Population control policies are proposed and in some places employed as a means towards curbing population growth. This paper is concerned with a disturbing side-effect of such policies, namely, the potential risk of societal fragmentation due to changes in the distribution of family sizes. This effect is illustrated in some simple settings and demonstrated by simulation. In addition, the dependence of societal fragmentation on family size distribution is analyzed. In particular, it is shown that under the studied model, any population control policy that disallows families of 3 or more children incurs the possible risk of societal fragmentation.

1. THE EFFECT OF
POPULATION CONTROL
ON SOCIETAL
FRAGMENTATION
By David Peleg, The Weizmann Institute of Science
ISRAEL,
Zvi Lotker, Ben-Gurion University of the Negev
ISRAEL

2. What are the 10 most influential events in the 20th century?
1. World War I
2. Russian Revolution and Communism
3. The Depression
4. The rise of dictatorships
5. World War II (1939–1945)
6. The Nuclear Age
7. The Cold War (1947–1991)
8. The space race
9. The end of the Cold War
10. Information and communications technology

3. What are the 5 biggest global challenges 21st century?
 Food security
 Climate change
 Gender inequality
 Poverty
 Overpopulation

4. Single child policy has many effects
 Population planning policy of China from 1979
 Disparity in sex ratio at birth
 30 million more men than women in 2020 (wiki)
 Quality of life for women
 women have increased opportunities to receive education (wiki)
 "Four-two-one" problem

5. “Four-two-one” Problem
Chinese
New Year
Eid al-Fitr
Christmas dinner

6. One-child policy- 1979-2017
 War on poverty
 1981: 42% poverty worldwide
 2017: 11% poverty worldwide
 The single biggest reason for this delightful trend is that China. In 1981, almost unbelievably, 88% of
Chinese (and 96% of rural Chinese) seem to have lived below the poverty line. In 2013 only 2%

7. What are the 10 most influential events in the 20th century?
1. World War I
2. Russian Revolution and Communism
3. The Depression
4. The rise of dictatorships
5. World War II (1939–1945)
6. The Nuclear Age
7. The Cold War (1947–1991) and the Space Race
8. The end of the Cold War
9. China’s One-Child Policy (1978)
10. Information and communications technology

8. Population Control:
Simple Model Galton–Watson process
 A Galton–Watson process is a stochastic process {Xn} which evolves according to
the recurrence formula X0 = 1 and 𝜉𝑗
𝑛
: 𝑗, 𝑛 ∈ ℕ is a set of IID natural number-
valued random variables.
 𝝃𝒋
𝒏
is the number of descendents (children)
 Let 𝝁 =E[𝝃𝒋
𝒏
]
 𝑋 𝑛+1 = 𝑗=1
𝑋 𝑛
𝜉𝑗
𝑛
 The extinction probability is: lim
𝑛→∞
Pr 𝑋 𝑛 = 0
 Theorem
 if 𝝁 ≤ 1 then lim
𝑛→∞
Pr 𝑋 𝑛 = 0 =1
 strictly less than one if 𝝁 > 1, lim
𝑛→∞
Pr 𝑋 𝑛 = 0 ≤1

9. Theological puzzle
 According to Genesis 12:2 of the Bible.
 "I will make you into a great nation, and I will bless you; I will
make your name great, and you will be a blessing.“
 Abraham has two children: Isaac and Ishmael.
 Isaac has two sons as well: Jacob and Esau
 Jacob and Esau each have many children
 The puzzle is: why God promised many children to Abraham
and Isaac, yet gave them only two?

10. A possible answer to the Theological puzzle
 God promised to make Abraham a great nation.
 To achieve this, two conditions are needed
1. The children of Abraham need to disconnect from the old nation
2. After the first step Jacob can have a lot of children.

11. “Strong” and “Weak” Ties
 Strong ties are family ties
 Weak ties do not influence connectivity
 Only look at social networks where edges indicate siblings or marriages
 Only looked at single generations.

