Statistics of Social and Economic Processes: Contributions from Mathematics and Physics

1
Statistics of Social and Economic Processes
Contributions from Mathematics and Physics

• Damián H. Zanette
• Alexander von Humboldt Stiftung and Fritz Haber
  Institut der Max Planck Gesellschaft, Germany
• Centro Atómico Bariloche, Argentina

2
OUR TWO TOPICS TODAY

• I.DISTRIBUTIONS WITH POWER-LAW TAILS
• II.NETWORK MODELS OF SOCIAL PROCESSES

3
Distribution of incomes

• V.Pareto, 1897
• i monthly income
• ffraction of population
• f 1/is

Paretos law
4
Distribution of city sizes

• p population
• n number of cities
• n 1/ps
• s 2

D.H. Zanette and S.C. Manrubia, Phys. Rev. Lett.
79, 523 (1997).
5
Frequency of family names

• p number of persons with a given surname
• (family size)
• n number of surnames
• n 1/ps
• s 2

D.H. Zanette and S.C. Manrubia, J. Theor. Biol.
216, 461 (2002).
6
Frequency of citations

• A. J. Lotka, 1926
• m number of citations
• n number of authors
• n 1/ms
• s 2

7
Word frequency in written texts

• r word n
• 1 THE 18234
• 2 AND 9859
• 3 OF 8116
• 4 A 7427
• 5 TO 7097
  

G. Zipf, 1936
n 1/rz z 1
Zipfs law
8
MULTIPLICATIVE STOCHASTIC PROCESSES

• X(t), dX/dt a(t)X(t)
• a(t) stochastic process
• lta(t)gt 0, lta(t)a(t)gt D
• What is the probability f(X) of each value of X?
• A log-normal distribution, f(X) 1/X

9
A modified stochastic process

• dX/dt a(t)-a0X(t)b
• What is the probability f(X) of each value of X?
• A distribution f(X) 1/X1a0/D

10
Simons model of text generation (1950s)

• One word added at each step
• Prob. q new word
• Prob. 1-q Already used word, with probability
  proportional to the number of occurrences.
• f(n) 1/n11/(1-q)

11
CONCLUSIONS (I)

• Multiplicative processes capture the feedback
  that drives many socio-economic phenomena.
• They explain power-law distributions.
• We should not discard other (more specific)
  explanations.

12
COMPLEX NETWORKS

• SCALE-FREE NETWORKS Power-law distribution
  of the number of links per site
• SMALL-WORLD NETWORKS Small-world effect
  (Milgram, 1940s)

13
SMALL-WORLD NETWORKS

• RECIPE
• Take ordered lattice
• Add pN shortcuts

• Large clustering (if A is friend of B and C, B
  and C are likely to be friends)
• Small distance (log N)
• Disorder p

14
A model of rumor propagation

• Three states
• SUSCEPTIBLE (has not heard the rumor yet)
  
• INFECTED (is willing to propagate the rumor)
  
• REFRACTORY (has lost interest)
  

15
A model of rumor propagation

• Evolution rules
• At each step and infected agent I contacts one of
  her neighbors J
• If J is susceptible, she becomes infected
  
• If J is infected or refractory, I becomes
  refractory

16
Rumor propagation

• After NR steps,the process terminates
• For plt0.19, NR has exponential distrib.
• For p0.19, NR has power-law distrib.
• For pgt0.19, NR has bimodal distrib.

17
Average duration of propagation

• Normalized average duration
• r ltNRgt/N
• r has a critical transition at p0.19

D.H. Zanette, Phys. Rev. E 65, 041908 (2002).
18
CONCLUSIONS (II)

• Structure of underlying network may play an
  essential role in social processes.
• Empirical data?