Predictions of localized Majorana modes, and ideas for manipulating these degrees of freedom, are the two key ingredients in proposals for physical platforms for Majorana quantum computation. Several proposals envisage a scalable network of such Majorana modes coupled bilinearly to each other by quantum-mechanical mixing amplitudes. Here, we develop a theoretical framework for characterizing collective topologically protected zero-energy Majorana fermion excitations of such networks of localized Majorana modes. A key ingredient in our work is the Gallai-Edmonds decomposition of a general graph, which we use to obtain an alternate ``local'' proof of a ``global'' result of Lov\'asz and Anderson on the dimension of the topologically protected null space of real skew-symmetric (or pure-imaginary Hermitian) adjacency matrices of general graphs. Our approach to Lov\'asz and Anderson's result constructs a maximally localized basis for the said null space from the Gallai-Edmonds decomposition of the graph. Applied to the graph of the Majorana network in question, this gives a method for characterizing basis-independent properties of these collective topologically protected Majorana fermion excitations, and relating these properties to the correlation function of monomers in the ensemble of maximum matchings (maximally packed dimer covers) of the corresponding network graph. Our approach can also be used to identify signatures of zero-energy excitations in systems modeled by a free-fermion Hamiltonian with a hopping matrix of this type; an interesting example is provided by vacancy-induced Curie tails in generalizations (on nonbipartite lattices) of Kitaev's honeycomb model.

Decomposition of cubic graphs with cyclic connectivity 5

Let G be a cyclically 5-connected cubic graph with a 5-edge-cut separating G into two cyclic components G1 and G2. We prove that each component Gi can be completed to a cyclically 5-connected cubic graph by adding three vertices, unless Gi is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity 4 from 1988.

MANAGEMENT IN HIGH-DIMENSIONAL MARKOV SYSTEMS

A problem of analyzing Markov systems along with a large number of states has been considered. The conventional computational procedure for obtaining analytical ratios for calculating the distribution of system states is based on the use of a system of Kolmogorov differential equations. The system of linear algebraic equations being formed later can be easily solved numerically. However, the complexity of obtaining an analytical solution increases rapidly with the increase in the problem dimension. In this regard, the purpose is to develop an effective method for studying Markov systems, the computational procedure of which ensures the possibility of obtaining solutions for high-dimensional models. The method is based on the decomposition of states graph and system transitions. The obtained analytical expressions allow to set and solve the problem of rational resource distribution for changing the values of the system parameters to increase its efficiency. The method ensures the possibility of solving management problems in Markov systems along with a large number of possible states. An example of method application has been considered.

Impact of COVID-19 Pandemic on Vertical Price Transmission in the U.S. Fresh Banana Market

The U.S. imports about two billion dollars of fresh bananas, accounting for over 99 percent of domestic banana consumption annually. The COVID-19 pandemic disrupted the fresh banana supply chain and caused unexpected price movements along the marketing channel. This research investigated the impact of the COVID-19 pandemic on price adjustments in the U.S. fresh banana market. A Vector Error Correction (VEC) model was employed to evaluate the speeds of price adjustments along the U.S. banana marketing channel at the import and retail levels, and historical decomposition graphs were used to investigate the magnitude of price adjustments caused by the COVID-19 shock. The results show that the deviation from the long-run equilibrium caused by the shock was corrected faster for the import prices than retail prices. Hence, the speeds of price adjustments were asymmetric in the period of the COVID-19 shock. Additionally, the magnitudes of price changes caused by the pandemic shock were different, leading to increased price margins. These results point to the inefficiency of the banana marketing channel with welfare, policy, and agribusiness implications.

