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					   Self-Organization of Speech
     Sound Inventories in the
framework of Complex Networks


           Animesh Mukherjee
          (Roll No.: 05CS9405)
Department of Computer Science & Engg.
Indian Institute of Technology, Kharagpur
    Overview of the Presentation
• Basics of Speech Sound Inventories
• Motivation & Objective
• Occurrence Network of Consonants
• Co-occurrence Network of Consonants
• Co-occurrence Patterns in Consonant Inventories
• Network Methods applied to Vowel Inventories
• Conclusions and Future Directions
                 Sound Inventories
• A repertoire of unique sounds (aka phonemes)
  that the speakers of a language use for
  communication
English Consonants    /p/   /b/   /s/   /z/    /r/     ……………

         As in        pit   bit send    zip    rat


                      /p/   /b/   /ɖ/   /ɖʱ/   /r/     ……………
Bangla Consonants

        As in        pAn    bAn ɖAl ɖhol       rAtri
     Representation of Phonemes

• Articulatory feature based representation
  – Place of Articulation (labial, velar, alveolar, dental
    etc.)
  – Manner of Articulation (plosive, fricative,
    affricate, nasal etc.)
  – Phonation (voiced, voiceless)



                                                Mermelstein’s Model
          Choice of Phonemes
• Given a set of phonemes how likely is it that
  the set corresponds to real language inventory?

• Does any random subset of phonemes qualify
  as a real inventory?

     • Certainly Not!

• What are the forces governing the structure of
  an inventory?
  Forces Governing the Structure
         A Linguistic System – How does it look?

                                 /a/                      /a/




                       Speaker                 Listener / Learner



Desires “ease of articulation”         Desires “perceptual contrast” / “ease of learnability”


Forces shaping the structure are opposing – There has to be a non-trivial solution
 Motivation – Choice of the Problem
• Vowel inventories
  – Linguistic arguments (Wang 1971)
  – Numerical simulations (Liljencrants & Lindblom
    1972, Lindblom 1986, Schwartz et al. 1997)
  – Genetic algorithms (Ke et al. 2003)
  – Multi-agent simulations (Boer 2000)
• Organized based on the principle of maximal
  perceptual contrast (mainly smaller
  inventories)
  – For instance if a language has three vowels then in
    more than 95% of the cases they are /a/,/i/, and /u/.
 Motivation – Choice of the Problem
• Consonant Inventories
  – Linguistic arguments (Clements 2003, Clements
    2008, Boersma 1998, Hockett 1974, Lindblom &
    Maddieson 1988)
• Studies limited to certain specific properties
  – They are much larger in size with many more
    articulatory/acoustic features
  – No single force is sufficient to explain their
    organization
  – A complex interplay of forces collectively shape
    their structure.
Motivation – Modeling Methodology

• We adopt a complex network approach to
  capture the self-organization of the consonant
  inventories
  – A versatile modeling methodology  view &
    solve the problem from an alternative perspective
  – Enormous success in explaining various dynamical
    properties of language (Adilson 2002, Ferrer-i-
    Cancho & Sole 2001, Gruenenfelder & Pisoni
    2005, Kapatsinski 2006, Sigman & Cecchi 2002)
  – Easy applicability in modeling this particular
    problem pertaining to sound inventories
                   Objective
• Representation of the Inventories
  – How can the structure of the consonant inventories
    be accurately represented within the framework of
    complex networks?
• Analysis of the Inventory Structure
  – How to conduct the analysis of the network(s)
    constructed in order to extract meaningful results
• Analysis of the Inventory Structure
  – Explain the emergence of the different statistical
    properties (obtained from the analysis) by means
    of generative mechanisms usually based on models
    of network growth
Occurrence Network of Consonants
• Phoneme-Language Network (PlaNet)
  – Bipartite                                                       /θ/
                                                          L1
  – VL (set of nodes in the language partition)                     /ŋ/

