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Netscal: A Network Scaling Algorithm for Nonsymmetric Proximity Data

Published online by Cambridge University Press:  01 January 2025

J. Wesley Hutchinson
J. Wesley Hutchinson*
Affiliation:
University of Florida
*
Requests for reprints should be sent to Wes Hutchinson, Department of Marketing, 205 Matherly Hall, University of Florida, Gainesville, FL 32611.
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Abstract

A simple property of networks is used as the basis for a scaling algorithm that represents nonsymmetric proximities as network distances. The algorithm determines which vertices are directly connected by an arc and estimates the length of each arc. Network distance, defined as the minimum pathlength between vertices, is assumed to be a generalized power function of the data. The derived network structure, however, is invariant across monotonic transformations of the data. A Monte Carlo simulation and applications to eight sets of proximity data support the practical utility of the algorithm.

Keywords

networksproximity scalingsimilaritygraph theory

Information

Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 1 , March 1989 , pp. 25 - 51
Copyright
Copyright © 1989 The Psychometric Society

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Footnotes

I am grateful to Roger Shepard and Amos Tversky for their helpful comments and guidance throughout this project. The work was supported by National Science Foundation Grant BNS-75-02806 to Roger Shepard and a National Science Foundation Graduate Fellowship to the author. Parts of this paper were drawn from a doctoral dissertation submitted to Stanford University (Hutchinson, 1981).

