Graph/Network

 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) 1: 2 2: 1, 3 3: 2 1 2 3 1 1 0 1 0 2 1 0 1 3 0 1 0 2 3 Drawing Adjacency

2 Graph Terminology

 Graphs can have cycles  Graph edges can be directed or undirected  The of a vertex is the number of edges connected to it – In-degree and out-degree 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

4 Issues in Graph visualization

 Layout and positioning  Scale: large scale graphs are difficult  Navigation: changing focus and scale

5 Vertex Issues

 Shape  Color  Size  Location  Label

6 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  -- minimize towards efficiency  Maximum Edge Length -- minimize longest edge  Uniform Edge Lengths -- minimize variances  Total Bends -- minimize orthogonal towards straight-line

8 Graph drawing optimization 3D-Graph Drawing Graph visualization techniques

 Node-link approach – (Sugiyama) – Force-directed layout – Multi-dimensional scaling (MDS) 

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 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 NP-hard – need heuristics. Sugiyama : Building Hierarchy Force layout

 No natural hierarchy or order  Based on principles of physics The Spring Model

 Using springs to represent node-node relations.  Minimizing energy function to reach energy equilibrium.  Initial layout is important  Local minimal problem Spring Layout

17 Energy Function

18 Network of character co-occurrence in Les Misérables

Multi-dimensional 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 ( , = , ) −𝑞𝑞 𝑖𝑖 𝑗𝑗 = w( ,𝑤𝑤)( , 𝛿𝛿𝑖𝑖 𝑗𝑗 , ) 2 𝑌𝑌 � 𝑖𝑖 𝑗𝑗 𝑑𝑑𝑖𝑖 𝑗𝑗 − 𝛿𝛿𝑖𝑖 𝑗𝑗 𝑖𝑖<𝑗𝑗 MDS for graph layout Other Node-Link 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 Arc Les Misérables character relations

EU Financial Crisis:

http://www.bbc.co.uk/news/business-15748696

Summary: Node-Link

 Pros – Intuitive – Good for global structure – Flexible, with variations  Cons – Complexity >O(N2) – 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. – Adjacency Matrix

 Edge weight  Directional edges  Sorting: node order  Path searching and path tracking ? 30 Node Order Path Tracking Adjacency matrix summary Avoid edge crossing, suitable for dense graphs Visually more scalable Visualization is not intuitive Hard to track a path MatLink Hybrid Layout

 Using adjacency matrix to represent small communities  Node-link for relationships between communities NodeTrix GMap

 Visualizing graphs and clusters as geographic maps to represent node relations (geographic neighbors) simplification

 Reducing amount of data – Reducing nodes: clustering – Reducing edges: minimal spanning – Edge bundling  Problems: – Loss of data Clustering Edge Bundling 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. Multilevel Agglomerative Edge Bundling

 Bottom-up merging approach, similar to hierarchical clustering  Minimize amount of ink used to render a graph. Skeleton-based Edge Bundling

 Skeletons: medial axes of edges which are similar in terms of positions information.  Iteratively attracting edges towards the skeletons. Comparison Interaction

 Viewing – Pan, Zoom, Rotate  Interacting with graph nodes and edges – Pick, highlight, delete, move  Structural interaction – Local re-order and re-layout – Focus+Context – Roll-up & Drill-down Fisheye  Focus+Context; Overviews + details-on-demand  Distortion to magnify areas of interest: zoom factors of 3-5  Multi-scale spaces: Zoom in/out & Pan left/right Interaction with Social Networks

Need to consider the social factors and behaviors related to nodes and edges

50 Graph Visualization Tools

 Prefuse (Java)  Pajek  UCINET / NetDraw  Sentinel Visualizer  JUNG (Java Universal Network/Graph framework)   TouchGraph  D3 http://prefuse.org/ Pajek

http://prefuse.org/ Sentinel Visualizer UCINET / NetDraw  Link Analysis, Data  Analysis and visualization Visualization, Geospatial of networks and graphs Mapping, and Analysis (SNA) Example: trade

55 Example: email traffic

56 Example: Subway map

57 Web page connections

58 Communication Networks Definition

 A is a generalization of a graph, where an edge can connect any number of vertices.  A hypergraph H is a pair H = (V,E) where V is a set of nodes/vertices, and E is a set of non-empty subsets of V called hyperedges/links. 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 Graph Representations Edge Nodes: Representative Graph