A Comparison of Graphical Techniques for the Display of Co-Occurrence Data

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A Comparison of Graphical Techniques for the
Display of Co-Occurrence Data

• Jan W. Buzydlowski, Xia Lin, Howard D. White
• College of Information Science and Technology
• Drexel University
• Philadelphia, PA 19104
• USA

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Information Visualization

• (Data) Visualization allows for the revelation of
  intricate structure which cannot be absorbed in
  any other way. Cleveland, 1993
• (Information) Visualization has two aspects,
  structural modeling and graphic
  representation.C. Chen, 1999
• data - model - display

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Visualization Overview

• Model - Display
• Co-Occurrence Model
• 3 Graphical Displays
• Data
• Co-citation counts from the Institute for
  Scientific Information, Philadelphia, PA
• Obtained from a 10-year Arts Humanities
  Citation Index database given Drexel by ISI for
  research purposes

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Co-Occurrence Model

• Examples
• Derivation
• Metrics

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Co-Occurrence Data - Example 1

• Market Basket Analysis
• a shopping cart holds items purchased
• e.g., milk, bread, razor blades, newspaper
• Over all the sales for one day
• what items are purchased together
• how can we arrange the items in the store
• Pampers and beer on Thursdays...

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Co-Occurrence Data - Example 2

• Author Co-citation Analysis (ACA)
• Bibliographic data on a given article holds,
  e.g.,
• title, keywords, abstract, citations to other
  documents
• An article might cite, e.g.
• Plato, Aristotle, Smith, Brown
• Over a given set of many citing articles
• Count how many times each pair of authors were
  cited together
• Resulting co-citation count shows common
  intellectual interest

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Co-Occurrence Derivation

• For a given data set (N 4 unique terms)
• Article 1 Plato, Aristotle, Smith
• Article 2 Plato, Smith
• Article 3 Plato, Aristotle, Smith, Brown
• The following co-citations (C(4,2) 6) are found
• COMBINATION COUNT ARTICLES
• Plato and Smith 3 1, 2, 3
• Plato and Aristotle 2 1, 3
• Plato and Brown 1 3
• Aristotle and Smith 2 1, 3
• Aristotle and Brown 1 3
• Smith and Brown 1 3

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Co-Occurrence Measures

• Raw counts
• Additional information
• Correlations
• Replace each cell by correlation measure of each
  pair-wise column
• Conditional Probability
• Compute each cell by dividing each unique
  combination by total occurring

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Co-Occurrence Structure -Example
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Graphical Techniques

• Three Methodologies
• Multi-dimensional scaling
• Self-organizing maps
• Pathfinder networks

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MDS
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MDS Methodology

• Given original distances (similarities) estimate
  coordinates that could give those distances
• The computed distances should correspond to the
  original distances
• Stress
• Added dimensions

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SOM
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Self-Organizing Maps (SOMs)

• Also known as Kohonen Maps
• Based on Neural Networks
• Related to wetware
• robust techniques
• If categories are known
• supervised technique
• backproprogating learning
• If categories are sought
• unsupervised technique
• competitive learning

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SOMs

• Given a 2-D grid of nodes
• each node has N weights
• each vector (row) has N terms
• map each input vector to a node
• Similar to vector quantization (VQ)

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SOMs Generation

• nodes initially given random weights
• randomly sample an input vector
• row of co-occurrence matrix
• with replacement
• find a node closest to vector
• Euclidean distance
• update node weights
• node weight node weight gain term distance
• update neighborhood
• cool gain term and neighborhood
• repeat

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PF Nets
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Pathfinder Networks

• Uses on graph notation
• nodes authors
• edges co-citation counts
• Co-occurrence is a complete network (weighted,
  undirected)

Plato
3
Smith
2
2
Aristotle
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Pathfinder Networks Generation

• Pathfinder Network is generated by varying the
  parameters
• distance (r)
• triangle inequality (q)

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Pathfinder Distance

• Uses Minkowski metric
• d (? eir )1/r
• Example
• e1 3, e2 4
• r 1 gt d 7 3 4
• Driving distance / ratio data
• r 2 gt d 5 (9 16)1/2
• Euclidean Distance
• r (approaches) infinity gt d 4 max( 3, 4)
• ordinal data
• rank rather than value

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Pathfinder Triangle Inequality

• A required property of a metric definition
• d(i,j) lt d(i,k) d(k,j)
• But may not be justified
• in personal judgments
• If a is similar to b, and b is similar to c,
  there may be no transitive judgment of similarity
  from a to c
• in set intersections
• Even though Smith and Jones appear 12 times, and
  Jones and Brown appear 5 times, the overlap
  between Smith and Brown cannot be predicted

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Pathfinder Triangle Inequality

• Defines q-triangular
• check paths of length q to determine if
  inequality is met
• minimum is 2
• maximum is n -1
• full compliance
• the longer the length, the fewer the connections

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Pathfinder Example
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Pathfinder Network Creation

• PFNet (r, q)
• Examine all paths of length q or less.
• Use Minkowski Metric with parameter r to compute
  path length.
• If a path of less weight is found, then remove
  the edge.

