1 / 56

Phonological neighbors in a small world: What can graph theory tell us about word learning?

Phonological neighbors in a small world: What can graph theory tell us about word learning?. Michael S. Vitevitch Department of Psychology University of Kansas NIH-NIDCD R03 DC 04259 NIH-NIDCD R01 DC 006472. Graph Theory. Graphically represent complex systems Graph or Network

misha
Download Presentation

Phonological neighbors in a small world: What can graph theory tell us about word learning?

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Phonological neighbors in a small world: What can graph theory tell us about word learning? Michael S. Vitevitch Department of Psychology University of Kansas NIH-NIDCD R03 DC 04259 NIH-NIDCD R01 DC 006472

  2. Graph Theory • Graphically represent complex systems • Graph or Network • Vertices or Nodes • Edges or Links/Connections • Examples of systems • Diamond (crystals) • WWW • power grid • Interstate highway system

  3. Ordered Graph

  4. Random Graph

  5. Graph Theory • Between ordered and random graphs are small-world graphs • Small path length (Six Degrees of Separation) • High clustering coefficient (relative to random) • “Probability” of my two friends being friends with each other. • 0 to 1

  6. Movie Actors(Watts & Strogatz, 1998) • Large and complex system • 225,226 actors • Internet Movie Database circa 1998 • Node = Actor • Connection = Co-starred in a movie

  7. Movie Actors • Despite large size, the network of actors exhibits small-world behavior • Average of 3 links between any two actors • Clustering coefficientrandom = .00027 • Clustering coefficientactors = .79 • Over 2000 times larger

  8. Graph Theory • Some small-world graphs also are scale-free. • Degree = number of links per node • These systems exhibit interesting characteristics • Efficient processing • Development/growth • Robustness of the system to attack/failure

  9. Graph Theory • A randomly connected network has a bell-shaped degree distribution. • Most nodes have the average number of links • Few nodes have more links than average • Few nodes have less links than average • Scale = stereotypical node (characterized by mean)

  10. Graph Theory • In a network with a scale-free degree distribution there is no stereotypical node. • The degree-distribution follows a power-law • Many nodes have few connections • Few nodes have many connections

  11. Scale-free network The power-law degree distribution of a scale-free network emerges as a result of: • Growth • New nodes are added to the system over time. • Preferential Attachment • New nodes tend to form links with nodes that are highly connected.

  12. Implications of graph structure • The structure of a network constrains the processes that operate in the system and influences how well the system withstands damage.

  13. Viewing the mental lexicon as a graph • Does the mental lexicon have a small-world structure? • Does the mental lexicon have a scale-free structure? • How does the structure of the mental lexicon influence various processes?

  14. The mental lexicon as a graph • Nodes = 19,340 word-forms in database • Nusbaum, Pisoni & Davis (1984) • Links = phonologically related • One phoneme metric (Luce & Pisoni, 1998)

  15. Links in the mental lexicon • One phoneme metric • cat has as neighbors • scat, at, hat, cut, can, etc. • dog is NOT a phonological neighbor of cat • Steyvers & Tenenbaum (2003) • Ferrer i Cancho & Solé (2001)

  16. The mental lexicon as a graph • Small-world network • Relatively small path-length • Relatively high clustering coefficient • Scale-free topology • Power-law degree distribution • Growth • Preferential attachment

  17. The mental lexicon as a graph Pajek (Batagelj & Mrvar, 2004) • Program for analysis and visualization of large networks

  18. Graph of Adult Lexicon

  19. The mental lexicon as a graph

  20. Path-length Average distance between two nodes • cat-mat-mass-mouse •  = 6.05

  21. Diameter Longest path length • D = 29 • connect & rehearsal • connect, collect, elect, affect, effect, infect, insect, inset, insert, inert, inurn, epergne, spurn, spin, sin, sieve, live, liver, lever, leva, leaven, heaven, haven, raven, riven, rivet, revert, reverse, rehearse, rehearsal

  22. Clustering Coefficient A small-world network has a larger clustering coefficient (by orders of magnitude) than a random network • C = .045 • Over 250 times greater