References

Andersen, J. R. (1976). Language, memory and thought, Hillsdale, NJ: Erlbaum.Google Scholar
Anderson, J. R. (1983). The architecture of cognition, Cambridge, MA: Harvard University Press.Google Scholar
Anderson, J. R., Bower, G. H. (1973). Human associative memory, Washington, DC: Winston.Google Scholar
Bobrow, D. G., & Fraser, J. B. (1969). An augmented state transition network analysis procedure. Proceedings of the International Joint Conference on Artificial Intelligence, Washington, DC, 557–567.Google Scholar
Burt, R. S. (1980). Models of network structure. Annual Review of Sociology, 6, 79–141.CrossRefGoogle Scholar
Chandler, J. D. (1969). STEPIT—Finds local minima of a smooth function of several parameters. Behavioral Science, 14, 81–82.Google Scholar
Chino, N. (1978). A graphical technique for representing the asymmetric relationships between N objects. Behaviormetrika, 5, 23–40.CrossRefGoogle Scholar
Cohen, B. H., Bousefield, W. A., Whitmarsh, G. A. (1957). Cultural norms for verbal items in 43 categories, Storrs: University of Connecticut.Google Scholar
Collins, A. M., Loftus, E. F. (1975). A spreading-activation theory of semantic processing. Psychological Review, 82, 407–428.CrossRefGoogle Scholar
Cunningham, J. P. (1978). Free trees and bidirectional trees as representations of psychological distance. Journal of Mathematical Psychology, 17, 165–88.CrossRefGoogle Scholar
Curry, R. E. (1975). A random search algorithm for laboratory computers. Behavioral Research Methods and Instrumentation, 7, 369–376.Google Scholar
Deese, J. (1962). On the structure of associative meaning. Psychological Review, 69, 161–175.CrossRefGoogle ScholarPubMed
Deese, J. (1965). The structure of association in language and thought, Baltimore, MD: John Hopkins.Google Scholar
De Soete, G., DeSarbo, W. S., Furnas, G. W., Carroll, J. D. (1984). The estimation of ultrametric and path length trees from rectangular proximity data. Psychometrika, 49, 289–310.CrossRefGoogle Scholar
Edgell, S. E., Geisler, W. S., Zinnes, J. L. (1973). A note on a paper by Rumelhart and Greeno. Journal of Mathematical Psychology, 10, 86–90.10.1016/0022-2496(73)90006-0CrossRefGoogle Scholar
Feigenbaum, E. A. (1963). The simulation of verbal learning behavior. In Feigenbaum, E. A., Feldman, J. (Eds.), Computers and thought (pp. 297–309). New York: McGraw-Hill.Google Scholar
Fish, R. S. (1981). Color: Studies of its perceptual, memory and linguistic representation. Unpublished doctoral dissertation, Stanford University.Google Scholar
Floyd, R. W. (1962). Algorithm 97, shortest path. Communications ACM, 5, 345–345.CrossRefGoogle Scholar
Gondran, M., Minoux, M. (1984). Graphs and algorithms, New York: Wiley.Google Scholar
Goldman, A. J. (1966). Realizing the distance matrix of a graph. Journal of Research of the National Bureau of Standards, Series B: Mathematics and Mathematical Physics, 70, 153–154.CrossRefGoogle Scholar
Gower, J. C. (1977). The analysis of asymmetry and orthogonality. In Barra, J. R., Brodeau, F., Romier, G., van Custems, B. (Eds.), Recent developments in statistics (pp. 109–123). Amsterdam: North-Holland.Google Scholar
Hakimi, S. L., Yau, S. S. (1964). Distance matrix of a graph and its realizability. Quarterly Journal of Applied Mathematics, 22, 305–317.CrossRefGoogle Scholar
Harary, F., Normal, R. Z., Cartwright, D. (1965). Structural models: An introduction to the theory of directed graphs, New York: Wiley.Google Scholar
Harshman, R. A., Green, P. A., Wind, Y., Lundy, M. E. (1982). A model for the analysis of asymmetric data in marketing research. Marketing Science, 1, 205–242.CrossRefGoogle Scholar
Hayes-Roth, B., Hayes-Roth, F. (1975). Plasticity in memorial networks. Journal of Verbal Learning and Verbal Behavior, 14, 506–522.CrossRefGoogle Scholar
Hebb, D. O. (1949). Organization of behavior, New York: Wiley.Google Scholar
Henley, N. M. (1969). A psychological study of the semantics of animal terms. Journal of Verbal Learning and Verbal Behavior, 8, 176–184.CrossRefGoogle Scholar
Hintzman, D. L. (1968). Explorations with a discrimination net model for paired-associate learning. Journal of Mathematical Psychology, 5, 123–162.CrossRefGoogle Scholar
Holman, E. W. (1979). Monotone models for asymmetric proximities. Journal of Mathematical Psychology, 20, 1–15.10.1016/0022-2496(79)90031-2CrossRefGoogle Scholar
Hutchinson, J. W. (1981). Network representations of psychological relations. Unpublished doctoral dissertation, Stanford University.Google Scholar
Kaufman, A. (1975). Introduction to the theory of fuzzy subsets (Vol. I), New York: Academic Press.Google Scholar
Krantz, D. H., Luce, R. D., Suppes, P., Tversky, A. (1971). Foundations of measurement (Vol. I), New York: Academic Press.Google Scholar
Krumhansl, C. L. (1978). Concerning the applicability of geometric models to similarity data: The interrelationship between similarity and spatial density. Psychological Review, 85, 445–463.CrossRefGoogle Scholar
Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7, 48–50.CrossRefGoogle Scholar
Kruskal, J. B., Young, F. W., Seery, J. B. (1973). How to use KYST: A very flexible program to do multidimensional scaling and unfolding, Murray Hill, NJ: AT&T Bell Labs.Google Scholar
Luce, R. D. (1959). Individual choice behavior, New York: Wiley.Google Scholar
Marshall, G. R., Cofer, C. N. (1970). Single-word free-association norms for 328 responses from the Connecticut norms for verbal items in categories. In Postman, L., Keppel, G. (Eds.), Norms of word association (pp. 321–360). New York: Academic Press.CrossRefGoogle Scholar
Mehler, J. (1963). Some effects of grammatical transformations on the recall of English sentences. Journal of Verbal Learning and Verbal Behavior, 2, 346–351.CrossRefGoogle Scholar
Miller, G. A. (1962). Some psychological studies of grammar. American Psychologist, 17, 748–762.CrossRefGoogle Scholar
Moeser, S. D., Tarrant, B. L. (1977). Learning a network of comparisons. Journal of Experimental Psychology: Human Learning and Memory, 3, 643–659.Google Scholar
Morrison, H. W. (1963). Testable conditions for triads of paired comparison choices. Psychometrika, 28, 369–390.CrossRefGoogle Scholar
Newell, A., Simon, H. (1972). Human problem solving, Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Pruzansky, S., Tversky, A., Carroll, J. D. (1982). Spatial versus tree representations of proximity data. Psychometrika, 47, 3–24.CrossRefGoogle Scholar
Restle, F. (1961). Psychology of judgment and choice, New York: Wiley.Google Scholar
Rips, L. J., Stubbs, M. E. (1980). Genealogy and memory. Journal of Verbal Learning and Verbal Behavior, 19, 705–721.CrossRefGoogle Scholar
Rosch, E. (1978). Principles of categorization. In Rosch, E., Lloyd, B. B. (Eds.), Cognition and categorization (pp. 27–48). Hillsdale, NJ: Erlbaum Press.Google Scholar
Rosch, E., Mervis, C. B., Gray, W., Johnson, D., Boyes-Braem, P. (1976). Basic objects in natural categories. Cognitive Psychology, 3, 382–439.CrossRefGoogle Scholar
Rumelhart, D. L., Greeno, J. G. (1971). Similarity between stimuli: An experimental test of the Luce and Restle choice models. Journal of Mathematical Psychology, 8, 370–381.CrossRefGoogle Scholar
Scott, A. (1977). Neurodynamics: A critical survey. Journal of Mathematical Psychology, 15, 1–45.CrossRefGoogle Scholar
Sjobërg, L. (1975). Choice frequency and similarity. Scandinavian Journal of Psychology, 18, 103–115.CrossRefGoogle Scholar
Smith, E. E., Shoben, E. J., Rips, L. J. (1974). Structure and process in semantic memory: A feature model for semantic decisions. Psychological Review, 81, 214–241.CrossRefGoogle Scholar
Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Tobler, W. (1976). Spatial interaction patterns. Journal of Environmental Systems, 6, 271–301.CrossRefGoogle Scholar
Tversky, A. (1972). Elimination by aspects: A theory of choice. Psychological Review, 79, 281–299.CrossRefGoogle Scholar
Tversky, A. (1977). Features of similarity. Psychological Review, 84, 327–352.CrossRefGoogle Scholar
Tversky, A., Sattah, S. (1979). Preference trees. Psychological Review, 86, 542–573.CrossRefGoogle Scholar