Slides from our PacificVis 2015 presentation. The paper tackles the problems of the “giant hairballs”, the dense and tangled structures often resulting from visualiza- tion of large social graphs. Proposed is a high-dimensional rotation technique called AGI3D, combined with an ability to filter elements based on social centrality values. AGI3D is targeted for a high-dimensional embedding of a social graph and its projection onto 3D space. It allows the user to ro- tate the social graph layout in the high-dimensional space by mouse dragging of a vertex. Its high-dimensional rotation effects give the user an illusion that he/she is destructively reshaping the social graph layout but in reality, it assists the user to find a preferred positioning and direction in the high- dimensional space to look at the internal structure of the social graph layout, keeping it unmodified. A prototype im- plementation of the proposal called Social Viewpoint Finder is tested with about 70 social graphs and this paper reports four of the analysis results.

1. Interactive High-Dimensional
Visualization of Social Graphs
Ken Wakita1, 2, Masanori Takami1, Hiroshi Hosobe3
1 Tokyo Institute of Technology
2 CREST/JST
3 Hosei University

2. When I find a complex object,
✤ I come closer to the object (zoom),
✤ look at it from different directions (rotation), and
✤ focus on some part to get clearer view of it (focus).

3. Findings of 杭州
via zoom/directions/focus

4. The complex things that I am
playing with: Social Networks
✤ Social Network
✤ Small World: short diameter with high clustering
coefficient
✤ Scale Free: hubs, long tail
✤ High-dimensional

5. PROBLEMS IN LARGE SCALE
NETWORKVISUALIZATION
Computational complexity
• KK, MDS – Simulation overhead: O(V
2
)
• MDS – All-pair distance:
O(V
3
), O(EV+V
2
log logV)
Display Resolution: #pixels << #vertices
Hairball problem – we tend to see hairbalsl
from social network visualization

6. Taming Hair Balls
✤ Edge bundling (Holten+09, Peysakhovich15, Bouts15)
✤ Graph Simplification (Sparsification – Satuluri+11, Simmerian backbones – Nick+13,
Motifs – Dunne+13)
✤ Multiscale (Auber+03, Li+05, Abello+06, Elmqvist+08, von Landesberger+11,
Zinsmaier+12, Hong+05)
✤ MDS-based approaches: Observing the graph in its unmodified, unsimplified form.
✤ Variation of CMDS: Distorted focal view (Klimenta+12), Distance scaling
(Gransner+04, Hu+12)
✤ CMDS with massively high dimensional interaction (Hosobe04, Hosobe07, this
work): High-dimensional rotation

7. Proposal

8. SocialView Point Finder
Overview
Social Graph
(2) X: Projection to 3D
X = V・P
(3) Presentation
OpenGL
(1) V: Massively High Dimensional Layout
Classical MDS: 500D∼15000D
V X

9. AGI3D: Untangling Effect from
High-dimensional Rotation

10. ChangingView Points
in HD

11. Interaction
http://vimeo.com/channels/pvis2015/

12. vi
vi
P
P’
xi
xi’
xi
xi’

13. vi
vi
P
P’
xi
xi’
xi
xi’
X = V・P
➡
X’ = V・P’

14. Lens-effect from
Adjustment of Projection Factor

15. Lens-effect Demo: StudentTwitter
Connection among 4 Universities
http://vimeo.com/channels/pvis2015/

16. H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
2
1 0 0
0
1
2
2 0
0 0
1
2
3
1
2
4 0 0
0
1
2
5 0
0 0
1
2
6
1
2
7 0 0
0
1
2
8 0
...
...
...
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V T
3D projection in AGI3D makes use of all
the positive eigenvalues by merging them

17. ✤ Projection factor (α)
controls the contribution
of minor eigenvectors.
✤ Smaller α makes minor
eigenvectors become
influential and shows
trim structure of the
network.
H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
↵
1 0 0
0 ↵
2 0
0 0 ↵
3
↵
4 0 0
0 ↵
5 0
0 0 ↵
6
↵
7 0 0
0 ↵
8 0
...
...
...
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V T
Projection Factor (α)

18. Centrality-based Filtering of
Vertices and Edges

19. Filtering is necessary for successful
high dimensional exploration
✤ Node centralities
✤ Betweenness centrality, Closeness centrality,
Clustering coefficient, Degree centrality, PageRank
✤ Edge centralities
✤ Simpson coefficient, Extended Simpson coefficient,
Betweenness coefficient