  – VC (set of nodes in the consonant




                                              Languages
                                                          L2




                                                                          Consonants
                                                                    /m/
    partition)
  – There is an edge e Є E between vl Є VL                L3        /d/
    and vc Є VC iff the consonant c occurs in
                                                                    /s/
    the language l                                        L4
  – PlaNet constructed from the UCLA                                /p/
    Phonological Segment Inventory                             PlaNet
    Database (UPSID)  317 languages with
    541 unique consonants appearing across
    them
             Degree Distribution (DD)
  .08
                                                    DD of the language nodes follows a
                  pk = beta(k) with α = 7.06,       β-distribution
  .06             and β = 47.64

                       Γ(54.7) k6.06(1-k)46.64
pk.04             pk =
                          Γ(7.06) Γ(47.64)
                                                              DD of the consonant nodes follows a
  .02         kmin= 5, kmax= 173, kavg= 21                    power-law with an exponential cut-off
                                                        1
                                                        1
        0    50           100         150        200
                     Degree (k)                         .1
                                                       0.1


 pk  Fraction of nodes with degree = k
                                                                  Pk = k -0.71
                                                   .01
                                                  0.01
                                                 Pk
 Pk  Fraction of nodes with degree >= k                               Exponential Cut-off

                                                  .001
                                                  0.001
                                                             11            10
                                                                            10           100
                                                                                         100   1000
                                                                                                 1000
                                                                                 Degree (k)
              Growth of PlaNet

                           Rules of the game:
   Phonemes




               Languages
                           • A new language is born




Coling-ACL, 2006
              Growth of PlaNet
                           Rules of the game:
                           • A new language is born
   Phonemes




               Languages
                           • Chooses μ distinct
                             phonemes from the set of
                             existing phonemes
                             preferentially based on the
                             degree

                                            γ k +1
                                         (γ k + 1)
                                all phonemes not
                                 already chosen


Coling-ACL, 2006
Analytical Solution for the Growth Model
     Notations
    t – #nodes in VL
    N – #nodes in VC (fixed and finite)
    pk,t – pk after adding t nodes
    Markov Chain Formulation (μ=1)




     where


Europhysics Letters, 2007
     The Hard Part of the Analysis
• Average degree of the VC partition, i.e., (μt)/N
  diverges as t∞
• Methods based on steady-state and continuous time
  assumptions fail (pk,t ≠ pk,t+1as t∞)
 Closed-form solution using linear algebra tricks




                                     where η=N/γ

Europhysics Letters, 2007
       Fitted Degree Distribution




                                               Theory




       t=317, N=541, μ=21. Best fit for γ=14
Coling-ACL, 2006
Co-occurrence Network of Consonants
• Phoneme-Phoneme Network                          1
                                                           /s/
                                                                      1
  (PhoNet)                                                   1
                                         /k/                              /n/
  – One-mode projection of PlaNet                      1          2

    onto the consonant nodes (VC)        2     1                          1

                                                                  1
  – Two nodes in this n/w are            /t/
                                                       2
                                                             1
                                                                          /d/

    connected by an edge if they co-               2                  1
    occur in the inventory of at least                      /p/

    one language. The number of
                                                       PhoNet
    languages they co-occur in
    defines the weight of the edge.
 Degree of the nodes in One-mode
• Easy to calculate if each node v in growing
  partition enters with exactly  (> 1) edges
• Consider a node u in the non-growing partition
  having degree k
• u is connected to k nodes in the growing
  partition and each of these k nodes are in turn
  connected to -1 other nodes in the non-
  growing partition
• Hence degree q=k(-1)
Submitted to Europhysics Letters
            Degree Distribution
• The degree distribution pu(q) of the nodes in the one-
  mode should be



                   Not a good match at all!!
        What if  is not fixed??
• Relax the assumption that the size of the
  consonant inventories is a constant ()
• Assume these sizes to be random variables
  being sampled from a distribution fd
• It is easy to show that, while the one-mode
  degree (q) for a node u is dependent on fd, its
  bipartite n/w degree (k) is not (the kernel of
  attachment roughly remains the same)
    Analysis of Degree Distribution
 • If fd varies as a Normal Distribution N(μ, σ2)


 • If fd varies as a Delta function δ(d, μ)

 • If fd varies as an Exponential function E(λ=1/μ)

