Spectrum-preserving sparsification for visualization of big graphs

We study a problem motivated by rectilinear schematization of geographic maps. Given a biconnected plane graph G and an integer $$k\ge 0$$, does G have a strict-orthogonal drawing with at most k reflex angles per face? For $$k=0$$ the problem is equivalent to realizing each face as a rectangle. The problem can be reduced to a max-flow problem in some linear-size nonplanar network, but the best solutions require $$\varOmega n^{1.5} \log n\log k$$ time. We describe a graph matching approach that can decide strict-orthogonal drawability for arbitrary reflex complexity k in $$Onk^{1.5}$$ time, which is faster for constant values of k. In contrast, if the embedding is not fixed, we prove that it is NP-complete to decide whether a planar graph admits a strict-orthogonal drawing with reflex face complexity 4.

/pdf/orthogonal-layout-with-optimal-face-complexity-1wgfhq2q2g.pdf

Orthogonal Layout with Optimal Face Complexity

Complex non-convex ad hoc networks (CNCAH) contain intersecting polygons and edges. In many instances, the layouts of these networks are not entirely convex in shape. In this article, we propose a Kamada-Kawai-based algorithm called W-KK-MS for boundary node detection problems, which is capable of aligning node positions while achieving high sensitivity, specificity, and accuracy in producing a visual drawing from the input network topology. The algorithm put forward in this article selects and assigns weights to top-k nodes in each iteration to speed up the updating process of nodes. We also propose a novel approach to detect and unfold stacked regions in CNCAH networks. Experimental results show that the proposed algorithms can achieve fast convergence on boundary node detection in CNCAH networks and are able to successfully unfold stacked regions. The design and implementation of a prototype system called ELnet for analyzing CNCAH networks is also described in this article. The ELnet system is capable of generating synthetic networks for testing, integrating with force-directed algorithms, and visualizing and analyzing algorithms’ outcomes.

Boundary Node Detection and Unfolding of Complex Non-Convex Ad Hoc Networks

This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition between planarity and non-planarity for edge density $d=0.5$. However, this asymptotic property does not clarify what happens for graphs of reduced size. We show that an unexpectedly sharp transition is also exhibited by small and medium sized graphs. Also, we show that the same “tipping point” behavior can be observed for some restrictions or relaxations of planarity (we considered outerplanarity and near-planarity, respectively).

A Tipping Point for the Planarity of Small and Medium Sized Graphs

This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition between planarity and non-planarity for edge density d=0.5. However, this asymptotic property does not clarify what happens for graphs of reduced size. We show that an unexpectedly sharp transition is also exhibited by small and medium sized graphs. Also, we show that the same "tipping point" behavior can be observed for some restrictions or relaxations of planarity (we considered outerplanarity and near-planarity, respectively).

Open graph drawing framework (OGDF)

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Open graph drawing framework (OGDF)