# The Role of Child Gender in the Formation of Parents’ Social
Networks††thanks: We thank Vincent Boucher for valuable comments. Asad Islam
would like to acknowledge the funding support from FCDO/ESRC grant (grant
number: ES/N010221/1: Investing in our Future: The Early Childhood
Intervention and parental involvement in Bangladesh).

Aristide Houndetoungan Asad Islam Michael Vlassopoulos Yves Zenou Department
of Economics, Thema, Cy Cergy Paris Université. Email:
<EMAIL_ADDRESS>for Development Economics and
Sustainability (CDES) and Department of Economics, Monash University, and
J-PAL. Email<EMAIL_ADDRESS>Department, Social Sciences,
University of Southampton, and IZA. Email:
<EMAIL_ADDRESS>of Economics, Monash University, CEPR,
and IZA. Email<EMAIL_ADDRESS>
###### Abstract

Social networks play an important role in various aspects of life. While
extensive research has explored factors such as gender, race, and education in
network formation, one dimension that has received less attention is the
gender of one’s child. Children tend to form friendships with same-gender
peers, potentially leading their parents to interact based on their child’s
gender. Focusing on households with children aged 3-5, we leverage a rich
dataset from rural Bangladesh to investigate the role of children’s gender in
parental network formation. We estimate an equilibrium model of network
formation that considers a child’s gender alongside other socioeconomic
factors. Counterfactual analyses reveal that children’s gender significantly
shapes parents’ network structure. Specifically, if all children share the
same gender, households would have approximately 15% more links, with a
stronger effect for families having girls. Importantly, the impact of
children’s gender on network structure is on par with or even surpasses that
of factors such as income distribution, parental occupation, education, and
age. These findings carry implications for debates surrounding coed versus
single-sex schools, as well as policies that foster inter-gender social
interactions among children.

Keywords: Social networks, early childhood, network formation, gender.

JEL classification: C57, D85, J16, O12.

##  1 Introduction

Social networks play a fundamental role in shaping various behaviors and
influencing multiple aspects of our lives, from educational choices, to job
opportunities, and access to information (Jackson, 2008; Jackson et al., 2017;
Jackson, 2019). Previous research has extensively explored the formation of
social networks, emphasizing factors such as gender, race, ethnicity, and
social and geographic proximity (homophily) as key determinants of network
connections (McPherson et al., 2001; Currarini et al., 2009; Boucher, 2015;
Fafchamps and Gubert, 2007). However, one dimension that has received
relatively less attention, despite its potential significance, is the gender
of children within households. Indeed, during early childhood, girls and boys
spend much of their time at home with their families and seek guidance from
parents. In this paper, we delve deeper into this relatively unexplored
dimension of parental networks.

Why might the gender of a child be an important determinant of parents’ social
links? Several underlying mechanisms are plausible. Firstly, parents may
actively seek opportunities to connect with other parents who have children of
the same gender. Secondly, children of the same gender tend to engage in more
frequent social interactions (Maccoby and Jacklin, 1987), which could
naturally lead to even more interactions among their parents, as they come
together for playdates, school and social events, and various extracurricular
activities.111In the psychology literature, when researchers observed
correlations between parenting practices and children’s behavior,
traditionally they inferred that parents were influencing their children;
however, developmental psychologists now recognize that children also
influence their parents’ behavior, particularly in the context of parental
socialization (Leaper, 2014). These connections can serve as valuable sources
of information and support, particularly in matters related to child rearing.
For instance, parents may exchange insights on gender-specific upbringing
challenges and share strategies for addressing them. This gender-specific
information exchange can further strengthen social ties among parents. A
better understanding of the role of children’s gender in parental network
formation is important as these parental connections have the potential to
impact children’s access to resources, information, social networks, and
opportunities, making it a critical component of children’s human capital
development (Guo et al., 2023).

We leverage a rich dataset of household networks across 222 villages in rural
Bangladesh, encompassing about 4,500 households with children aged 3-5. The
network data are based on a detailed pairwise network census of participating
households, capturing various dimensions of their interactions. This dataset
provides a unique opportunity to investigate the role of children’s gender in
parental social network formation in a context where social networks play an
important role and can act as substitutes for formal institutions, for
example, in the provision of insurance against income or health shocks
(Fafchamps and Lund, 2003; De Weerdt and Dercon, 2006; Breza, 2016; Breza et
al., 2019). We estimate a version of the equilibrium model of network
formation developed by Mele (2017, 2020, 2022) in which individuals may
receive distinct utility benefits from forming direct, mutual, and indirect
links. In estimating the model, we allow the influence of a child’s gender on
the likelihood of parents forming connections, alongside other socioeconomic
factors known to influence the formation of network links, such as occupation
and education.

Our structural estimates reveal that children’s gender is an important
determinant of parents’ networks. Moreover, we find evidence of indirect
homophily by children’s gender, that is, the influence of children’s gender
extends beyond direct links: families prefer to form links with other families
that are in turn connected to families with same-gender children. We then use
the model to perform counterfactual simulations to gain insight on how
changing the gender composition of children affects the structure of the
network. This analysis reveals that children’s gender is an important factor
for the formation of network links. Our simulations show that if all children
have the same gender, then households would have about 15% more links.
Interestingly, the effect of gender on link formation of families is stronger
for pairs of families that have girls rather than boys. We also examine,
through policy experiments, the importance of other factors for parents’
network structure, such as income inequality, parents’ occupation, education,
and age. Remarkably, we find that, in comparison to children’s gender, most of
these other key factors carry similar or even less quantitative importance in
shaping parents’ networks.

