interactive high-dimensional visualization of social graphs
TRANSCRIPT
Interactive High-Dimensional Visualization of Social GraphsKen Wakita1, 2, Masanori Takami1, Hiroshi Hosobe3
1 Tokyo Institute of Technology 2 CREST/JST3 Hosei University
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).
Findings of 杭州via zoom/directions/focus
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
PROBLEMS IN LARGE SCALE NETWORK VISUALIZATION
Computational complexity
• KK, MDS – Simulation overhead: O(V2)
• MDS – All-pair distance:O(V3), O(EV+V2log log V)
Display Resolution: #pixels << #vertices
Hairball problem – we tend to see hairbalsl from social network visualization
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
Proposal
Social View Point FinderOverview
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
AGI3D: Untangling Effect fromHigh-dimensional Rotation
Changing View Pointsin HD
vi
vi
P
P’
xi
xi’
xi
xi’
vi
vi
P
P’
xi
xi’
xi
xi’
X = V・P
⬇
X’ = V・P’
Lens-effect fromAdjustment of Projection Factor
Lens-effect Demo: Student Twitter Connection among 4 Universities
http://vimeo.com/channels/pvis2015/
HD̃(2)H = V
0
BBBBBBBBBBBBBBBBB@
�121 0 0
0 �122 0
0 0 �123
�124 0 0
0 �125 0
0 0 �126
�127 0 0
0 �128 0
......
...
1
CCCCCCCCCCCCCCCCCA
V T
3D projection in AGI3D makes use of all the positive eigenvalues by merging them
✤ Projection factor (α) controls the contribution of minor eigenvectors.
✤ Smaller α makes minor eigenvectors become influential and shows trim structure of the network.
HD̃(2)H = V
0
BBBBBBBBBBBBB@
�↵1 0 00 �↵
2 00 0 �↵
3
�↵4 0 00 �↵
5 00 0 �↵
6
�↵7 0 00 �↵
8 0...
......
1
CCCCCCCCCCCCCA
V T
Projection Factor (α)
Centrality-based Filtering of Vertices and Edges
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
Example: References among Wikipedia Math-related Pages
Evaluation
Efficiency
(a) The initial view of the Les Miserables example
(b) High-dimensional rotation reveals cluster structure of characters.
Figure 1: The “Les Miserables” example
matrix. The dimension dH typically is about half the num-ber of vertices in the social graph. The optional labels datais used to show the vertices’ labels in the Labels pane ofthe software. A node centrality data, V ! R, is a mappingthat associates a vertex centrality value to each vertex. Anedge centrality data, V ⇥ V ,! R is a partial mapping thatassociates an edge centrality value to each edge.
Figure 1(a) presents a three-dimensional visualization ofa co-appearance network of 77 characters from “Les Miser-ables.” [16] The selected vertex is painted in magenta, itsadjacent vertices in blue, and all the rest in light blue. Tothe right of the visualization is a pane that presents variousinformation (the frame rate, the name of the social graph,and its size); controls to arrange the projection factor value;selectors for node- and edge-centrality, and sliders to specifyfilters according to the centrality values; and a list of la-bels of the selected node and its neighbors. The reader mayfind popular characters like the main character Jean Valjean,his adopted daughter Cosette, and the persistent police in-spector Javert from the list. When a label is selected, itscorresponding vertex is highlighted in red.
Social Viewpoint Finder supports the basic three-dimensional user interactions. The view magnification iscontrolled by the mouse wheel, the parallel translationby mouse dragging holding the left button, and (three-dimensional) rotation by mouse dragging holding the rightbutton.
High-dimensional rotation is activated by dragging a ver-tex. The projection matrix is updated continuously whilethe vertex is being dragged, and the user is given real timefeedback for the high-dimensional rotation e↵ects. The viewchanges dramatically after a few dragging operations, (Fig-ure 1(b)) and reveals some interesting properties of the char-acters. Characters from the first volume of the novel includ-
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 networkamong astrophisical publication, Internet: Internet routingnetwork, Enron: Corporate-wide email message exchangenetwork).