12. Family parameters
 Distribution of family sizes
 𝐹 = 𝐹0, 𝐹1, 𝐹2 … distribution of family sizes
 𝐹𝑖 ≥ 0, is the fraction of families with i children
𝑖 𝐹𝑖 = 1
 The marriage ratio 𝛼; sibling links are blue; marital links are red

13. Data for China and India family size
China India

14. Data for China and India number of children
China India

15. Simulation: 0/3-Children Policy, 𝛼 =0.92
China India

16. POPULATION CONTROL POLICIES
 A population control policy is defined as a real vector
 𝑃 = 𝑃0, 𝑃1, 𝑃2 …
 𝑃0: the percentage of families that can have up to 0 children according to policy P,
 𝑃1: the percentage of families that can have up to 1 children according to policy P,
 𝑃𝑖: the percentage of families that can have up to i children according to policy P,
 The distribution of family size F respects the population control P if for all 𝑘 ≥
0,
𝑖=1
𝑘
𝐹𝑖 ≥
𝑖=1
𝑘
𝑃𝑖

17. Example: Comparison of Two Basic Policies, 𝛼 = 1
 One-Child Policy (Chinese population control
policy)
 P=(0,1), F=(0,1)
 F0+F1 ≥1
 The main effect
 strongly curbs the population size
 0/2-children policy
 P=(1/2,0,1/2), F=(1/2,0,1/2)
All family
can have 1
Child of
more
All family
mast have
1
Child or
less
All
family
have 1
Child
Half of the
family can
have 1
Child of
more and
half have
zero
Half of the
family can
have 2 or
lass

18. Example 0/3-Children Policy, 𝛼 = 1
 P=(1/3,0,0,2/3), F=(1/3,0,0,2/3)
 Identical to that of the 2-Child Policy in
terms of influence on population control
According to this policy vector 1/3 of
couples are not allowed to have a child,
and 2/3 of population have 3 children. F
denotes the implementation of this policy,
and in this example indeed all population
behaves according to policy vector. The
result is a population which maintains its
size.

19. General Policy, 𝛼 < 1
 We Denote The Probability for family to have children that have married by 𝐴𝑗
 𝐴0 = 𝑘=1
∞
𝐹𝑘 1 − 𝛼 𝑘−1
 𝐴𝑗 = 𝑘=𝑗+1
∞ 𝑘 − 1
𝑗
𝐹𝑘 𝛼 𝑗 1 − 𝛼 𝑘−1−𝑗
 𝜇 = 𝑖=0
∞
𝑖 𝐴𝑖
 Theorem
 if 𝝁 ≤ 1 then lim
𝑛→∞
Pr 𝑋 𝑛 = 0 =1
 strictly less than one if 𝝁 > 1, lim
𝑛→∞
Pr 𝑋 𝑛 = 0 ≤1

20. Example 0/3-Children Policy, 𝛼 < 1
 We Denote The Probability for family to have children that have married by 𝐴𝑗
 𝐴0 = 𝑘=1
∞
𝐹𝑘 1 − 𝛼 𝑘−1
 𝐴𝑗 = 𝑘=𝑗+1
∞ 𝑘 − 1
𝑗
𝐹𝑘 𝛼 𝑗 1 − 𝛼 𝑘−1−𝑗
 𝜇 = 𝑖=0
∞
𝑖 𝐴𝑖
 Assume 0/3 0/3-Children Policy, it follows that
𝐴0 = 1 − 𝛼 2
𝐴1 = 2 1 − 𝛼 𝛼
𝐴2 = 𝛼2
𝐴𝑗 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ≥ 3
 Therefore
𝜇 = 0 ∗ 1 − 𝛼 2 + 1 ∗ 2 1 − 𝛼 𝛼 + 2 ∗ 𝛼2 = 1
Is the Phase transition at 𝛼∗
= 1/2

21. Simulation: 0/3-Children Policy, 𝛼 < 1
𝛼=0.4 𝛼=0.5

22. Simulation: 0/3-Children Policy, 𝛼 < 1
𝛼=0.2 𝛼=0.3

23. Simulation: 0/3-Children Policy, 𝛼 < 1
𝛼=0.6 𝛼=0.7

24. General Policy, 𝛼 < 1
 Our result
 Theorem
Having a nonzero fraction of families with 3 or more children
(namely, Pj > 0 for some 𝑗 ≥ 3) is a necessary condition for the
emergence of an infinite size component in the strong-ties branching proces

25. Is China shattering like glass?