This paper contributes to the empirical literature on network formation (for
overviews, see Graham (2015), Chandrasekhar (2016), and De Paula (2017, 2020))
by providing novel insights into the so far overlooked role of children’s
gender within parental networks. Methodologically, the paper most closely
related to ours is Mele (2020), which estimates the network formation
framework developed in Mele (2017) to study racial segregation in schools and
assess how student reallocation policies impact segregation through
simulations. We take advantage of our rich dataset to address the non-
identification issue in exponential random graphs pointed out by Mele (2017).
As we observe networks across a large number of villages, we can leverage
variation in network features across these villages to identify the model
parameters. Other notable contributions in this literature include works by
Currarini et al. (2009), Leung (2015), Boucher (2015), Boucher and Mourifié
(2017), Graham (2017), Christakis et al. (2020), Boucher et al. (2021), Mele
(2022), Hsieh et al. (2022), and Boucher et al. (2023). The majority of these
studies emphasize factors such as gender, race, ethnicity, and social and
geographic proximity as key determinants of network connections. To the best
of our knowledge, our paper is the first to explicitly consider the role of
children’s gender within parental networks.222Heizler and Kimhi (2017) study
the effect of family composition (e.g., number of children and age gap between
the oldest and youngest child) on parents’ social interactions. The analysis
is very different since they do not have network data and thus, do not model
the network formation of parents and the role of children’s gender on their
parents’ networks. There is also a literature on the interaction between
parents and children in terms of education (for an overview, see Doepke et al.
(2019)) but the focus is very different since they look at how various
parenting styles affect children’s education outcomes.

Our results speak to the ongoing debate around coed (coeducational) schools
versus single-sex (same-sex) schools. A large literature has documented that a
single-sex environment might facilitate more effective learning for females;
this is particularly true at a younger age (Sullivan et al., 2010).
Specifically, reduced gender stereotype threat might lead to increased self-
confidence (Spencer et al., 1999; Booth et al., 2014; Eisenkopf et al., 2015)
and, in the absence of male peers, females might also feel less anxiety
pursuing non-stereotypical courses, such as math (Mael et al., 2004; Hahn et
al., 2020). Our counterfactual exercises suggest that single-sex schools could
lead to parental networks that are more dense, especially in the case of
girls, potentially influencing parents’ decisions regarding their children’s
education. Beyond schools, opportunities for social interactions among
children arise in participation to youth clubs and organizations, and various
religious and community events. Our findings indicate that promoting social
interactions among children, irrespective of their gender, can have a
meaningful impact on parental networks, potentially influencing the flows of
information, social support, and cohesion in these communities, which might
even feedback into the upbringing of the children.

The rest of the paper is organized as follows. In Section 2, we introduce the
context and present the dataset, offering descriptive evidence that highlights
the influence of children’s gender on parental networks. Section 3 presents a
dynamic network model in which each parent’s network decisions are influenced
by observable characteristics and the networks of other parents. The
estimation results are presented in Section 4. Additionally, we present
counterfactual analyses in this section to examine how specific
characteristics can impact the network. Finally, Section 5 offers some
concluding remarks.

##  2 Context, Data, and Descriptive Evidence

Our data collection was conducted in 222 villages located in two rural
districts of Bangladesh, Khulna and Shatkhira. These villages had been
selected to be part of an early childhood intervention targeting 3-5 year old
children over a 2-year period (see Guo et al. (2023) for details). In each of
these participating villages, we recruited all eligible families, defined as
those having at least one child in the 3-5 year old age group, residing in the
vicinity where an early childhood center was planned to be established for the
intervention. Consequently, we recruited an average of 20 households in each
village, with numbers ranging from 7 to 37, resulting in a total sample of
4,473 families. The parents’ social network information employed in the
current study was collected at the baseline from April to May 2017, preceding
the implementation of the intervention. During this data collection, we
gathered a comprehensive array of information from surveyed households
concerning their education, occupation, land ownership, income, expenditures
and savings, and child-rearing practices.

Furthermore, we asked several questions that allow us to discern the social
ties among all surveyed households within each group of eligible households.
Specifically, the survey included questions on the extent to which each pair
of families interact with each other. This included whether each household
borrows from each of the other households in the same group and whether they
seek help from each other when a family member is sick. We make use of these
questions to define network links among families, following a similar approach
as in Banerjee et al. (2013, 2023). Given the rural context of these
communities, it is reasonable to expect that families relying on each other
for risk-sharing and support would have close-knit social bonds. Therefore,
our definition of a link is designed to capture these meaningful and important
relationships within the community.

#### Defining links and networks

The focus of our study is the formation of social networks among parents
within these villages. To be more precise, the network is defined among the
group of eligible households that were surveyed within each village.
Hereafter, we use the term “village networks” to refer to the networks
established within this group of households in each village. Formally,
consider any pair of surveyed parents, denoted by $i$ and $j$, belonging to
the set of eligible parents $\mathcal{V}_{r}$ in village $r$. We define a
binary variable, $a_{r,ij}$, such that $a_{r,ij}=1$ if parent $i$ lists parent
$j$ as a friend and $a_{r,ij}=0$ otherwise. Family $j$ is connected to family
$i$ if at least one of the following conditions holds: $i$ can ask for help
from $j$ when $i$ needs to borrow 100 Taka; $i$ can ask for help from $j$ when
someone in family $i$ falls ill.

It is important to note that we observe the entire network among households
within each group $\mathcal{V}_{r}$. Furthermore, households are not
considered to be connected to themselves, that is, $a_{r,ii}=0$ for any $r$
and $i$. The network is directed, meaning that if $j$ is listed as a
connection by $i$, it does not imply that $i$ is a connection to $j$. This
configuration matches our dataset and is consistent with existing literature
(see Graham and De Paula, 2020). We also restrict interactions to the same
village: household $i$ can only seek help in their own village and not in
neighboring ones. Finally, the social network is independent across villages.
Thus, the network in the village $r$ can be represented by an adjacency matrix
$A_{r}=[a_{r,ij}]_{\begin{subarray}{c}1\leq i\leq n_{r}\\\ 1\leq j\leq
n_{r}\end{subarray}}\in\mathcal{A}_{r}$, where $\mathcal{A}_{r}$ is the set of
possible directed adjacency matrices in village $r$.

#### Descriptives of networks and households

Making use of these data, we construct individual networks for each village at
the household level.333See Banerjee et al. (2013) and Breza and Chandrasekhar
(2019) for a related approach to defining a network among village households.
Summary statistics for both the networks and the households are presented in
Table 1. The average network size (number of network members) is approximately
20.15 households, with a range from 7 to 37 households. The average degree
(number of links) is 6.56, with the number ranging from 0 to 28 for some
households. The high clustering coefficient of 0.72 indicates a propensity for
triads to form clusters within these networks.

