• Visualizes relational information
  • Weighted/unweighted, directional/bidirectional
• Applications
  • Social networks
  • Biological networks (genetic)
  • Computer networks
  • Railway/bus networks
  • …

• Space-filling methods (sunburst chart, treemaps, circle packing)
• Non-space filling methods (trees)
• Aesthetics: node color/ size, link color/size

Treemaps, Circle packing and Sunburst diagrams

• Recursively divides the space into areas according to the hierarchy (e.g. catalogue structure)
• Aesthetics: size and color
  • Assuming: \(size(parent)=\sum size(kids)\)

• Analysis: Comparing size/color of hierarchies between/within different levels

Example: World population

Example: Hierarchy for Flare ActionScript Library

Example: Hierarchy for Flare ActionScript Library

• Aesthetics: Node/link size, node/link color
• Can be used for classification result analysis
• Analysis
  • Analyse each branch and terminal nodes
  • Compare information between nodes
  • Compare information between branches

Picture here

• Layout has great impact (influences understanding a lot)

• Different layouts proposed

  • Force-directed layout - Repulsive and Spring forces - Optimization used to find equilibrium
  • Fixed layouts: circular, line,…
  • Minimizing other objectives (e.g. symmetry, edge crossing)

Example: Network of hyperlinks and mentions among news sources

See Network example 1

• Now adding node_size=“audience size”, link_size=amount of references

See Network example 2

Analysis:

• Which nodes are hubs?
• Which links are strongest?
• Which nodes are \(n\) steps away from some node?
• Components with strong connections?
  • Relationships to groups?
• How are different groups connected?
• Shortest path between nodes
• Community detection (densely connected nodes)
• Other interesting substructures

Example: Community detection based on edge betweeness

See Network example 3

• Graph can be represented as adjacency matrix
• Seriation methods can be used
• Communities can be detected by heatmaps

• A plenty of packages available
  • igraph: rich functionality, poor graphics
  • visNetwork: interactive graph visualization, can use igraph objects
  • ggraph: static graphs, based on the grammar of graphics, can use igraph
  • ggnet2 in GGally: static graphs
  • …
• Describing graph objects - very package dependent
  • igraph/visNetwork/ggraph:
    • Nodes: data frame with “id” column (and other attrs)
    • Edges: data frame with “from” and “to” columns (and other attrs)

• Eye is drawn to similar motions and outlying motions
• Advantages
  • Effective at attracting attention
  • Time=one extra aesthetics
  • Easily perceived in peripheral vision ->many features can be captured at one time point
• Disadvantages
  • Unappropriate transformation (transition) -> false conclusions
  • Speed of the graphics may hide important details/make boring

• Maintain valid data during transitions
  • Example: using splines
• Be careful when using interpolations: use appropriate models
• Group similar transitions
• Minimize occlusion
• Use simple transitions
• If trajectory is stable, use ease-in, ease-out
• Make transition as long as needed but no longer.

See http://www.gapminder.org

• Gapminder is
  • A database that stores important world statistics (by country)
  • Each dataset can be analysed online with Motion Chart
  • Popularized by Hans Rosling
• Check “Life expectancy” (years) in Gapminder

Life expectancy set

• How the minimum and maximum life expectancies changed after 200 years
• Which countries had in general the highest life expectancy? Lowest lifexpectancy? Highest income level? Lowest income level?
• What happened after 1946 and why?
• Compare development of China and USA (mark this countries and use “trace”)

• Limited amount of timepoints

• Smooth transition is desired

• For each aesthetics \(x\), consider interpolation \(x'(t)\)

  • How to create?
    • Linear interpolation, splines, kernel smoothers…
    • Compute path length \(s(t)\)
    • Insert intermediate frames \(t_1, \ldots, t_n\) with \(x'(t_1), \ldots x'(t_n)\)

Inserting intermediate values:

• Smooth interpolation
• Ease in, ease out (focuses user on main frames)
• … ( https://easings.net/)

• Motion charts : googleVis package
• Plotly: various animation, set time variable to frame aesthetics
  • Motion charts are simpler than googleVis

Example: power function

library(plotly)

m=matrix(nrow=0,ncol=3)
for (a in seq(0,3,by=0.03)) {
  x<-seq(0,2,0.01)
  y<-x^a
  m<-rbind(m,cbind(x,y,a))
}

df=as.data.frame(m)

plot_ly(df, x=~x, y=~y, frame =~a)%>%add_lines()%>%animation_opts(
  100, easing = "cubic", redraw = F
)

Example: power function

• MDS and PCA projections
  • Is one projection enough to understand data?

##2D-tours

• Idea
  • Investigate various projections
  • Connect them into animation
• What to analyze?
  • Same as in scatterplots plus
  • Which variables contribute to projections
    • Patterns in lower dimensional manifold?
• Types of 2d-tours: grand tour and guided tour

• Grand tour for Car data

• Data matrix \(X=[n \times p]\)

• Projection matrix \(A=[p \times d]\), \(d=2\)

  • Rows in \(A\): contribution of \(i\)th coordinate to projection
  • Columns in \(A\): coordinate of projecting vectors in \(R^p\)

• Projected data \(Z=X\cdot A\)

• Example: \(X=\left( \begin{matrix} 2 & 4 &3\\ 6 & 2 & 1\\ 2 & 9 & 9 \end{matrix} \right)\), \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix} \right)\)

• Idea: explore projections randomly

1. Select some projection \(A_1\)
2. Select next projection randomly \(A_2\)
3. Create shortest path projections \(A_1^1, \ldots, A_1^k\) from \(A_1\) to \(A_2\)
4. Select next projection randomly \(A_3\)
5. …

• Idea: focus on “interesting” projections
  • Interest measure: projection pursuit index \(I(XA)\)
  • Go through local \(\max{I(XA)}\)

- Various PP indices: Holes, central mass,…

Algorithm:

• Select some projection \(A_a\), calculate \(I_a\)
• For i=1,2,…
  • \(A_i=A_a +c^i B\), \(B\)- random direction, \(c^i\)- “cooling parameter”
  • Move to \(A_i\) with probability depending on \(\Delta I\) (and create shortest path intermediate projections)
  • Set \(A_a=A_i\) if moved.

• Note: path goes often through same points…

• Chapter 9, 12.7
• Cook, D., Buja, A., Lee, E. K., & Wickham, H. (2008). Grand tours, projection pursuit guided tours, and manual controls. In Handbook of data visualization (pp. 295-314). Springer, Berlin, Heidelberg.
• Plotly book Ch 5.1
• http://kateto.net/networks-r-igraph
• https://datastorm-open.github.io/visNetwork/
• https://www.data-imaginist.com/2017/ggraph-introduction-layouts