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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.

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What is a Graph?

Vertices (nodes)

Edges (links)

1 2 30 1 01 0 10 1 0

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1: 22: 1, 33: 2

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Adjacency matrix

Adjacency list

Drawing

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Graph Terminology

Graphs can have cyclesGraph edges can be directed or

undirectedThe degree 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)

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Trees are Different

Subcase of general graph

No cycles

Typically directed edges

Special designated root vertex

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Issues in Graph visualization Graph drawing Layout and positioning Scale: large scale graphs are difficult Navigation: changing focus and scale

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Vertex Issues

Shape

Color

Size

Location

Label

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Edge Issues

Color

Size

Label

Form– Polyline, straight line, orthogonal, grid, curved,

planar, upward/downward, ...

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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 straight-line

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Graph drawing optimization

3D-Graph Drawing

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Graph visualization techniques

Node-link approach– Layered graph drawing (Sugiyama)

– Force-directed layout

– Multi-dimensional scaling (MDS)

Adjacency Matrix

Sugiyama (layered) method

Suitable for directed graphs with natural hierarchies: All edges are oriented in a consistent direction and no pairs of edges cross

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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

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Force directed graph layout

No natural hierarchy or orderBased on principles of physics

The Spring Model

Using springs to represent node-node relations.

Minimizing energy function to reach energy equilibrium.

Initial layout is importantLocal minimal problem

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Spring Layout

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Energy Function

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Network of character co-occurrence in Les Misérables

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Multi-dimensional Scaling

Dimension reduction to 2D Graph distance of two nodes are as close to 2D

Euclidean distance as possibleMDS is a global approach Distance between two nodes: shortest path

(classical scaling).Weighted distances ( , ,

w , , ,

MDS for graph layout

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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 Diagrams

Arc DiagramLes Misérables character relations

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Arc Diagram

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

EU Financial Crisis:

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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.

–

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Adjacency Matrix

Edge weight

Directional edges

Sorting: node order

Path searching and path tracking ?

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Node Order

Path Tracking

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Adjacency matrix summary

Avoid edge crossing, suitable for dense graphsVisually more scalableVisualization is not intuitiveHard to track a path

MatLink

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Hybrid Layout

Using adjacency matrix to represent small communities

Node-link for relationships between communities

NodeTrix

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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

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Clustering

Edge Bundling

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Edges are modeled as flexible springs that are able to attract each other.

Force Directed Edge Bundling

Edges clusters are found based on a geometric control mesh.

Geometry Based Edge Bundling

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Bottom-up merging approach, similar to hierarchical clustering

Minimize amount of ink used to render a graph.

Multilevel Agglomerative Edge Bundling

Skeletons: medial axes of edges which are similar in terms of positions information.

Iteratively attracting edges towards the skeletons.

Skeleton-based Edge Bundling

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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

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Fisheye Focus+Context; Overviews + details-on-demand Distortion to magnify areas of interest: zoom

factors of 3-5Multi-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

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Graph Visualization Tools

Prefuse (Java)

Pajek

UCINET / NetDraw

Sentinel Visualizer

JUNG (Java Universal Network/Graph framework)

Graphviz

Gephi

TouchGraph

D3

http://prefuse.org/

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http://prefuse.org/

Pajek

Sentinel Visualizer Link Analysis, Data

Visualization, Geospatial Mapping, and Social Network Analysis (SNA)

UCINET / NetDraw Analysis and visualization

of networks and graphs

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Example: trade

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Example: email traffic

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Example: Subway map

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Web page connections

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Communication Networks

Hypergraphs

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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) 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)}

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Applications Data Mining Biological Interactions Social Networks Circuit Diagrams

Graph Representations

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Edge Nodes: Representative Graph