Figure 1 illustrates the networks within a selected 10% subset of villages
from our sample. Each node in this graph corresponds to a family, with
families having girls represented by red nodes, those having boys represented
by blue nodes, and those with both girls and boys depicted as green nodes. It
is noteworthy that nodes of the same color tend to cluster together in the
graph, indicating that families with children of the same gender tend to form
connections within the network. Notably, the formation of these clusters
appears to be more prominent among families with girls.

Turning to household characteristics, children are balanced in terms of
gender. Given the presence of young children in these households, fathers have
an average age of about 34 years, while mothers average around 27 years. In
terms of education, a significant portion of fathers, roughly 73%, have
completed primary or secondary education, while 85% of mothers have attained
similar levels of education. In terms of occupation, fathers in the sample are
mostly daily laborers, self-employed, or farmers. In contrast, the majority of
mothers in this sample do not work outside of the household.

Table 1: Data summary

| Mean | SD | Min | Max
---|---|---|---|---
Network |  |  |  |
Degree | 6.56 | 4.41 | 0 | 28
Density | 0.31 |  |  |
Clustering | 0.72 |  |  |
Asymmetry | 0.22 |  |  |
Group’s Size | 20.15 | 5.42 | 7 | 37
Household |  |  |  |
Male children | 0.50 | 0.50 | 0 | 1
Female children | 0.50 | 0.50 | 0 | 1
Father’s age | 34.19 | 6.12 | 20 | 82
Mother’s age | 27.18 | 5.00 | 20 | 55
Income ($10^{3}$ Taka) | 11.20 | 5.06 | 0 | 66
Homestead land ($10^{3}$ Ha) | 0.084 | 0.114 | 0 | 0.230
Farming land ($10^{3}$ Ha) | 0.317 | 1.700 | 0 | 100
Father’s education: none | 0.16 | 0.37 | 0 | 1
Father’s education: primary | 0.32 | 0.47 | 0 | 1
Father’s education: secondary | 0.41 | 0.49 | 0 | 1
Father’s education: university | 0.11 | 0.31 | 0 | 1
Mother’s education: none | 0.08 | 0.27 | 0 | 1
Mother’s education: primary | 0.23 | 0.42 | 0 | 1
Mother’s education: secondary | 0.62 | 0.49 | 0 | 1
Mother’s education: university | 0.08 | 0.27 | 0 | 1
Daily laborer fathers | 0.35 | 0.48 | 0 | 1
Farmer fathers | 0.21 | 0.41 | 0 | 1
Self-employed fathers | 0.31 | 0.46 | 0 | 1
Mother without a job | 0.93 | 0.26 | 0 | 1

Figure 1: Networks of Selected Villages

#### Descriptives of link formation

To gain an initial, descriptive insight into the factors influencing the
formation of links between households, Table 2 presents results of logit
regressions. These regressions estimate the probability of a link between
household $i$ and $j$ based on various household characteristics, including
whether the children in the two households share the same gender and various
other socio-economics characteristics of the households. Specifically, we
estimate two models defined as follows:

$\frac{\mathbb{P}(a_{r,ij}=1)}{1-\mathbb{P}(a_{r,ij}=1)}=\begin{cases}\exp(\theta_{1}Gender_{r,ij}+\boldsymbol{x}_{r,ij}^{\prime}\boldsymbol{\theta}_{2}),\\\
\exp(\theta_{1}^{m}Male_{r,ij}+\theta_{1}^{f}Female_{r,ij}+\boldsymbol{x}_{r,ij}^{\prime}\boldsymbol{\theta}_{2}),\\\
\end{cases}$ (1)

where $Gender_{r,ij}$, $Male_{r,ij}$, and $Female_{r,ij}$ are dummy variables,
such that $Gender_{r,ij}=1$ if $i$ and $j$ have children of same gender,
$Male_{r,ij}=1$ if they both have boys and $Female_{r,ij}=1$ if they have
girls. The second equation in (1) is a more flexible version of the first one.
The first specification in (1) imposes that
$\theta_{1}^{m}=\theta_{1}^{f}=\theta_{1}$. The variable
$\boldsymbol{x}_{r,ij}$ is a vector of control factors susceptible to
influence the network, and $\theta_{1}$, $\theta_{1}^{m}$, $\theta_{1}^{f}$,
and $\boldsymbol{\theta}_{2}$ are unknown parameters to be estimated.

In Column (1), we estimate a single parameter for the effect of same gender
corresponding to the first specification in Equation (1), whereas in Column
(2) (second specification in Equation (1)) we allow the gender of children to
have different effects. In Column (1), we find that having same gender
children has a positive effect on the probability of a link, but the effect is
not statistically significant. In Column (2), we observe gender differences,
with having boys in both households having a positive and statistically
significant effect on link formation, while having girls has a positive but
non-significant effect. Beyond the gender of the children, these regressions
highlight that households with similar characteristics, such as age,
education, occupation, income, and land ownership of parents, are more likely
to be linked. These results provide a preliminary understanding of the factors
influencing social network formation among parents in the studied villages.