 • If fd varies as Power-law function (power = –λ)

Submitted to Europhysics Letters
            Results of the Analysis
Bipartite
                                 One-Mode
Networks
                                 Networks



                                 N = 1000,
                                 t = 1000,
                                 γ= 2,
                                 μ=22
    Degree Distribution of PhoNet
                                   fd = consonant inventory
                    Real PhoNet         size distribution



                                   fd = constant




Submitted to Europhysics Letters
  Clustering Coefficient of PhoNet
• The Clustering Coefficient (CC) for a node i is
  the proportion of links between the nodes that
  are the neighbors of i divided by the number of
  links that could possibly exist between them.
• CC of PhoNet is 0.89
• CC of the synthesized n/w obtained from our
  model is 0.35
• The model needs to be refined to increase the
  number of triangles in the emergent network to
  match CC
      Improving CC – Triad Model
              L1         L2    L3    L4    L5


       IF
            L1       L2       L3    L4    L5    L6


Then (triad step – pt)
            L1           L2   L3    L4    L5    L6
                          Results
   • The triad model produces CC = 0.85 (within
     3.5% of the real network) [0.8<= pt <=0.9]

                         The degree distribution also
                             remains unaffected




Journal of Quantitative Linguistics, 2009
        Patterns of Co-occurrence
• Consonants tend to co-occur in groups or communities

• These groups tend to be organized around a few
  distinctive features (based on: manner of articulation,
  place of articulation & phonation) – Principle of
  feature economy


              plosive        voiced          voiceless
                                                         If a language has
             bilabial          /b/            /p/        in its inventory


             dental            /d/           /t/

            then it will also tend to have
           Automatic Identification of
            Co-occurrence Patterns
• Community structure analysis of PhoNet
• Employ modified Radicchi et al. algorithm
    – Look for triangles, where the weights on the edges
      are comparable. If comparable, then the group of
      consonants co-occur highly else it is not so.
    – Calculate strength S of each edge
                wuv
    S=                                 if   √Σi Є Vc-{u,v}(wui – wvi)2>0 else S = ∞
         √Σi Є Vc-{u,v}(wui – wvi   )2

    – Remove edges with S less than a threshold η
International Journal of Modern Physics C, 2007
Consonant Communities
                    η=0.35

           η=0.60

                    η=0.72
  η=1.25
 Feature Economy: The Binding Force
• pf – number of consonants in a community (C) in
  which feature f is present
• qf – number of consonants in C in which feature f is
  absent
• The probability that a consonant chosen at random
                  p
  form C has f is N f and that is does not have f is qf
  (1- p)f                                            N
      N
• If F denote the set of all features,
                                   q     q
             FE= –∑fєF N log2 N + N log2 N
                         p  f  p  f    f    f



• FE Total discriminative capacity of the
  features in an inventory
              Comparison between
             PhoNet and PhoNetrand




                                  PhoNetrand



                         PhoNet



International Journal of Modern Physics C, 2007
          Network Methods for
          the Vowel Inventories
• Construct two networks
  – VlaNet (Vowel-Language Network): Bipartite
    network with one partition of languages (VL) and
    the other of vowels (VV); an edge signifies the a
    particular vowel occurs in a particular language;
    317 languages and 151 vowels
  – VoNet (Vowel-Vowel Network): One-mode
    projection of VlaNet where to vowel nodes are
    connected as many times as they co-occur across
    different languages
Degree Distribution (VlaNet)

    β-distribution as in                Theory
   the case of consonants




            Simulation      Real Data
Degree Distribution (VoNet)


                       fd = consonant
                        inventory size
                         distribution



                          Real Data




      fd = constant
  Clustering Coefficient (VoNet)
• CC for VoNet is 0.86
• Using triad model on can achieve a CC of 0.83
  (within 3.5%) of the real data

                    The degree distribution also
                         not much affected
Community Analysis of VoNet
                  Two forces acting together