20. Example: References among
Wikipedia Math-related Pages

21. Evaluation

22. Efficiency
Table 1: Larger datasets (USPol: Citation among US Po-
litical blog sites [2], 4Univ: see ..., Yeast: Protein, Math:
References in Mathematics related Wikipedia pages, PGP:
Key exchange in PGP network, Arxiv: Citation network
among astrophisical publication, Internet: Internet routing
network, Enron: Corporate-wide email message exchange
network).
Dataset |V | |E| dH t (s) FPS
USPol 1,222 16,714 5/8 680 57
4Univ 1,896 26,183 5/10 991 35
Yeast 2,224 6,609 7/11 1,354 0.001 58
Math 3,608 48,315 4/7 1,930 0.005 30
PGP 10,680 24,316 14/19 6,931 0.057 26
Arxiv 17,903 196,972 7/14 10,912 0.083 9.6
Internet 22,463 48,436 14/19 17,573 0.193 15
Enron 33,696 180,811 5/8 24,943 0.379 7.5
Below are giant hairballs obtained from larger datasets.

23. Political Blog Network in
Y2004 (US Presidential Poll)
Support for
Republicans
Support for
Democrats

24. Mathematical shapes
Cone, Great icosahedron,
Dual polyhedron,
Johnson solid,
Pillar
Conway's game of life
hertz oscillator,
traffic light,
spaceship,
glider gun
Game theory
prisoner's dillemma,
normal-form game,
zero-sum game,
minimax
Mathematicians
Enrico Bombieri,
Vladimir Drinferd,
Keisuke Hironaka,
Vaughan Jones,
Alain Connes,
Terence Tao
Stochastic Distribution
β-distr.,
Pareto distr.,
logistic distr.,
Dirichlet distr.,
hyperbolic secant distr.
Wikipedia Math Pages

25. Summary
✤ High dimensional graph interaction method extended
to 3D (and higher-dimensional) visualization
✤ Effectiveness of the proposal, combined with
centrality based filters and projection factor control, in
visual analytics of various small world networks has
been tested.

26. FutureWork
✤ GPU implementation: Shaders (rendering) & GPGPU (filter/projection)
✤ More user interface supports: bookmarking, labeling, …
✤ Integration with graph clustering system
✤ Data compaction
✤ High dim. layout ∼ O(|V|
2
), Node centrality: O(|V|) each/Edge
centrality: O(|E|
2
) each
✤ Scalable implementation: combination with a multi scale visualization
system

27. “thank you for listening”

28. Backup slides

29. Hair balls

30. HAIR BALL PROBLEM
Visualization of networks often results in hairballs.
They are beautiful and powerful.
But are they useful?
What we need is insight.
Not a picture.

31. CAUSES OF HAIR BALLS
Several Causes
Multi-Layered/Multiplexed
nature (Nocaj+, GD14)
Small World Nature
The objective of graph drawing is
to finding a graph layout that
whose Euclidean distance most
closely maintains the topological
distance of the graph.
Duncan Watts’s curse:
Diameter < 6
Classical graph layout results in
embedding of graph in a sphere
whose diameter is < 6!

32. A Lesson from Complete Graph
✤ 1 Node: Dot
✤ 2 Nodes: Line
✤ 3 Nodes: Triangle
✤ 4 Nodes: Tripod
✤ 5 Nodes: 4D Tripod
✤ k Nodes: (k-1)-D Tripod
✤ Social Graphs?: Not so
bad as complete graphs
but it seems that they
are naturally expressed
in very high
dimensional space.

33. Efficiency
Table 1: Larger datasets (USPol: Citation among US Po-
litical blog sites [2], 4Univ: see ..., Yeast: Protein, Math:
References in Mathematics related Wikipedia pages, PGP:
Key exchange in PGP network, Arxiv: Citation network
among astrophisical publication, Internet: Internet routing
network, Enron: Corporate-wide email message exchange
network).
Dataset |V | |E| dH t (s) FPS
USPol 1,222 16,714 5/8 680 57
4Univ 1,896 26,183 5/10 991 35
Yeast 2,224 6,609 7/11 1,354 0.001 58
Math 3,608 48,315 4/7 1,930 0.005 30
PGP 10,680 24,316 14/19 6,931 0.057 26
Arxiv 17,903 196,972 7/14 10,912 0.083 9.6
Internet 22,463 48,436 14/19 17,573 0.193 15
Enron 33,696 180,811 5/8 24,943 0.379 7.5
Below are giant hairballs obtained from larger datasets.