Table 2: Standard logit model for network formation

| (1) | (2)
---|---|---
Intercept | $-0.998^{***}$ | (0.069) | $-1.004^{***}$ | (0.070)
$\text{Male}_{ij}$ |  |  | $0.043^{***}$ | (0.017)
$\text{Female}_{ij}$ |  |  | $0.012$ | (0.017)
$\text{Gender}_{ij}$ | $0.021$ | (0.014) |  |
Adiff(Father’s Age) | $0.002$ | (0.002) | $0.002$ | (0.002)
$\text{Age Father}_{i}$ | $-0.002$ | (0.002) | $-0.002$ | (0.002)
$\text{Age Father}_{j}$ | $-0.005^{**}$ | (0.002) | $-0.005^{**}$ | (0.002)
Adiff(Mother’s Age) | $-0.005^{**}$ | (0.002) | $-0.005^{**}$ | (0.002)
$\text{Age Mother}_{i}$ | $0.002$ | (0.002) | $0.002$ | (0.002)
$\text{Age Mother}_{j}$ | $0.006^{**}$ | (0.002) | $0.006^{**}$ | (0.002)
Adiff(Income) | $-0.016^{***}$ | (0.002) | $-0.016^{***}$ | (0.002)
$\text{income}_{i}$ | $0.012^{***}$ | (0.002) | $0.012^{***}$ | (0.002)
$\text{Income}_{j}$ | $0.013^{***}$ | (0.002) | $0.013^{***}$ | (0.002)
Adiff(Homestead land) | $-0.411^{***}$ | (0.119) | $-0.412^{***}$ | (0.119)
$\text{Homestead land}_{i}$ | $0.300^{***}$ | (0.112) | $0.300^{***}$ | (0.112)
$\text{Homestead land}_{j}$ | $0.191^{*}$ | (0.112) | $0.192^{*}$ | (0.112)
Adiff(Farming land) | $-0.185^{***}$ | (0.017) | $-0.185^{***}$ | (0.017)
$\text{Farming land}_{i}$ | $0.179^{***}$ | (0.017) | $0.179^{***}$ | (0.017)
$\text{Farming land}_{j}$ | $0.189^{***}$ | (0.017) | $0.189^{***}$ | (0.017)
$\text{Father\textquoteright s education}_{ij}$ | $0.055^{***}$ | (0.015) | $0.055^{***}$ | (0.015)
$\text{Mother\textquoteright s education}_{ij}$ | $0.031^{**}$ | (0.015) | $0.031^{**}$ | (0.015)
$\text{Father is daily laborer}_{ij}$ | $0.268^{***}$ | (0.021) | $0.268^{***}$ | (0.021)
$\text{Father is farmer}_{ij}$ | $0.151^{***}$ | (0.029) | $0.152^{***}$ | (0.029)
$\text{Father is self-employed}_{ij}$ | $-0.098^{***}$ | (0.023) | $-0.098^{***}$ | (0.023)
$\text{Mother stays home}_{ij}$ | $-0.073^{***}$ | (0.021) | $-0.072^{***}$ | (0.021)

  * Note: ***: pvalue $<1\%$, **: pvalue $<5\%$, *: pvalue $<10\%$
This table presents estimation results on link formation using a logit model.
The first column for each model indicates the coefficient estimate followed by
their standard error in brackets. For any variable denoted "var", "varij"
takes one if var${}_{i}=$ varj and zero otherwise; “Adiff(var)" if the
absolute difference between vari and varj. The network is directed and the
relationship we model is the link from $j$ to $i$. There are 4,473 families,
and 92,152 potential links in the sample.

The aforementioned results impose that $(a_{r,ij},\leavevmode\nobreak\
Gender_{r,ij},\leavevmode\nobreak\ Male_{r,ij},\leavevmode\nobreak\
Female_{r,ij},\leavevmode\nobreak\ \boldsymbol{x}_{r,ij})$ are independent
across $r$, $i$, and $j$. This assumption implies that family $i$ does not
consider the potential connections between $j$ and other families when
deciding to form a link with $j$. Given the definition of the network used in
this paper, this assumption may be overly restrictive. For instance, family
$i$ is more likely to interact with family $j$ if the latter has access to
resources or has friends with such access. Therefore, we need to consider a
more flexible model to account for these complexities.

##  3 A Model of Parent Network Formation

This section presents the model and the estimation strategy. We adapt the
framework of Mele (2017) to our specific context in which we have many
villages (groups) ($\bar{r}\in\mathbb{N}^{\ast}$). From a theoretical point of
view, the model remains the same because we assume that the network is
independent across villages (see Mele, 2022). Let
$\mathcal{V}_{r}=\\{1,\leavevmode\nobreak\ \dots,n_{r}\\}$ denote a set of
$n_{r}$ parents (families) within village $r$, for $r=1,\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \bar{r}$. Each family $i\in\mathcal{V}_{r}$ is
characterized by a vector of observed characteristics assumed to be exogenous
and denoted by $\boldsymbol{x}_{r,i}$. This vector $\boldsymbol{x}_{r,i}$
encompasses various attributes such as the presence of a male child, parents’
ages, father’s occupation, family income, etc.

###  3.1 Microeconomic foundations

#### Family preferences

The model is dynamic and time is discrete. The realization of the network at
time $t$ is denoted by $A_{r}^{t}=[a_{r,ij}^{t}]_{\begin{subarray}{c}1\leq
i\leq n_{r}\\\ 1\leq j\leq n_{r}\end{subarray}}$, where $a_{r,ij}^{t}=1$ if
parent $j$ is connected with parent $i$ at time $t$ and $a_{r,ij}^{t}=0$
otherwise. In each village $r$, at the beginning of each period, a family $i$
is selected at random to meet family $j$ according to a meeting probability
conditional on the set of family characteristics
$\mathcal{X}_{r}:=\\{\boldsymbol{x}_{r,i},\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \boldsymbol{x}_{r,n_{r}}\\}$ and the realization of
the network at time $t-1$. For instance, family $i$ is likely to meet family
$j$ if they have common friends at time $t-1$ or if they share similar
characteristics such as being both farmers or having attended the same school.

It is important to note that a meeting does not necessarily result in the
formation of a link. Family $i$ can meet family $j$, but if the interaction
doesn’t result in a strong connection, $i$ may not consider $j$ as someone
they would ask for help. Upon meeting, family $i$ decides whether to update
their link $a_{r,ij}^{t}$. In this model, information is complete, in the
sense that families observe the entire network (in its dynamic form) and the
set of family characteristics $\mathcal{X}_{r}$ before making their choice
regarding forming a link. When family $i$ selects their friends, they derive
utility given by

$U_{r,i}(A_{r},\mathcal{X}_{r})=\underbrace{\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n_{r}}a_{r,ij}u_{r,ij}}_{\text{direct
friends}}+\underbrace{\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n_{r}}a_{r,ij}a_{r,ji}m_{r,ij}}_{\text{mutual
friends}}+\underbrace{\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n_{r}}\sum_{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}^{n_{r}}a_{r,ij}a_{r,jk}v_{r,ik}}_{\text{friends of
friends}}+\underbrace{\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n_{r}}\sum_{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}^{n_{r}}a_{r,ij}a_{r,ki}w_{r,kj}}_{\text{popularity}},$ (2)

where $u_{r,ij}\equiv u(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})$,
$m_{r,ij}\equiv m(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})$, $v_{r,ik}\equiv
v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,k})$, and $w_{r,kj}\equiv
w(\boldsymbol{x}_{r,k},\boldsymbol{x}_{r,j})$ are bounded real-valued
functions of the observed family characteristics $\boldsymbol{x}_{r,i}$,
$\boldsymbol{x}_{r,j}$, and $\boldsymbol{x}_{r,k}$.444We omit the superscript
$t$ to ease the notational burden.

