10,950 research outputs found
Multidimensional Scaling Using Majorization: SMACOF in R
In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it is implemented in an R package of the same name which is presented in this article. We extend the basic SMACOF theory in terms of configuration constraints, three-way data, unfolding models, and projection of the resulting configurations onto spheres and other quadratic surfaces. Various examples are presented to show the possibilities of the SMACOF approach offered by the corresponding package.
Anisotropic Radial Layout for Visualizing Centrality and Structure in Graphs
This paper presents a novel method for layout of undirected graphs, where
nodes (vertices) are constrained to lie on a set of nested, simple, closed
curves. Such a layout is useful to simultaneously display the structural
centrality and vertex distance information for graphs in many domains,
including social networks. Closed curves are a more general constraint than the
previously proposed circles, and afford our method more flexibility to preserve
vertex relationships compared to existing radial layout methods. The proposed
approach modifies the multidimensional scaling (MDS) stress to include the
estimation of a vertex depth or centrality field as well as a term that
penalizes discord between structural centrality of vertices and their alignment
with this carefully estimated field. We also propose a visualization strategy
for the proposed layout and demonstrate its effectiveness using three social
network datasets.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Embedding Graphs under Centrality Constraints for Network Visualization
Visual rendering of graphs is a key task in the mapping of complex network
data. Although most graph drawing algorithms emphasize aesthetic appeal,
certain applications such as travel-time maps place more importance on
visualization of structural network properties. The present paper advocates two
graph embedding approaches with centrality considerations to comply with node
hierarchy. The problem is formulated first as one of constrained
multi-dimensional scaling (MDS), and it is solved via block coordinate descent
iterations with successive approximations and guaranteed convergence to a KKT
point. In addition, a regularization term enforcing graph smoothness is
incorporated with the goal of reducing edge crossings. A second approach
leverages the locally-linear embedding (LLE) algorithm which assumes that the
graph encodes data sampled from a low-dimensional manifold. Closed-form
solutions to the resulting centrality-constrained optimization problems are
determined yielding meaningful embeddings. Experimental results demonstrate the
efficacy of both approaches, especially for visualizing large networks on the
order of thousands of nodes.Comment: Submitted to IEEE Transactions on Visualization and Computer Graphic
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