A Regularized Graph Layout Framework for Dynamic Network Visualization

Many real-world networks, including social and information networks, are dynamic structures that evolve over time. Such dynamic networks are typically visualized using a sequence of static graph layouts. In addition to providing a visual representation of the network structure at each time step, the sequence should preserve the mental map between layouts of consecutive time steps to allow a human to interpret the temporal evolution of the network. In this paper, we propose a framework for dynamic network visualization in the on-line setting where only present and past graph snapshots are available to create the present layout. The proposed framework creates regularized graph layouts by augmenting the cost function of a static graph layout algorithm with a grouping penalty, which discourages nodes from deviating too far from other nodes belonging to the same group, and a temporal penalty, which discourages large node movements between consecutive time steps. The penalties increase the stability of the layout sequence, thus preserving the mental map. We introduce two dynamic layout algorithms within the proposed framework, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL). We apply these algorithms on several data sets to illustrate the importance of both grouping and temporal regularization for producing interpretable visualizations of dynamic networks.Comment: To appear in Data Mining and Knowledge Discovery, supporting material (animations and MATLAB toolbox) available at http://tbayes.eecs.umich.edu/xukevin/visualization_dmkd_201

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

Grid Representations and the Chromatic Number

A grid drawing of a graph maps vertices to grid points and edges to line segments that avoid grid points representing other vertices. We show that there is a number of grid points that some line segment of an arbitrary grid drawing must intersect. This number is closely connected to the chromatic number. Second, we study how many columns we need to draw a graph in the grid, introducing some new \NP-complete problems. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures asked by David Flores-Pe\~naloza and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure

Node-attribute graph layout for small-world networks

Small-world networks are a very commonly occurring type of graph in the real-world, which exhibit a clustered structure that is not well represented by current graph layout algorithms. In many cases we also have information about the nodes in such graphs, which are typically depicted on the graph as node colour, shape or size. Here we demonstrate that these attributes can instead be used to layout the graph in high-dimensional data space. Then using a dimension reduction technique, targeted projection pursuit, the graph layout can be optimised for displaying clustering. The technique out-performs force-directed layout methods in cluster separation when applied to a sample, artificially generated, small-world network