Utility function (2) is quite general and accounts for multiple factors that
can influence the network formation process. The first term captures the
utility derived from direct friendships. Family $i$ receives a utility of
$u_{r,ij}$ if they form a link with family $j$. Given that $u_{r,ij}$ depends
on both $\boldsymbol{x}_{r,i}$ and $\boldsymbol{x}_{r,j}$, then $u_{r,ij}$ can
be defined as a distance between $\boldsymbol{x}_{r,i}$ and
$\boldsymbol{x}_{r,j}$; that is, a measure of social distance between families
$i$ and $j$. Consequently, (2) serves as a generalized framework that can
encompass specific cases where network formation is driven by homophily
(McPherson et al., 2001).

The second term in (2) represents the additional utility ($m_{r,ij}$) derived
from friends who reciprocate the connection by also considering family $i$ as
a friend (mutual relationships). This term holds particular significance
within our framework for several reasons. Firstly, it addresses the
reciprocity of assistance. When family $i$ seeks help, their friend $j$ may be
hesitant to provide aid. However, if $j$ is aware that they can also turn to
$i$ for assistance in the future, they are more likely to agree to provide
assistance. On the other hand, because of the assumption of perfect
information, family $i$ will not hesitate to request help if they are aware
that their friend $j$ can also seek help from them. Therefore, mutual
relationships offer family $i$ a distinct benefit compared to direct
friendships without such mutual connections.

The third term in (2) represents the additional utility derived from indirect
relationships (friends of friends). In this term, family $i$ accrues a utility
benefit $v_{r,ik}$ for every pair of families $j$ and $k$ where $j$ is a
direct friend of $i$ and $k$ is a friend of $j$, irrespective of whether $k$
is a direct friend of $i$ or not. For simplicity, we assume, as in Mele
(2017), that $v_{r,ik}$ depends only on the observable characteristics of
family $i$ and family $k$, but not those of intermediary family $j$. In our
context, these indirect relationships are relevant because they facilitate the
potential for family $i$ to receive assistance from family $k$ through the
intermediary connection of family $j$, even if family $k$ is not a directly
connected to family $i$.

Finally, the last term in (2) accounts for popularity within the network. In
this term, family $i$ receives a utility benefit $w_{r,kj}$ for every pair of
families $j$ and $k$ where $j$ is a direct friend of $i$ and $i$ is a direct
friend of $k$. Being popular can have both beneficial and costly implications
for $i$. The utility term $w_{r,kj}$ serves to capture the combined impact of
receiving assistance from family $j$ and the potential cost associated with
providing support to family $k$. It reflects the intricate dynamics of network
formation, where the benefits of popularity are counterbalanced by the
obligations that come with maintaining connections.

#### A potential game

The terms accounting for benefits arising from indirect relationships and
popularity introduce a significant complication: it makes family $i$’s linking
decision dependent on decisions made by other families in the village.
Although this feature of the model is relevant for our setting, it also poses
significant challenges when attempting to estimate the econometric model
without imposing additional constraints on preferences. The observational unit
in the econometric model is the entire network at the village level
(consisting of the group of surveyed households) rather than the entry
$a_{r,ij}$, as is the case in classical dyadic models. This perspective
exponentially expands the number of potential network structures as the number
of possible network structures in village $r$ is $2^{(n_{r}-1)n_{r}}$. For
instance, in a village with just $n_{r}=10$ families, the number of possible
network configurations reaches a staggering $1.24\times 10^{27}$, thereby
rendering the estimation of the model computationally infeasible, even using
supercomputers. To address this computational challenge, it is common practice
to introduce restrictions for the game to have a potential function (see
Graham and De Paula, 2020).

######  Assumption 3.1 (Preferences).

For any $i$, $j$, $k$, the preferences satisfy the restrictions
$m(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})=m(\boldsymbol{x}_{r,j},\boldsymbol{x}_{r,i})$,
and
$v(\boldsymbol{x}_{r,k},\boldsymbol{x}_{r,j})=w(\boldsymbol{x}_{r,k},\boldsymbol{x}_{r,j})$.

The first restriction is necessary for the identification of the model. If two
families $i$ and $j$ are mutually linked, they should receive the same
additional payoff in the second term of (2). The second restriction requires
that family $i$ values their popularity as much as family $k$ values the
indirect relationship. This restriction ensures the model’s coherency in the
sense of Tamer (2003). Under Assumption 3.1, Mele (2017) shows that the
network formation process in each village is a potential game (Monderer and
Shapley, 1996), where a potential function in village $r$ is given by

$Q_{r}(A_{r},\mathcal{X}_{r})=\sum_{i=1}^{n_{r}}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{n_{r}}a_{r,ij}u_{r,ij}+\sum_{i=1}^{n_{r}}\sum_{\begin{subarray}{c}j>i\end{subarray}}^{n_{r}}a_{r,ij}a_{r,ji}m_{r,ij}+\sum_{i=1}^{n_{r}}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n_{r}}\sum_{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}^{n_{r}}a_{r,ij}a_{r,jk}v_{r,ik}.$ (3)

The existence of a potential function greatly simplifies the characterization
of families’ incentives to form a link within the network. Consider two
networks $A_{r,ij}$ and $B_{r,ij}$, which are identical in all respects except
for a single entry $(i,\leavevmode\nobreak\ j)$. The difference in utility for
family $i$ between these two networks,
$U_{r,i}(A_{r,ij},\mathcal{X}_{r})-U_{r,i}(B_{r,ij},\mathcal{X}_{r})$, is
equivalent to the difference in the potential function,
$Q_{r}(A_{r,ij},\mathcal{X}_{r})-Q_{r}(B_{r,ij},\mathcal{X}_{r})$. This
equivalence implies a straightforward criterion for family $i$: it prefers
$A_{r,ij}$ over $B_{r,ij}$ if and only if
$Q_{r}(A_{r,ij},\mathcal{X}_{r})>Q_{r}(B_{r,ij},\mathcal{X}_{r})$.
Importantly, the function $Q_{r}$ is an aggregated measure of the entire
network and is not specific to any particular family, as is the case with the
utility function $U_{r,i}$. This fact has practical implications for empirical
strategies. It allows us to sidestep the daunting computational challenge of
analyzing $2^{(n_{r}-1)n_{r}}$ distinct networks for each village.