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.
PGP Arxiv Internet Enron
ing Bishop Myriel form a cluster to the left of the visualiza-tion, characters from the second volume (Cosette’s motherand her co-workers) form another cluster to its right, and thebig cluster at the top right of the pane contains Cosette’sboyfriend Marius and his comrades. The main charactersValjean, Cosette, and Javert appear ubiquitously through-out the book, and therefore they seem not to belong to aspecific group.
3.2 Visual Analytics Using Social Viewpoint Finder
We have collected about 70 graph datasets for evaluationof Social Viewpoint Finder (see Table 1 for a list of largerdatasets). Among them are mathematical topology data(e.g., hypercube, hyper-lattice, Sierpinski gasket), classicdatasets from social network analysis (e.g., Karate, Dol-phin, co-authorship, Enron, PGP), social networking system(e.g., Facebook and Twitter), bio-informatics (e.g., neuron,metabolic network, and Yeast RNA), and Web archive (e.g.,Internet and Wikipedia). For each dataset, the following aregiven: the number of vertices and edges (|V |, |E|), the net-work diameter (�), the dimension of the layout (dH), timeconsumed for processing each interaction (t), frames per sec-ond (FPS). The numbers, `/L, given for � are 99%-diameter(`) and diameter (L). The time t is served for constraint res-olution and rendering. If 99%-diameter for a network is `,more than 99% pairs of vertices have distance `.
Below the table, there are three-dimensional visualiza-tion of four datasets. These are examples of hairballs thatare commonly generated from visualization of large socialgraphs.
Given a social graph, the user typically practice the fol-lowing Social Viewpoint Finder workflow:
1. Arrange projection factor to reveal clustering substruc-tures.
2. Adjust the scale factor for the graph layout to fit thewindow pane.
3. 3D-rotate the graph to examine its appearance.
4. Choose and adjust the vertex- and edge-filters accord-ing to the characteristics of the social graph.
Political Blog Network inY2004 (US Presidential Poll)
Support forRepublicans
Support forDemocrats
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
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.
Future Work
✤ 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
“thank you for listening”
Backup slides
Hair balls
HAIR BALL PROBLEMVisualization of networks often results in hairballs. They are beautiful and powerful. But are they useful?
What we need is insight. Not a picture.
CAUSES OF HAIR BALLSSeveral 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!
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.
Efficiency
(a) The initial view of the Les Miserables example
(b) High-dimensional rotation reveals cluster structure of characters.
Figure 1: The “Les Miserables” example
matrix. The dimension dH typically is about half the num-ber of vertices in the social graph. The optional labels datais used to show the vertices’ labels in the Labels pane ofthe software. A node centrality data, V ! R, is a mappingthat associates a vertex centrality value to each vertex. Anedge centrality data, V ⇥ V ,! R is a partial mapping thatassociates an edge centrality value to each edge.
Figure 1(a) presents a three-dimensional visualization ofa co-appearance network of 77 characters from “Les Miser-ables.” [16] The selected vertex is painted in magenta, itsadjacent vertices in blue, and all the rest in light blue. Tothe right of the visualization is a pane that presents variousinformation (the frame rate, the name of the social graph,and its size); controls to arrange the projection factor value;selectors for node- and edge-centrality, and sliders to specifyfilters according to the centrality values; and a list of la-bels of the selected node and its neighbors. The reader mayfind popular characters like the main character Jean Valjean,his adopted daughter Cosette, and the persistent police in-spector Javert from the list. When a label is selected, itscorresponding vertex is highlighted in red.
Social Viewpoint Finder supports the basic three-dimensional user interactions. The view magnification iscontrolled by the mouse wheel, the parallel translationby mouse dragging holding the left button, and (three-dimensional) rotation by mouse dragging holding the rightbutton.
High-dimensional rotation is activated by dragging a ver-tex. The projection matrix is updated continuously whilethe vertex is being dragged, and the user is given real timefeedback for the high-dimensional rotation e↵ects. The viewchanges dramatically after a few dragging operations, (Fig-ure 1(b)) and reveals some interesting properties of the char-acters. Characters from the first volume of the novel includ-
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 networkamong astrophisical publication, Internet: Internet routingnetwork, Enron: Corporate-wide email message exchangenetwork).
