56,551 research outputs found
Out-Tournament Adjacency Matrices with Equal Ranks
Much work has been done in analyzing various classes of tournaments, giving a partial characterization of tournaments with adjacency matrices having equal and full real, nonnegative integer, Boolean, and term ranks. Relatively little is known about the corresponding adjacency matrix ranks of local out-tournaments, a larger family of digraphs containing the class of tournaments. Based on each of several structural theorems from Bang-Jensen, Huang, and Prisner, we will identify several classes of out-tournaments which have the desired adjacency matrix rank properties. First we will consider matrix ranks of out-tournament matrices from the perspective of the structural composition of the strong component layout of the adjacency matrix. Following that, we will consider adjacency matrix ranks of an out-tournament based on the cycles that the out-tournament contains. Most of the remaining chapters consider the adjacency matrix ranks of several classes of out-tournaments based on the form of their underlying graphs. In the case of the strong out-tournaments discussed in the final chapter, we examine the underlying graph of a representation that has the strong out-tournament as its catch digraph
SLACID - Sparse Linear Algebra in a Column-Oriented In-Memory Database System
Scientific computations and analytical business applications are often based on linear algebra operations on large, sparse matrices. With the hardware shift of the primary storage from disc into memory it is now feasible to execute linear algebra queries directly in the database engine. This paper presents and compares different approaches of storing sparse matrices in an in-memory column-oriented database system. We show that a system layout derived from the compressed sparse row representation integrates well with a columnar database design and that the resulting architecture is moreover amenable to a wide range of non-numerical use cases when dictionary encoding is used. Dynamic matrix manipulation operations, like online insertion or deletion of elements, are not covered by most linear algebra frameworks. Therefore, we present a hybrid architecture that consists of a read-optimized main and a write-optimized delta structure and evaluate the performance for dynamic sparse matrix workloads by applying workflows of nuclear science and network graphs
Optimizing an Organized Modularity Measure for Topographic Graph Clustering: a Deterministic Annealing Approach
This paper proposes an organized generalization of Newman and Girvan's
modularity measure for graph clustering. Optimized via a deterministic
annealing scheme, this measure produces topologically ordered graph clusterings
that lead to faithful and readable graph representations based on clustering
induced graphs. Topographic graph clustering provides an alternative to more
classical solutions in which a standard graph clustering method is applied to
build a simpler graph that is then represented with a graph layout algorithm. A
comparative study on four real world graphs ranging from 34 to 1 133 vertices
shows the interest of the proposed approach with respect to classical solutions
and to self-organizing maps for graphs
NodeTrix: Hybrid Representation for Analyzing Social Networks
The need to visualize large social networks is growing as hardware
capabilities make analyzing large networks feasible and many new data sets
become available. Unfortunately, the visualizations in existing systems do not
satisfactorily answer the basic dilemma of being readable both for the global
structure of the network and also for detailed analysis of local communities.
To address this problem, we present NodeTrix, a hybrid representation for
networks that combines the advantages of two traditional representations:
node-link diagrams are used to show the global structure of a network, while
arbitrary portions of the network can be shown as adjacency matrices to better
support the analysis of communities. A key contribution is a set of interaction
techniques. These allow analysts to create a NodeTrix visualization by dragging
selections from either a node-link or a matrix, flexibly manipulate the
NodeTrix representation to explore the dataset, and create meaningful summary
visualizations of their findings. Finally, we present a case study applying
NodeTrix to the analysis of the InfoVis 2004 coauthorship dataset to illustrate
the capabilities of NodeTrix as both an exploration tool and an effective means
of communicating results
An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width
We provide a doubly exponential upper bound in on the size of forbidden
pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field
of linear rank-width at most . As a corollary, we obtain a
doubly exponential upper bound in on the size of forbidden vertex-minors
for graphs of linear rank-width at most . This solves an open question
raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear
rank-width at most . European J. Combin., 41:242--257, 2014]. We also give a
doubly exponential upper bound in on the size of forbidden minors for
matroids representable over a fixed finite field of path-width at most .
Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on
the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series
B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded
path-width. To adapt this notion into linear rank-width, it is necessary to
well define partial pieces of graphs and merging operations that fit to
pivot-minors. Using the algebraic operations introduced by Courcelle and
Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we
define boundaried -labelled graphs and prove similar structure theorems for
pivot-minor and linear rank-width.Comment: 28 pages, 1 figur
HARP: Hierarchical Representation Learning for Networks
We present HARP, a novel method for learning low dimensional embeddings of a
graph's nodes which preserves higher-order structural features. Our proposed
method achieves this by compressing the input graph prior to embedding it,
effectively avoiding troublesome embedding configurations (i.e. local minima)
which can pose problems to non-convex optimization. HARP works by finding a
smaller graph which approximates the global structure of its input. This
simplified graph is used to learn a set of initial representations, which serve
as good initializations for learning representations in the original, detailed
graph. We inductively extend this idea, by decomposing a graph in a series of
levels, and then embed the hierarchy of graphs from the coarsest one to the
original graph. HARP is a general meta-strategy to improve all of the
state-of-the-art neural algorithms for embedding graphs, including DeepWalk,
LINE, and Node2vec. Indeed, we demonstrate that applying HARP's hierarchical
paradigm yields improved implementations for all three of these methods, as
evaluated on both classification tasks on real-world graphs such as DBLP,
BlogCatalog, CiteSeer, and Arxiv, where we achieve a performance gain over the
original implementations by up to 14% Macro F1.Comment: To appear in AAAI 201
A Regularized Graph Layout Framework for Dynamic Network Visualization
Many real-world networks, including social and information networks, are
dynamic structures that evolve over time. Such dynamic networks are typically
visualized using a sequence of static graph layouts. In addition to providing a
visual representation of the network structure at each time step, the sequence
should preserve the mental map between layouts of consecutive time steps to
allow a human to interpret the temporal evolution of the network. In this
paper, we propose a framework for dynamic network visualization in the on-line
setting where only present and past graph snapshots are available to create the
present layout. The proposed framework creates regularized graph layouts by
augmenting the cost function of a static graph layout algorithm with a grouping
penalty, which discourages nodes from deviating too far from other nodes
belonging to the same group, and a temporal penalty, which discourages large
node movements between consecutive time steps. The penalties increase the
stability of the layout sequence, thus preserving the mental map. We introduce
two dynamic layout algorithms within the proposed framework, namely dynamic
multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL). We
apply these algorithms on several data sets to illustrate the importance of
both grouping and temporal regularization for producing interpretable
visualizations of dynamic networks.Comment: To appear in Data Mining and Knowledge Discovery, supporting material
(animations and MATLAB toolbox) available at
http://tbayes.eecs.umich.edu/xukevin/visualization_dmkd_201
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