Graph/Network Visualization


 Derrick Casey
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1 Graph/Network Visualization Data model: graph structures (relations, knowledge) and networks. Applications: Telecommunication systems, Internet and WWW, Retailers distribution networks knowledge representation Trade Collaborations literature citations, etc. 1 What is a Graph? Vertices (nodes) Edges (links) Adjacency list : 2 2: 1, 3 3: Adjacency matrix Drawing 2 1
2 Graph Terminology Graphs can have cycles Graph edges can be directed or undirected The degree of a vertex is the number of edges connected to it Indegree and outdegree for directed graphs Graph edges can have values (weights) on them (nominal, ordinal or quantitative) 3 Trees are Different Subcase of general graph No cycles Typically directed edges Special designated root vertex Spring 2002 CS
3 Issues in Graph visualization Graph drawing Layout and positioning Scale: large scale graphs are difficult Navigation: changing focus and scale 5 Vertex Issues Shape Color Size Location Label 6 3
4 Edge Issues Color Size Label Form Polyline, straight line, orthogonal, grid, curved, planar, upward/downward,... 7 Aesthetic Considerations Crossings  minimize number of edge crossings Total Edge Length  minimize total length Area  minimize towards efficiency Maximum Edge Length  minimize longest edge Uniform Edge Lengths  minimize variances Total Bends  minimize orthogonal towards straightline 8 4
5 Graph drawing optimization 3DGraph Drawing 5
6 Graph visualization techniques Nodelink approach Layered graph drawing (Sugiyama) Forcedirected layout Multidimensional scaling (MDS) Adjacency Matrix Attribute based approach 11 Sugiyama (layered) method Suitable for directed graphs with natural hierarchies: All edges are oriented in a consistent direction and no pairs of edges cross 6
7 Sugiyama : Building Hierarchy Assign layers according to the longest path of each vertex Dummy vertices are used to avoid path across multiple layers. Vertex permutation within a layer to reduce edge crossing. Exact optimization is NPhard need heuristics. Sugiyama : Building Hierarchy 7
8 Force directed graph layout No natural hierarchy or order Based on principles of physics The Spring Model Using springs to represent nodenode relations. Minimizing energy function to reach energy equilibrium. Initial layout is important Local minimal problem 8
9 Network of character cooccurrence in Les Misérables 9
10 Multidimensional Scaling Dimension reduction to 2D Graph distance of two nodes are as close to 2D Euclidean distance as possible MDS is a global approach Distance between two nodes: shortest path (classical scaling). Weighted distances (,, w,,, MDS for graph layout 10
11 Other NodeLink Methods Orthogonal layout Suitable for UML graph Radial graph Often used in social networks Nested graph layout Apply graph layout hierarchically Suitable for graphs with hierarchy Arc Diagrams Arc Diagram Les Misérables character relations 11
12 Arc Diagram EU Financial Crisis: 12
13 Summary: NodeLink Pros Intuitive Good for global structure Flexible, with variations Cons Complexity >O(N 2 ) Not suitable for large graphs Adjacency Matrix matrix, for a graph with N nodes. (i, j) position represent the relationship of the ith node and jth node. 13
14 Adjacency Matrix Edge weight Directional edges Sorting: node order Path searching and path tracking? 28 14
15 Node Order Path Tracking 15
16 Adjacency matrix summary Avoid edge crossing, suitable for dense graphs Visually more scalable Visualization is not intuitive Hard to track a path MatLink 16
17 Hybrid Layout Using adjacency matrix to represent small communities Nodelink for relationships between communities NodeTrix 17
18 GMap Visualizing graphs and clusters as geographic maps to represent node relations (geographic neighbors) Topological graph simplification Reducing amount of data Reducing nodes: clustering Reducing edges: minimal spanning tree Edge bundling Problems: Loss of data 18
19 Clustering Edge Bundling 19
20 Force Directed Edge Bundling Edges are modeled as flexible springs that are able to attract each other. Geometry Based Edge Bundling Edges clusters are found based on a geometric control mesh. 20
21 Multilevel Agglomerative Edge Bundling Bottomup merging approach, similar to hierarchical clustering Minimize amount of ink used to render a graph. Skeletonbased Edge Bundling Skeletons: medial axes of edges which are similar in terms of positions information. Iteratively attracting edges towards the skeletons. 21
22 Comparison Interaction Viewing Pan, Zoom, Rotate Interacting with graph nodes and edges Pick, highlight, delete, move Structural interaction Local reorder and relayout Focus+Context Rollup & Drilldown 22
23 Fisheye Focus+Context; Overviews + detailsondemand Distortion to magnify areas of interest: zoom factors of 35 Multiscale spaces: Zoom in/out & Pan left/right Interaction with Social Networks Need to consider the social factors and behaviors related to nodes and edges 23
24 Graph Visualization Tools Prefuse (Java) UCINET / NetDraw Sentinel Visualizer JUNG (Java Universal Network/Graph framework) Graphviz Gephi TouchGraph Flare: ActionScript Library 24
25 Pajek 25
26 Sentinel Visualizer Link Analysis, Data Visualization, Geospatial Mapping, and Social Network Analysis (SNA) UCINET / NetDraw Analysis and visualization of networks and graphs Example: trade 52 26
27 Example: traffic 53 Example: Subway map 54 27
28 Web page connections 55 Communication Networks 28
29 Hypergraphs Definition A hypergraph is a generalization of a graph, where an edge can connect any number of vertices. A hypergraph H is a pair H = (V,E)whereV is a set of nodes/vertices, and E is a set of nonempty subsets of V called hyperedges/links. 29
30 The Hypergraph H = (V,E) where V = (1,2,3,4,5) and E = {(1,2) (2,3,5) (1,3), (5,4) (2,3)} Applications Data Mining Biological Interactions Social Networks Circuit Diagrams 30
31 Graph Representations Edge Nodes: Representative Graph 31
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