Bipartite graphs model the key relations in many large scale real-world data: customers purchasing items, legislators voting for bills, people's affiliation with different social groups, faults occurring in vehicles, etc. However, it is challenging to visualize large scale bipartite graphs with tens of thousands or even more nodes or edges. In this paper, we propose a novel visual summarization technique for bipartite graphs based on the minimum description length (MDL) principle. The method simultaneously groups the two different set of nodes and constructs aggregated bipartite relations with balanced granularity and precision. It addresses the key trade-off that often occurs for visualizing large scale and noisy data: acquiring a clear and uncluttered overview while maximizing the information content in it. We formulate the visual summarization task as a co-clustering problem and propose an efficient algorithm based on locality sensitive hashing (LSH) that can easily scale to large graphs under reasonable interactive time constraints that previous related methods cannot satisfy. The method leads to the opportunity of introducing a visual analytics framework with multiple levels-of-detail to facilitate interactive data exploration. In the framework, we also introduce a compact visual design inspired by adjacency list representation of graphs as the building block for a small multiples display to compare the bipartite relations for different subsets of data. We showcase the applicability and effectiveness of our approach by applying it on synthetic data with ground truth and performing case studies on real-world datasets from two application domains including roll-call vote record analysis and vehicle fault pattern analysis. Interviews with experts in the political science community and the automotive industry further highlight the benefits of our approach.
Fig. 1. The visualization interface of GraphQ contains: (1) A query editing panel to specify the subgraph patterns and initiate the search.(2.1) (2.2) Query result panels to display the retrieved results. The graph thumbnails can be displayed in overview and detail modes.(3) A statistics and filtering panel that helps users select a graph to construct example-based query, and visualizes the distribution of the query results in the database. (4) A query option control panel to specify whether fuzzy-pattern search is enabled and whether the node-match should be highlighted. (5) A popup window for pairwise comparison between the query pattern and the returned result.The figure shows a case study on program workflow graph pattern search and the details are described in Section 5.1.
Figure 1: The overview (left) of an interior scene illuminated by traditional shadow mapping and close look (right) to certain areas. Both constant depth bias (first column) and slope scale depth bias (second column) suffer from shadow acne and shadow detachment to different extent. Our method (third column) has no visible acne and preserves more shadow details. Dual depth layers depth bias (fourth column) is used as reference to compare our method against. AbstractShadow aliasing due to limited storage precision has been plaguing discrete shadowing algorithms for decades. We present a simple method to eliminate false self-shadowing through adaptive depth bias. Unlike existing methods which simply set the weight of the bias based on surface slope or utilize the second nearest surface, we evaluate the bound of bias for each fragment and compute the optimal bias within the bound. Our method introduces small overhead, preserves more shadow details than widely used constant bias and slope scale bias and works for common 2D shadow maps as well as 3D binary shadow volumes.
Figure 1: (Left) A scene with no participating media, media with no shadows, and with imperfect voxelized shadow volumes. (Right) A more complex scene without and with volumetric shadows. All images also use imperfect voxelized shadow volumes for surface shadows. AbstractVoxelized shadow volumes [Wyman 2011] provide a discretized view-dependent representation of shadow volumes, but are limited to point or directional lights. We extend them to allow dynamic volumetric visibility from area light sources using imperfect shadow volumes. We show a coarser visibility sampling suffices for area lights. Combining this coarser resolution with a parallel shadow volume construction enables interactive rendering of dynamic volumetric shadows from area lights in homogeneous single-scattering media, at under 4x the cost of hard volumetric shadows.
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