                       Feature Economy
                           Perceptual
                           Contrast
 VoNethub, VoNetrest and VoNetrest'
• VoNethub
   – All vowel nodes having frequency of occurrence < 120
     removed from VoNet along with all edges  A network of
     hub nodes.
• VoNetrest
   – All vowel nodes in VoNet are retained. Only edges
     between hub & non-hub nodes removed.
• VoNetrest'
   – All vowel nodes in VoNet are retained. Only edges that
     connect a hub with a non-hub where the non-hub occurs
     more than 95% of times with the hub are retained
Advances in Complex Systems, 2008
            Vowel Communities

• VoNethub


• VoNetrest


• VoNetrest'


Advances in Complex Systems, 2008
       Comparison with Randomly
         Generated Inventories
         Perceptual Contrast               Feature Economy



                         VoNethub
                                                             VoNetrest




                                    Feature Economy


                                           VoNetrest'




Advances in Complex Systems, 2008
Consonant Vs. Vowel Inventories
• Topological properties are qualitatively similar
   preferential attachment plays the key role in
  the emergence of the structure
• Community and redundancy ratio analysis
  however shows differences
  – Consonants  Feature economy is the key driving
    force
  – Vowels  Smaller inventories are driven by
    perceptual contrast while the larger ones are driven
    by feature economy
Conclusions and Future Directions
• Complex Network based modeling allowed us
  to excavate various interesting universal
  properties of sound inventories
• We do not claim that all the inferences that we
  draw are sacrosanct; rather they are indicative
• Trends are more important than exact values
• Results should help propelling future research
  in self-organizing phonology.
Conclusions and Future Directions
• Quite a few theoretical problems that might
  attract statistical physicists
• Network methods highly instrumental in doing
  computational linguistics
  – Unsupervised NLP (Distributional Similarity N/ws
    for learning syntactic and semantic categories)
  – IR (Blog and Query-log analysis)
     Publications from the Thesis
[1] M. Choudhury, A. Mukherjee, A. Basu, and N. Ganguly. Analysis and
   synthesis of the distribution of consonants over languages: A complex
   network approach. In Proceedings of COLING–ACL, 128–135, 2006.
[2] F. Peruani, M. Choudhury, A. Mukherjee, and N. Ganguly.
   Emergence of a nonscaling degree distribution in bipartite networks: A
   numerical and analytical study. Euro. Phys. Lett., 79(2):28001, 2007.
[3] A. Mukherjee, M. Choudhury, A. Basu, and N. Ganguly. Modeling
   the cooccurrence principles of the consonant inventories: A complex
   network approach. Int. Jour. of Mod. Phy. C, 18(2):281–295, 2007.
[4] A. Mukherjee, M. Choudhury, A. Basu, and N. Ganguly. Redundancy
   ratio: An invariant property of the consonant inventories of the world’s
   languages. In Proceedings of ACL, 104–111, 2007.
[5] A. Mukherjee, M. Choudhury, A. Basu, and N. Ganguly. Emergence
   of community structures in vowel inventories: An analysis based on
   complex networks. In Proceedings of ACL SIGMORPHON9, 101–108,
   2007.
[6] A. Mukherjee, M. Choudhury, S. Roy Chowdhury, A. Basu, and N.
   Ganguly. Rediscovering the co-occurrence principles of the vowel
   inventories: A complex network approach. Advances in Complex
   Systems, 11(3):371–392, 2008.
      Publications from the Thesis
[8] M. Choudhury, A. Mukherjee, A. Garg, V. Jalan, A. Basu, and N.
   Ganguly. Language diversity across the consonant inventories: A
   study in the framework of complex networks. In EACL workshop on
   Cogn. Aspects of Comp. Lang. Acquisition, 51–58, 2009.
[9] A. Mukherjee, M. Choudhury, A. Basu, and N. Ganguly. Self-
   organization of sound inventories: Analysis and synthesis of the
   occurrence and co-occurrence network of consonants. Journal of
   Quantitative Linguistics, 16(2):157–184, 2009.
[10] A. Mukherjee, M. Choudhury, and N. Ganguly. Analyzing the
   degree distribution of the one-mode projection of alphabetic
   bipartite networks (α − BiNs). preprint: arXiv.org:0902.0702.

				
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