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.
PGP Arxiv Internet Enron
ing Bishop Myriel form a cluster to the left of the visualiza-tion, characters from the second volume (Cosette’s motherand her co-workers) form another cluster to its right, and thebig cluster at the top right of the pane contains Cosette’sboyfriend Marius and his comrades. The main charactersValjean, Cosette, and Javert appear ubiquitously through-out the book, and therefore they seem not to belong to aspecific group.
3.2 Visual Analytics Using Social Viewpoint Finder
We have collected about 70 graph datasets for evaluationof Social Viewpoint Finder (see Table 1 for a list of largerdatasets). Among them are mathematical topology data(e.g., hypercube, hyper-lattice, Sierpinski gasket), classicdatasets from social network analysis (e.g., Karate, Dol-phin, co-authorship, Enron, PGP), social networking system(e.g., Facebook and Twitter), bio-informatics (e.g., neuron,metabolic network, and Yeast RNA), and Web archive (e.g.,Internet and Wikipedia). For each dataset, the following aregiven: the number of vertices and edges (|V |, |E|), the net-work diameter (�), the dimension of the layout (dH), timeconsumed for processing each interaction (t), frames per sec-ond (FPS). The numbers, `/L, given for � are 99%-diameter(`) and diameter (L). The time t is served for constraint res-olution and rendering. If 99%-diameter for a network is `,more than 99% pairs of vertices have distance `.
Below the table, there are three-dimensional visualiza-tion of four datasets. These are examples of hairballs thatare commonly generated from visualization of large socialgraphs.
Given a social graph, the user typically practice the fol-lowing Social Viewpoint Finder workflow:
1. Arrange projection factor to reveal clustering substruc-tures.
2. Adjust the scale factor for the graph layout to fit thewindow pane.
3. 3D-rotate the graph to examine its appearance.
4. Choose and adjust the vertex- and edge-filters accord-ing to the characteristics of the social graph.
Classical MDS
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.
minvi,vj2vertices
kDg(xi, xj)�Dt(vi, vj)k
Classical MDSTorgerson-Kruscal-Seeri (TKS)✤ Torgerson scaling & Projection
✤ Graph G = (V, E)
✤ Distance matrix: D = (di,j)
✤ D’: Centralized D via Young-Householder transformation
✤ VTD’V = Λ: Eigen decomposition
✤ Λ: Eigenvalues, V: Eigenvectors
✤ 2D projection in TKS respects two largest eigen-{values,vectors}
✤ X = Λ11/2V1 + Λ2
1/2V2
High-Dimensional Layout Based on Classical MDS
Classical MDSEigen Decomposition of the Distance Matrix
�1 � �2 � �3 � · · ·�H > 0 � �H+1 � · · · � �N
HD̃(2)H = V
0
BBBBBBBBBBBBBBB@
�121 0 0 0 0 0 0
0 �122 0 0 0 0 0
0 0 �123 0 0 0 0 0
0 0 0. . . 0 0 0 0
0 0 0 0 �12H 0 0 0
0 0 0 0 0 �12H+1 0 0
0 0 0 0 0 0. . . 0
0 0 0 0 0 0 0 �12N
1
CCCCCCCCCCCCCCCA
V T
X = V ⇤12
2D Projection in Torgerson-Kruscal-Keeri Method
HD̃(2)H = V
0
BBBBBBBBBBBB@
�121 0 0 0 0 0 0
0 �122 0 0 0 0 0
0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
0 0 0 0 0 0. . . 0
0 0 0 0 0 0 0 0
1
CCCCCCCCCCCCA
V T
HD̃(2)H = V
0
BBBBBBBBBBBBBBBBBBBB@
�121 0 0 0 0 0 0 0 0 0
0 �122 0 0 0 0 0 0 0 0
0 0 �123 0 0 0 0 0 0 0
0 0 0 �124 0 0 0 0 0 0
0 0 0 0 �125 0 0 0 0 0
0 0 0 0 0 �126 0 0 0 0
0 0 0 0 0 0 �127 0 0 0
0 0 0 0 0 0 0 �128 0 0
0 0 0 0 0 0 0 0. . . 0
0 0 0 0 0 0 0 0 0 �12H
1
CCCCCCCCCCCCCCCCCCCCA
V T
3D projection in AGI3D makes use of all the positive eigenvalues by …
HD̃(2)H = V
0
BBBBBBBBBBBBBBBBB@
�121 0 0
0 �122 0
0 0 �123
�124 0 0
0 �125 0
0 0 �126
�127 0 0
0 �128 0
......