34. Classical MDS

35. An Objective in Graph Drawing
✤ Graph Drawing for a graph (G) tries to find a
mapping from graph vertices to locations (X) in
Euclidean space,
(Graph G → Vertex Locations X)
✤ such that geographical (Euclidean) distance (Dg) for
X best simulates topological (shortest path) distance
(Dt) with respect to G.
min
vi,vj 2vertices
kDg(xi, xj) Dt(vi, vj)k

36. Classical MDS
Torgerson-Kruscal-Seeri (TKS)
✤ Torgerson scaling & Projection
✤ Graph G = (V, E)
✤ Distance matrix: D = (di,j)
✤ D’: Centralized D via Young-Householder transformation
✤ V
T
D’V = Λ: Eigen decomposition
✤ Λ: Eigenvalues, V: Eigenvectors
✤ 2D projection in TKS respects two largest eigen-{values,vectors}
✤ X = Λ1
1/2
V1 + Λ2
1/2
V2

37. High-Dimensional Layout Based
on Classical MDS

38. Classical MDS
Eigen Decomposition of the Distance Matrix
1 2 3 · · · H > 0 H+1 · · · N
H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
2
1 0 0 0 0 0 0
0
1
2
2 0 0 0 0 0
0 0
1
2
3 0 0 0 0 0
0 0 0
... 0 0 0 0
0 0 0 0
1
2
H 0 0 0
0 0 0 0 0
1
2
H+1 0 0
0 0 0 0 0 0
... 0
0 0 0 0 0 0 0
1
2
N
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V T
X = V ⇤
1
2

39. 2D Projection inTorgerson-
Kruscal-Keeri Method
H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
2
1 0 0 0 0 0 0
0
1
2
2 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0
... 0
0 0 0 0 0 0 0 0
1
C
C
C
C
C
C
C
C
C
C
C
C
A
V T

40. H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
2
1 0 0 0 0 0 0 0 0 0
0
1
2
2 0 0 0 0 0 0 0 0
0 0
1
2
3 0 0 0 0 0 0 0
0 0 0
1
2
4 0 0 0 0 0 0
0 0 0 0
1
2
5 0 0 0 0 0
0 0 0 0 0
1
2
6 0 0 0 0
0 0 0 0 0 0
1
2
7 0 0 0
0 0 0 0 0 0 0
1
2
8 0 0
0 0 0 0 0 0 0 0
... 0
0 0 0 0 0 0 0 0 0
1
2
H
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V T
3D projection in AGI3D makes use
of all the positive eigenvalues by …

41. H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
2
1 0 0
0
1
2
2 0
0 0
1
2
3
1
2
4 0 0
0
1
2
5 0
0 0
1
2
6
1
2
7 0 0
0
1
2
8 0
...
...
...
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V T
3D projection in AGI3D makes use of all
the positive eigenvalues by merging them

42. ✤ Projection factor (α)
controls the contribution
of minor eigenvectors.
✤ Smaller α makes minor
eigenvectors become
influential and shows
trim structure of the
network.
H ˜D(2)
H = V
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
↵
1 0 0
0 ↵
2 0
0 0 ↵
3
↵
4 0 0
0 ↵
5 0
0 0 ↵
6
↵
7 0 0
0 ↵
8 0
...
...
...
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V T
Projection Factor (α)

43. High Dimensional Rotation

44. Interpretation of Dragging as High-
Dimensional Rotation of theWhole Graph
✤ Initially, X = V P, where P = Λα
✤ The idea: according to user’s dragging operation, the
projection matrix changes.
✤ When the user moves a vertex v located at x to another
location x’
✤ We find a high-dimensional rotation rot such that
✤ x’ = rot(x) and X’ = V P’ where P = rot(P)

45. Formulation of HD rotation
✤ ei: Current basis
✤ e0: Additional axis to
increase degree of
freedom for rotation
✤ ei’: Updated basis
✤ bi: rotation axis
high-dimensional point, qH 2 RdH , or more formally
qH P and q0
d = qH P0
. We also assume that the reposi
the vertex qd to q0
d is caused by a high-dimensional ro
Because a projection matrix consists of normal orth
basis, the problem of finding the high-dimensional ro
can be paraphrased as translation of a normal orth
basis, {ei}, to another, {e0
i}:
e0 = qH
dX
i=1
xiei
e0
i =
dX
j=0
aijej (1  i  d)
ri =
dX
j=1
bijej (1  i  d 1)
qH · e0
i = x0
i (1  i  d)
e0
i · e0
j = i,j (1  i, j  d)
ri · rj = i,j (1  i, j  d 1)
ri · e0
j = ri · ej (1  i  d 1, 1  j  d)
The first two equations characterize the space wher

46. Full Demo

47. Citation among Political Blogs
Y2004 US Presidential Poll (1.2K/16.7K)
http://vimeo.com/channels/pvis2015/

48. Twitter Connection among Four
Universities Students (1.1K/26.2K)
http://vimeo.com/channels/pvis2015/

49. Page Reference among Math. Pages
fromWikipedia (3.6K/48.3K)
http://vimeo.com/channels/pvis2015/

50. Tree Example (1K/1K)
http://vimeo.com/channels/pvis2015/

51. Clusters Created from Planted
Model (1K/12.9K)