BC tree-based spectral sampling for big complex network visualization

Graph sampling methods have been used to reduce the size and complexity of big complex networks for graph mining and visualization. However, existing graph sampling methods often fail to preserve the connectivity and important structures of the original graph. This paper introduces a new divide and conquer approach to spectral graph sampling based on graph connectivity, called the BC Tree (i.e., decomposition of a connected graph into biconnected components) and spectral sparsification. Specifically, we present two methods, spectral vertex sampling $$BC\_SV$$ B C _ S V and spectral edge sampling $$BC\_SS$$ B C _ S S by computing effective resistance values of vertices and edges for each connected component. Furthermore, we present $$DBC\_SS$$ D B C _ S S and $$DBC\_GD$$ D B C _ G D , graph connectivity-based distributed algorithms for spectral sparsification and graph drawing respectively, aiming to further improve the runtime efficiency of spectral sparsification and graph drawing by integrating connectivity-based graph decomposition and distributed computing. Experimental results demonstrate that $$BC\_SV$$ B C _ S V and $$BC\_SS$$ B C _ S S are significantly faster than previous spectral graph sampling methods while preserving the same sampling quality. $$DBC\_SS$$ D B C _ S S and $$DBC\_GD$$ D B C _ G D obtain further significant runtime improvement over sequential approaches, and $$DBC\_GD$$ D B C _ G D further achieves significant improvements in quality metrics over sequential graph drawing layouts.

Studying the use and effect of graph decomposition in qualitative spatial and temporal reasoning

We survey the use and effect of decomposition-based techniques in qualitative spatial and temporal constraint-based reasoning, and clarify the notions of a tree decomposition, a chordal graph, and a partitioning graph, and their implication with a particular constraint property that has been extensively used in the literature, namely, patchwork. As a consequence, we prove that a recently proposed decomposition-based approach that was presented in the study by Nikolaou and Koubarakis for checking the satisfiability of qualitative spatial constraint networks lacks soundness. Therefore, the approach becomes quite controversial as it does not seem to offer any technical advance at all, while results of an experimental evaluation of it in a following work presented in the study by Sioutis become questionable. Finally, we present a particular tree decomposition that is based on the biconnected components of the constraint graph of a given large network, and show that it allows for cost-free utilization of parallelism for a qualitative constraint language that has patchwork for satisfiable atomic networks.