#### Network distribution

In addition to utility $U_{r,ij}$, family $i$ receives an idiosyncratic shock
$\varepsilon_{r,ij}(a_{r,ij})$ when they meet another family $j$. This shock
is specific to the particular link configuration $a_{r,ij}$ between families
$i$ and $j$ in village $r$. Based on their preferences at time $t$, family $i$
decides to establish a link with family $j$ if and only if it results in a
higher utility, taking into account the impact of the shock. The idiosyncratic
shock $\varepsilon_{r,ij}(a_{r,ij})$ is known to all families due to the
assumption of complete information. However, the econometrician does not
observe this shock, which makes the network at time $t$ a random variable due
to the inherent uncertainty associated with these idiosyncratic shocks..

Although the network is dynamic, we only observe a single snapshot at a
specific point in time. To estimate preferences using this limited observation
of the network, Mele (2017) demonstrated that the dynamic network generated by
the model followed a Markov Chain of networks that converges to a unique
stationary distribution $\pi$ given by

$\pi(A_{r}|\mathcal{X}_{r})=\dfrac{\exp(Q_{r}(A_{r},\mathcal{X}_{r}))}{\sum_{W\in\mathcal{A}_{r}}\exp(Q_{r}(W,\mathcal{X}_{r}))}.$
(4)

The denominator $\sum_{W\in\mathcal{A}_{r}}\exp(Q_{r}(W,\mathcal{X}_{r}))$
ensures that $\pi(.|\mathcal{X}_{r})$ is a proper probability—that is, it
satisfies $\sum_{W\in\mathcal{A}_{r}}\pi(W|\mathcal{X}_{r})=1$ for any $r$.
The notion of convergence to a stationary distribution implies that, for
practical estimation purposes, the focus can be shifted towards estimating
this stationary distribution $\pi$. This approach is underpinned by the
assumption that the single network observation available to us originates from
this distribution $\pi$. In essence, this perspective simplifies the
estimation task, as it allows us to concentrate on understanding the network’s
long-term equilibrium behavior encapsulated by $\pi$.

###  3.2 Econometric model and estimation strategy

In this section, we present the econometric specification of the model and
discuss its identification. We also outline a Bayesian approach for estimating
the model parameters.

#### Econometric specification

Our objective is to estimate $u(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})$,
$m(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})$, and
$v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,k})$. This estimation would be
sufficient to approximate the stationary distribution $\pi$ in Equation (4)
and quantify the impact of changes in $\boldsymbol{x}_{r,i}$ (e.g., change in
the proportion of families with a boy) on the distribution $\pi$. We assume
that

$\displaystyle\begin{split}u(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})&=\beta^{u}_{1}Gender_{r,ij}+\boldsymbol{x}^{u\prime}_{ij}\boldsymbol{\beta}^{u}_{2},\\\
m(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})&=\beta^{m}_{1}Gender_{r,ij}+\boldsymbol{x}^{m\prime}_{ij}\boldsymbol{\beta}^{m}_{2},\\\
v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,k})&=\beta^{v}_{1}Gender_{r,ik}+\boldsymbol{x}^{v\prime}_{ik}\boldsymbol{\beta}^{v}_{2},\end{split}$
(5)

where $Gender_{r,ij}$ is equal to 1 if families $i$ and $j$ have children of
the same gender, $\boldsymbol{x}^{u}_{ij}$, $\boldsymbol{x}^{m}_{ij}$, and
$\boldsymbol{x}^{v}_{ik}$ are vectors of control variables derived from family
observable characteristics, while $\beta^{s}_{1}$ and
$\boldsymbol{\beta}^{s}_{2}$, for $s\in\\{u,\leavevmode\nobreak\
m,\leavevmode\nobreak\ v\\}$, are the unknown parameters to be estimated. We
control for a wide range of explanatory variables that may influence the
network, such as parents’ age, income, occupation, education, etc. (see
household characteristics in Table 1). As $m_{r,ij}$ and $v_{r,ik}$ are
symmetric (Assumption 3.1), we have
$\boldsymbol{x}^{m}_{ij}=\boldsymbol{x}^{m}_{ji}$ and
$\boldsymbol{x}^{v}_{ik}=\boldsymbol{x}^{v}_{ki}$. However,
$\boldsymbol{x}^{u}_{ij}$ may not be symmetric. In other words,
$\boldsymbol{x}^{u}_{ij}$ can encompass variables that drive homophily in
linking decisions, as well as personal characteristics of families $i$ and
$j$. In contrast, personal characteristics cannot be included in
$\boldsymbol{x}^{m}_{ij}$ and $\boldsymbol{x}^{v}_{ik}$. For example,
$\boldsymbol{x}^{u}_{ij}$ may include family $i$’s income, family $j$’s
income, and the absolute difference in income, whereas
$\boldsymbol{x}^{m}_{ij}$ and $\boldsymbol{x}^{v}_{ik}$ can only include the
absolute difference in income. Additionally, as in Equation (1), we can also
decompose the gender effect in (5) into effects specific to boys and girls.