...
1
CCCCCCCCCCCCCCCCCA
V T
3D projection in AGI3D makes use of all the positive eigenvalues by merging them
✤ Projection factor (α) controls the contribution of minor eigenvectors.
✤ Smaller α makes minor eigenvectors become influential and shows trim structure of the network.
HD̃(2)H = V
0
BBBBBBBBBBBBB@
�↵1 0 00 �↵
2 00 0 �↵
3
�↵4 0 00 �↵
5 00 0 �↵
6
�↵7 0 00 �↵
8 0...
......
1
CCCCCCCCCCCCCA
V T
Projection Factor (α)
High Dimensional Rotation
Interpretation of Dragging as High-Dimensional Rotation of the Whole 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)
Formulation of HD rotation
✤ ei: Current basis
✤ e0: Additional axis to increase degree of freedom for rotation
✤ ei’: Updated basis
✤ bi: rotation axis
space. Standard graph manipulation techniques based ona�ne geometry allow the user to scale, rotate, and trans-form the graph layout, but those operations are restrictedin two- or three-dimensional spaces and they fail to makeuse of the rest of the dimensions that are available in thehigh-dimensional graph layout.
In contrast, the proposed system interprets user’s dragoperation as a command for rotation in the high-dimensional
space. Intuitively, this gives visual e↵ects similar to rotatinga three dimensional object and viewing the other side of theobject, which could not be seen in the initial layout. Becausethis rotation is performed in the high-dimensional space, thismechanism sometimes demonstrates e↵ects that occasionallytwist, distort, untangle, or decompose the network layout.
Note also that with this technique, dragging of a singlevertex triggers high-dimensional rotation of the whole net-work. That means the whole landscape can change in accor-dance with a single dragging action.
Following is the theory that supports high-dimensionalrotation. This formulation is a generalization of Hosobe’swhich deal with the case d = 2. Suppose that a user grabsa graph node placed at qd = (x1, . . . , xd) 2 Rd and tries toreposition it to another position, q0
d = (x01, . . . , x
0d) 2 Rd. We
consider both of these positions as projection of the identicalhigh-dimensional point, qH 2 RdH , or more formally, qd =qHP and q
0d = qHP 0. We also assume that the reposition of
the vertex qd to q
0d is caused by a high-dimensional rotation.
Because a projection matrix consists of normal orthogonalbasis, the problem of finding the high-dimensional rotationcan be paraphrased as translation of a normal orthogonalbasis, {ei}, to another, {e0
i}:
e0 = qH �dX
i=1
xiei (1)
e
0i =
dX
j=0
aijej (1 i d) (2)
ri =dX
j=1
bijej (1 i d� 1) (3)
qH · e0i = x0
i (1 i d) (4)
e
0i · e0
j = �i,j (1 i, j d) (5)
ri · rj = �i,j (1 i, j d� 1) (6)
ri · e0j = ri · ej (1 i d� 1, 1 j d) (7)
The first two equations characterize the space where rota-tion is performed, which is formed from the original basis,ei’s (1 i d), as well as the direction the vertex beingdragged to, e0. A new vector e0 is introduced to increase thedegree of freedom to enable high-dimensional rotation. Thisvector is arranged such that it is normal and orthogonal toall the basis. The system tries to find a rotation axis, ri, onthe plane spanned by the original basis (the third equation).The fourth states that the new projection properly translatesthe (high-dimensional) vertex being dragged to the target ofthe drag operation, where � denotes Kronecker’s delta. Thisrequirement guarantees that the projection gives consistentviewpoints with respect to user’s designation. Equations inthe third row state that the modified projection basis andthe rotation axis are normal orthogonal basis. The last equa-tion requires that the modification of the projection matrixpreserves the angles included by the rotation axis and thebasis, in other words the basis, {ri}, forms a rotation.
