Patterns often appear in a variety of large, real-world networks, and interesting physical phenomena are often explained by network topology as in the case of the bow-tie structure of the World Wide Web, or the small world phenomenon in social networks. The discovery and modelling of such regular patterns has a wide application from disease propagation to financial markets. In this work we describe a newly discovered regularly occurring striation pattern found in the PageRank ordering of adjacency matrices that encode real-world networks. We demonstrate that these striations are the result of well-known graph generation processes resulting in regularities that are manifest in the typical neighborhood distribution. The spectral view explored in this paper encodes a tremendous amount about the explicit and implicit topology of a given network, so we also discuss the interesting network properties, outliers and anomalies that a viewer can determine from a brief look at the re-ordered matrix.
The adjacency matrix is a common tool for visualizing graph structure, and has been shown to perform well in providing an intuitive understanding of dense or large graphs, except in the task of path finding @cite_35 . The drawback of matrices, however, is that nodes can appear in any arbitrary order. Structures can often be revealed by re-ordering the nodes of the matrix to reveal clusters of related nodes. A study by Mueller al provided a comparison of different methods of ordering the matrices, using Random, BFS, DFS, King's algorithm, Reverse Cuthill McKee, Degree, Spectral, Separator tree partitioning algorithm, and Sloan ordering @cite_4 . They evaluated the different orderings in terms of their consistency and ability to reveal structure on graphs generated by different algorithms. They did not, however, investigate PageRank or any of the other ranking-models discussed in the present work.
@cite_35 @cite_4
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In this article, we describe a taxonomy of generic graph related tasks along with a computer-based evaluation designed to assess the readability of two representations of graphs: matrix-based representations and node-link diagrams. This evaluation encompasses seven generic tasks and leads to insightful recommendations for the representation of graphs according to their size and density. Typically, we show that when graphs are bigger than twenty vertices, the matrix-based visualization outperforms node-link diagrams on most tasks. Only path finding is consistently in favor of node-link diagrams throughout the evaluation. In this study, we examine the use of graph ordering algorithms for visual analysis of data sets using visual similarity matrices. Visual similarity matrices display the relationships between data items in a dot-matrix plot format, with the axes labeled with the data items and points drawn if there is a relationship between two data items. The biggest challenge for displaying data using this representation is finding an ordering of the data items that reveals the internal structure of the data set. Poor orderings are indistinguishable from noise whereas a good ordering can reveal complex and subtle features of the data. We consider three general classes of algorithms for generating orderings: simple graph theoretic algorithms, symbolic sparse matrix reordering algorithms, and spectral decomposition algorithms. We apply each algorithm to synthetic and real world data sets and evaluate each algorithm for interpretability (i.e., does the algorithm lead to images with usable visual features?) and stability (i.e., does the algorithm consistently produce similar results?). We also provide a detailed discussion of the results for each algorithm across the different graph types and include a discussion of some strategies for using ordering algorithms for data analysis based on these results.