The influence of same-gender children is captured by the parameters
$\beta_{1}^{u}$, $\beta_{1}^{m}$, and $\beta_{1}^{v}$. Figure 2 presents
potential situations that may arise in the network based on the signs of these
parameters. In this visual representation, blue nodes represent families with
boys, while red nodes represent families with girls. We are likely to observe
structure (a) if $\beta_{1}^{u}>0$, meaning that the network would contain
more links (not necessarily reciprocal) between families with same-gender
children. Conversely, if $\beta_{1}^{m}>0$, structure (b) becomes more
prevalent. In this scenario, families with children of the same gender are
more inclined to form mutual connections. Finally, if $\beta_{1}^{v}>0$,
structures (c)–(f) are more probable. This suggests that families $i$ and $k$
are indirectly linked because they have boys, with this connection passing
through family $j$, irrespective of the gender of family $j$ children, because
$v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,k})$ is independent of
$\boldsymbol{x}_{r,j}$.555As pointed out by Mele (2017), it is possible to
include $\boldsymbol{x}_{r,j}$ in
$v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,k})$. However,
$v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j},\boldsymbol{x}_{r,k})$ must be
invariant to any permutation of $\boldsymbol{x}_{r,i}$,
$\boldsymbol{x}_{r,j}$, and $\boldsymbol{x}_{r,k}$ for the game to have a
potential function. In the model, the definition of indirect links from $i$ to
$k$ does not preclude the existence of a direct link. This is the case, for
example, in structures (e) and (f), where families $i$ and $k$ are connected
both directly and indirectly in the network.

$i$$j$$j^{\prime}$Structure (a)$i$$j$$j^{\prime}$Structure
(b)$i$$j$$k$Structure (c)$i$$j$$k$Structure (d)$i$$j$$k$Structure
(e)$i$$j$$k$Structure (f) Figure 2: Illustration of the specification

While the roles of the parameters $\beta_{1}^{u}$, $\beta_{1}^{m}$, and
$\beta_{1}^{v}$ in the network are well-defined, their interpretation can be
challenging. The complexity arises because the influence of same-gender
children on direct links is not solely captured by $\beta_{1}^{u}$; it is also
influenced by $\beta_{1}^{m}$ and $\beta_{1}^{v}$. For instance, when
$\beta_{1}^{u}=0$, it is not straightforward to conclude that same-gender
children do not foster the formation of direct links. Even with
$\beta_{1}^{u}=0$, a positive value for $\beta_{1}^{m}$ corresponds to
structure (b) in Figure 2, indicating a positive correlation between same-
gender children and the likelihood to form a relationship. This is because a
direct link in the network can also be involved in a mutual relationship. This
duality is evident from Equation (2), where $a_{r,ij}$ can take the value of 1
in both the direct and mutual utility terms for the same $i$ and $j$.

Similarly, if $\beta_{1}^{u}=0$ and $\beta_{1}^{m}=0$, but $\beta_{1}^{v}>0$,
an increase in the proportion of same-gender children may result in a higher
proportion of direct and mutual links. As illustrated in structures (e) and
(f), on the one hand, family $k$ can act as both a direct (or mutual) and an
indirect connection to $i$. Therefore, even if $\beta_{1}^{u}=0$ and
$\beta_{1}^{m}=0$, we might still observe an increased probability of $i$
being directly connected to $k$ when $\beta_{1}^{v}>0$. On the other hand, the
indirect connection between $i$ and $k$ can go through a family, $j$, with a
same-gender child, as is the case in structure (c). Increasing the prevalence
of this structure also raises both the proportion of direct links and that of
indirect links between families with same-gender children. The only scenario
where an indirect link between families with same-gender children would not
increase the proportion of direct links between these families is illustrated
in structure (c). The actual impact of the condition $\beta_{1}^{v}>0$ on the
proportion of direct links between families with same-gender children depends
on whether our dataset contains more instances of structures (c), (e), and (f)
or structures (d).

The preceding discussion implies that observing $\beta_{1}^{u}=0$ and
$\beta_{1}^{m}=0$ does not necessarily imply that same-gender children do not
influence the likelihood of their families forming direct or mutual
relationships. For this reason, we place less emphasis on interpreting the
individual parameters in Equation (5). Instead, our main interest lies in
understanding the global effect of children’s gender on the network. To
achieve this, we conduct counterfactual analyses, varying, for instance, the
proportion of boys in the sample. For each proportion, we measure specific
network characteristics and analyze how they are influenced by children’s
gender.666One condition that would facilitate parameter interpretation is to
define an indirect friend in the model as a friend of a friend who is not a
friend and to refer to a direct friend as a non-mutual friend. Under these
definitions, there should be no overlap between structure (c) and structures
(a) and (b), nor should structure (b) overlap with structure (a). However, by
imposing these restrictions, the game would have no potential function, making
the identification of parameters impossible.

#### Parameter identification

In our dataset, 65.43% of direct links between families with same-gender
children are involved in mutual links. Additionally, 51.73% of indirect links
between families with same-gender children pass through a family with a same-
gender child, and 66.45% of these indirect links are also direct.777These
proportions indicate that indirect links between families with same-gender
children generate a substantial number of direct links between these families.
On one hand, there are 116,668 indirect links between families with same-
gender children (note that we may observe more than one indirect link between
two families as the link can go through many families). Given this large
number of indirect links, the proportion of 51.73% is statistically greater
than 50%, suggesting that indirect links between families with same-gender
children often go through a family with a same-gender child. On the other
hand, the proportion of 66.45% indicates that these indirect links are also
direct links in most cases. This observation reinforces our earlier point that
$\beta_{1}^{v}$ captures the influence of same-gender children on the
probability of forming both direct and mutual links. Importantly, this
situation may also give rise to a non-identification issue. Specifically, we
are concerned that many combinations of $\beta_{1}^{u}$, $\beta_{1}^{m}$, and
$\beta_{1}^{v}$ could produce the same network structure. If this is the case,
it might be challenging to disentangle between the parameters.

The non-identification issue in this model has also been highlighted by Mele
(2017). For instance, in Figure 1, we can typically replicate structures
(c)–(f) using only direct and mutual links between multiple nodes. This
implies that, in certain cases, we can achieve the same network configuration
by increasing $\beta_{1}^{u}$ and $\beta_{1}^{m}$ while decreasing
$\beta_{1}^{v}$. Links that will break up in some indirect relationships due
to the reduction in $\beta_{1}^{v}$ could be regenerated as direct and mutual
links due to the increase in $\beta_{1}^{u}$ and $\beta_{1}^{m}$.