The formulae consists of 2d2 equations and contains
d(d+ 1) variables that correspond to aij , and (d� 1)d vari-ables to bij , totaling 2d2. A standard numerical algorithmcan therefore solve the equations. We have implemented aconstraint resolver to support two-dimensional and three-dimensional presentation (d = 2 and d = 3 cases) for SocialViewpoint Finder.
To improve computational e�ciency, it is favored thatthe constraint system contains smaller number of variables.For this purpose, we use equation 2 to represent variablesai0 (1 i d) by linear combinations of aij (1 i, j d). With this treatment, the number of variables inthe constraint system reduces by d. Therefore in the two-dimensional case, the number of variables reduces from 8down to 6, and in the three-dimensional case, from 18 to 15.
The constraint system includes equations that require or-thogonality of vectors (see equations 5 and 6, where i 6= j).A problem with these equations is that they are satisfied byzero vectors. On the other hand, the constraint system alsorequires these vectors be normal (equations 5 and 6, wherei = j). If a near-zero vector appears, a simple constraintresolver can stack in an oscillation state and fail to find thesolution. To deal with this problem, we replaced each equa-tion of orthogonality, u · v = 0, by its normalized version,u · v/kukkvk.
Hosobe resolved the constraint system of AGI2D by theNewton-Raphson method. Firstly, we attempted to followthe same approach for solving the AGI3D constraint satisfac-tion problem. Unfortunately, with this prototype implemen-tation, the determinant of the Jacobi matrix occasionallybecomes close to zero and in such case the system su↵eredfrom poor stability. Unstability leads to discontinuous userexperience, where a small dragging of a node may abruptlychange the whole graph layout.
To stabilize the process of constraint satisfaction, wehave relaxed the constraint system and applied a vari-ant of Broyden-Fletcher-Goldfarb-Shannon (BFGS) method,called L-BFGS [19]. In this relaxation, we have translatedthe constraint sytem expressed by a series of equations,fi(x) = 0, to the following minimization problem.
minmX
i=1
(fi(x))2 (8)
3 Social Viewpoint Finder
As explained in Section , Social Viewpoint Finder is a soft-ware that visualizes a social graph in three-dimensionalspace. The user can click on a vertex to find its label anda set of vertices it connects to. As we have explained inthe previous section, the user can drag a vertex to triggerhigh-dimensional rotation. Social Viewpoint Finder o↵ersstandard operations for manipulation of three-dimensionalvisualization, such as scaling and three-dimensional rotation.The user can choose to apply filters based on various cen-trality indices or adjust the projection factor. Accordingly,one can shift focus on influential vertices and edges, or onthe trunks (or twigs) of the network.
3.1 The functionality
The input of Social Viewpoint Finder for a social graph,G = (V,E), consists of five kinds of data: (1) distancematrix, (2) high-dimensional layout, (3) labels, (4) node-centrality, and (5) edge-centrality. The distance matrix is a|V |⇥ |V | symmetric matrix that represents all-pairs graph-theoretic distances in the largest connected component of thesocial graph. The high-dimensional layout X is a |V | ⇥ dH
Full Demo
Citation among Political BlogsY2004 US Presidential Poll (1.2K/16.7K)
http://vimeo.com/channels/pvis2015/
Twitter Connection among Four Universities Students (1.1K/26.2K)
http://vimeo.com/channels/pvis2015/
Page Reference among Math. Pages from Wikipedia (3.6K/48.3K)
http://vimeo.com/channels/pvis2015/
Clusters Created from Planted Model (1K/12.9K)