  23. The mental lexicon is asmall-world network • Relatively short path length • Large clustering coefficient

  24. Does the mental lexicon have a scale-free topology? • A degree distribution that follows a power law. • Growth • Preferential attachment

  25. Power-law degree distribution

  26. Power-law degree distribution Degree distribution for the lexicon:  = 1.96 (approaching 2 < < 3)

  27. Growth & Preferential Attachment • Growth • Children (and adults) learn new words • New words are added to the language over time • Preferential attachment

  28. Preferential Attachment • Words that are added to the lexicon early in life should have more links than words that are added to the lexicon latter in life. • Storkel (2004) • Relationship between AoA and Density

  29. Preferential Attachment • Phonological neighborhoods should become “denser” over time. • Charles-Luce and Luce (1990, 1995) • Analyzed words in adults and children 5- and 7-years old. • Neighborhood density for words in the adult lexicon were denser than the neighborhood density for those same words in the 5- and 7-years old lexicons.

  30. Preferential Attachment • Words with denser neighborhoods should be easier to learn/acquire. • Storkel (2001, 2003) • Pre-school age children learned novel words that had common sound sequences/dense neighborhoods more rapidly than novel words that had rare sound sequences/sparse neighborhoods. • Adults, too (Storkel, Armbrüster & Hogan, submitted)

  31. Advantage of this structure Topological robustness • Damage does not result in catastrophic failure

  32. Topological robustness • Damage tends to affect less connected nodes. • Hubs maintain integrity of the whole system. • Even if a hub is damaged, the presence of other hubs will absorb the extra processing load. • Only if everynode has been damaged will a scale-free network catastrophically fail.

  33. Topological robustness in the mental lexicon • Speech production errors occur more for words with sparse than dense neighborhoods. • Vitevitch (1997; 2002) and Vitevitch & Sommers (2003) • Same pattern for errors in patients with aphasia • Gordon & Dell (2001) • More errors in STM for words with sparse than dense neighborhoods. • Roodenrys et al. (2002)

  34. Mental lexicon as a scale-free network • The present analysis suggests the lexicon has a scale-free topology. • Evidence from several areas is consistent with predictions derived from a scale-free lexicon.

  35. Do the characteristics of graph theory have any psychological reality?

  36. Psychological reality of graph theory • k (degree) = Neighborhood Density • Luce & Pisoni (1998) • Vitevitch (2002) • Clustering Coefficient • Probability of two neighbors of a word being neighbors with each other.

  37. Clustering Coefficient Experiment • Auditory Lexical Decision Task (n = 57) • Words varying in clustering coefficient • Frequency • Familiarity • Neighborhood Density • NHF • Phonotactic Probability • Onsets

  38. hive wise

  39. Clustering Coefficient Experiment

  40. Clustering Coefficient Experiment In spoken word recognition • k (degree), neighborhood density • Words with sparse neighborhoods are responded to more quickly than words with dense neighborhoods. • Clustering Coefficient • Words with high CC are responded to more quickly than words with low CC.

  41. When does a scale-free lexicon emerge?

  42. When does a scale-free lexicon emerge? • Traditional benchmark for “vocabulary spurt” is 50 words (about 18 mo.) • (e.g., Goldfield & Reznick, 1996; Mervis & Bertrand, 1995). • Various mechanisms have been proposed for the vocabulary spurt • (e.g., Golinkoff et al. 2000; Nazzi & Bertoncini, 2003).

  43. MacArthur Communicative Development Inventory (CDI) Estimate known words in 16-30 m.o. children • The earliest age at which 50% of the children knew a given word at a particular age. • 16, 18, 19, 30 months of old

  44. Network Statistics

  45. Emergence of a scale-free lexicon “Vocabulary spurt” is often observed: • 18- to 19-months of age • 50-words • Signals reorganization in the lexicon.

  46. Emergence of a scale-free lexicon A scale-free network emerged at the same age/developmental milestone. • This may lead to highly efficient word learning and language processing.

  47. Emergence of a scale-free lexicon Variability in age/vocabulary size associated with this developmental milestone may be due to different initial starting states. • The first few sound patterns that are learned may play a large role in determining how easily subsequent words are acquired. • Mandel, Jusczyk & Pisoni (1995)

More Related