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Pathfinder - Example
Smith
Jones
5
q 2
4
3
Brown
r 1 gt Smith - Jones is kept
r 2 gt Smith - Jones is kept
r infinity gt Smith - Jones is removed
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Comparison of Techniques

• MDS
• Reduces dimensions / reveals clusters
• 2D may be insufficient
• measurement may not be Euclidean
• SOM
• robust
• no guarantee of convergence/unique solution
• Pathfinder
• does not assume ratio data/triangle inequality
• connections rather than position is important
• additional methodology needed for display

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Comparison of Techniques

• Similarities
• Spatial models
• Differences
• use of visual space
• semantic meaning
• as related to data
• research in progress

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Graphical Display of Methodologies

• MDS
• assume that 2 dimensions are sufficient
• x, y for each point already defined
• SOM
• grid defines the 2D surface
• plot each label with the appropriate node
• Pathfinder
• only defines the nodes and links
• need additional methodologies
• Spring-embedder models
• Kamada and Kawai (1989)
• Fruchterman and Reingold (1991)
• Davidson and Harel (1996)

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Graphical Comparison of Three Methods

• Data
• Institute for Scientific Information
• Arts and Humanities Database (AHCI)
• 1988 - 1997
• 1.26 million records
• Example
• Given Plato, find related authors
• Interface described in IV 2000 Paper
• CSNA 2000 Paper
• (Lin, Buzydlowski, White)

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25 Authors Co-cited with Plato

• PLATO (4928)
• ARISTOTLE (1861)
• PLUTARCH (838)
• CICERO (699)
• HOMER (627)
• BIBLE (552)
• EURIPIDES (515)
• ARISTOPHANES (474)
• XENOPHON (459)
• AUGUSTINE (432)
• HERODOTUS (425)
• KANT-I (385)
• AESCHYLUS (374)

• SOPHOCLES (363)
• THUCYDIDES (363)
• OVID (334)
• HESIOD (325)
• DIOGENES-LAERTIUS (317)
• HEIDEGGER-M (312)
• DERRIDA-J (304)
• PINDAR (292)
• NIETZSCHE-F (278)
• HEGEL-GWF (264)
• VERGIL (259)
• AQUINAS-T (255)

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300 Pair-wise co-citations

• 1 PLATO AND ARISTOTLE -1940 docs
• 2 PLATO AND PLUTARCH - 872 docs
• .
• .
• .
• 300 VERGIL AND AQUINAS-T - 38 docs

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Visualization allows for the revelation of
intricate structure which cannot be absorbed in
any other way...
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2D MDS map of 25 authors co-cited with Plato
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PFNet of 25 authors co-cited with Plato
AESCHYLUS
SOPHOCLES
EURIPIDES
HESIOD
AUGUSTINE
HOMER
PINDAR
BIBLE
ARISTOPHANES
PLATO
DIOGENES-LAERTIUS
ARISTOTLE
XENOPHON
KANT-I
CICERO
AQUINAS-T
PLUTARCH
HEIDEGGER-M
THUCYDIDES
DERRIDA-J
HEGEL-GWF
HERODOTUS
OVID
NIETZSCHE-F
VERGIL
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Conclusion

• Slides available at
• faculty.cis.drexel.edu/jbuzydlo/
• janb_at_drexel.edu

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Bibliography

• Chen, Chaomei, Information Visualization and
  Virtual Environments, 1999.
• Cleveland, William S., Visualizing Data, Hobart
  Press, 1993.
• Davidson, R, Harel, D, Drawing Graphs Nicely
  Using Simulated Annealing, ACM Transactions on
  Graphics, 15(4) 301-31 (1996).
• Fruchterman,TMJ, Reingold, EM, Graph Drawing by
  Force-Directed Placement, Software Practice and
  Experience, 21 1129-64 (1991).
• Kamada, T,Kawai, S, An Algorithm for Drawing
  General Undirected Graphs, Information Processing
  Letters, 31(1) 7-15, (1989).