Decomposition of hypercube graphs into paths and cycles of length four

By a -decomposition of a graph G, we mean a partition of the edge set of G into p paths of length 4 and q cycles of length 4. In this paper, we give conditions for a -decomposition of the n-dimensional hypercube graph Qn for all possible values of p and q. In particular, we find the necessary and sufficient conditions for such decomposition in Qn when n is even, and n is odd,

Non-zero sum Heffter arrays and their applications

In this paper we introduce a new class of partially filled arrays that, as Heffter arrays, are related to difference families, graph decompositions and biembeddings. A non-zero sum Heffter arrayNH(m,n;h,k) is an m×n p.f. array with entries in Z2nk+1 such that: each row contains h filled cells and each column contains k filled cells; for every x∈Z2nk+1∖{0}, either x or −x appears in the array; the sum of the elements in every row and column is different from 0 (in Z2nk+1). Here first we explain the connections with relative difference families and with path decompositions of the complete multipartite graph. Then we present a complete solution for the existence problem and a constructive complete solution for the square case and for the rectangular case with no empty cells when the additional, very restrictive, property of being “globally simple” is required. Finally, we show how these arrays can be used to construct biembeddings of complete graphs.

Structural Parameterizations with Modulator Oblivion

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph when parameterized by k. While this investigation fits naturally into the recent trend of what is called ‘structural parameterizations’, here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least \(k^{{{\mathcal {O}}}(k)}n^{{{\mathcal {O}}}(1)}\) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design \(2^{{{\mathcal {O}}}(k)}n^{{{\mathcal {O}}}(1)}\) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in \(2^{{{\mathcal {O}}}(k)}n^{{{\mathcal {O}}}(1)}\) time where each bag is a union of four cliques and \({{\mathcal {O}}}(k)\) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the strong exponential time hypothesis.

Distributed minimum vertex coloring and maximum independent set in chordal graphs

We give deterministic distributed (1+ϵ)-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O(1ϵlog⁡n) rounds, and our independent set algorithm has a runtime of O(1ϵlog⁡(1ϵ)log⁎⁡n) rounds. For coloring, existing lower bounds imply that the dependencies on 1ϵ and log⁡n are best possible. For independent set, we prove that Ω(1ϵ) rounds are necessary.Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log⁡n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O(log⁡1ϵ) layers are required as they already contain a large enough independent set. We develop a (1+ϵ)-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers.While tree decompositions have only played a minor role in distributed computing, our work demonstrates their potential for designing efficient distributed algorithms.

Efficient parallel graph trimming by arc-consistency

Given a large data graph, trimming techniques can reduce the search space by removing vertices without outgoing edges. One application is to speed up the parallel decomposition of graphs into strongly connected components (SCC decomposition), which is a fundamental step for analyzing graphs. We observe that graph trimming is essentially a kind of arc-consistency problem, and AC-3, AC-4, and AC-6 are the most relevant arc-consistency algorithms for application to graph trimming. The existing parallel graph trimming methods require worst-case \(\mathcal O(nm)\) time and worst-case \(\mathcal O(n)\) space for graphs with n vertices and m edges. We call these parallel AC-3-based as they are much like the AC-3 algorithm. In this work, we propose AC-4-based and AC-6-based trimming methods. That is, AC-4-based trimming has an improved worst-case time of \(\mathcal O(n+m)\) but requires worst-case space of \(\mathcal O(n+m)\); compared with AC-4-based trimming, AC-6-based has the same worst-case time of \(\mathcal O(n+m)\) but an improved worst-case space of \(\mathcal O(n)\). We parallelize the AC-4-based and AC-6-based algorithms to be suitable for shared-memory multi-core machines. The algorithms are designed to minimize synchronization overhead. For these algorithms, we also prove the correctness and analyze time complexities with the work-depth model. In experiments, we compare these three parallel trimming algorithms over a variety of real and synthetic graphs on a multi-core machine, where each core corresponds to a worker. Specifically, for the maximum number of traversed edges per worker by using 16 workers, AC-3-based traverses up to 58.3 and 36.5 times more edges than AC-6-based trimming and AC-4-based trimming, respectively. That is, AC-6-based trimming traverses much fewer edges than other methods, which is meaningful especially for implicit graphs. In particular, for the practical running time, AC-6-based trimming achieves high speedups over graphs with a large portion of trimmable vertices.