Consider a scenario where we observe a single village in which all families
are mutually connected. In this particular situation, it becomes impossible to
distinguish between the parameters $\beta_{1}^{u}$, $\beta_{1}^{m}$, and
$\beta_{1}^{v}$. However, if we observe a second village where no links are
mutual, it would not be possible to achieve the same network structures using
various combinations of $\beta_{1}^{u}$ and $\beta_{1}^{m}$. Therefore, a
sufficient condition for identifying the individual effects of $\beta_{1}^{u}$
and $\beta_{1}^{m}$ is the presence of enough variation in the proportion of
direct links between families with same-gender children that are also mutual
across villages. In addition, to identify $\beta_{1}^{v}$ from the rest of the
parameters, we also need the proportion of direct and mutual links between
families having same-gender children involved by indirect relation between
these families to vary sufficiently across villages. Fortunately, this
condition is likely to be met because we have a substantial dataset with 222
villages assumed to be independent. Table 3 provides an overview of the
distribution of link proportions among families with same-gender children
across different villages. The significant variability in these proportions
ensures the identification of model parameters. This identification analysis
can also be extended to parameters associated with other control variables in
Equation (5). Using variation across groups to achieve identification is also
employed by Mele (2022) in a context where networks from multiple schools are
assumed to be independent.

Table 3: Link proportions between families with same-gender children across
villages

| Min | Pctl 25 | Median | Pctl 75 | Max
---|---|---|---|---|---
Direct links that are also mutual | $0.133$ | $0.522$ | $0.644$ | $0.735$ | $1.000$
Indirect links through families with same-gender children | $0.000$ | $0.418$ | $0.483$ | $0.590$ | $1.000$
Indirect links that are also non-mutually direct | $0.000$ | $0.161$ | $0.211$ | $0.259$ | $0.643$
Indirect links that are also mutually direct | $0.048$ | $0.271$ | $0.398$ | $0.509$ | $1.000$
Indirect links that involve direct reciprocal (non-mutual) links | $0.000$ | $0.077$ | $0.109$ | $0.135$ | $0.301$

  * This table summarizes link proportions between families with same-gender children across villages. Pctl 25 and Pctl 75 represent the 25th and 75th percentiles of these proportions. “Indirect links that involve direct reciprocal (non-mutual) links" means that $i$ is indirectly connected to $k$ with a direct (non-mutual) link from $k$ to $i$.

#### Estimation strategy

Our estimation strategy follows Mele (2022). Specifically, we adapt the
Bayesian estimation approach proposed by Mele (2017) to our context where we
have a large number of villages. This approach relies on a Markov Chain Monte
Carlo (MCMC) process to simulate the posterior distribution of the parameters.
Importantly, this approach circumvents the need to compute the denominator in
the distribution of $A_{r}$ as given by Equation (4). This denominator
involves summing over all possible networks, and, as we pointed out earlier,
this can amount to millions of networks even with a relatively small number of
families per village.

In the MCMC process, a computationally intensive task is to simulate, in each
iteration, an artificial network $\tilde{A}_{r}$ with the same likelihood as
that of the observed network $A_{r}$. As the artificial network must be
independent across villages as the observed network, we perform this
simulation for multiple villages simultaneously. Our MCMC process consists of
100,000 iterations, with the last 50,000 iterations retained for estimating
the posterior distribution of the parameters. During each iteration, we
conduct 7,000 simulations of artificial networks and retain the final one, as
suggested by Mele (2017). To achieve an acceptance rate of 23.4%, we propose
an adaptive version, as detailed in Appendix A, following the approach
outlined in Atchadé and Rosenthal (2005).

For the counterfactual analysis, we perform 1,000 simulations for each
assigned value of the network explanatory variables.888To assign a value to
the explanatory variables, one approach is to generate, for example, the
variable “Child Gender” to achieve a specific proportion of same-gender
children in the sample, while keeping other control variables fixed. In each
simulation, we draw parameter values from the last 50,000 simulations of the
MCMC. Once these values are drawn, given the assigned values of the
explanatory variables, we perform 7,000 network simulations in each village.
We retain the last simulations to compute network characteristics, such as
degree, clustering density, diameter, etc. Therefore, given the value assigned
to the network explanatory variables, we have 1,000 simulations for each
network characteristic that can be used to estimate the average and confidence
interval of the characteristic (see Appendix B).

##  4 Results

###  4.1 Posterior estimates

Table 4 presents the estimated posterior means that we obtain from structural
estimation of the preference parameters of the model. This table contains two
specifications: the first specification only considers direct links, while the
second specification includes mutual and indirect links as well.999For
completeness, we present two additional specifications in Appendix Table A1:
one that only includes benefits from mutual links and another that includes
both direct and mutual links (but not indirect ones). Following the
presentation in Table 2, in columns (1) and (3), we model the effect of a pair
of families having same-gender children, while in columns (2) and (4), we
allow having boys to have different effect than having girls. As in our
previous descriptive analysis, the number of observations in these estimations
corresponds to the 92,152 possible pairs of links that the 4,473 families in
the sample can form. Each estimated coefficient that we report in Table 4
represents the marginal impact of the associated variable on the utility of
forming a direct, mutual, or indirect link, respectively, while other
variables are kept constant. Our primary focus in discussing these results
revolves around the effect of having same-gender children ($Gender_{ij}$),
which we then decompose to the effect of both families having boys and the
effect of both families having girls. All specifications also include various
demographic controls (parents’ age, income, occupation, and education) for
families $i$ and $j$.

Starting with the estimates reported in columns (1) and (2), we observe that
the sign and statistical significance of the coefficients are very similar to
those reported using a standard logit model of link formation in Table 2,
although the estimation approach is different. This provides some confidence
in the model and Bayesian estimation approach pursued here.

Our preferred specification, of course, is one which also accounts for
preferences for mutual and indirect links presented in columns (3) and (4).
This specification also captures equilibrium and network effects of network
formation that a standard logit model cannot address. In column (3), we
observe that the coefficients on “$\textit{Child Gender}_{ij}$” are positive
across all three terms of the specification, although only the coefficient on
indirect links is statistically significant. Note, however, that indirect
links encompass the benefits of both direct and mutual links, as these are
prerequisites for indirect links to be formed. In addition, direct links in
the model can be part of mutual relationships. As pointed out earlier, we
observe that 65.43% of direct links between families with same-gender children
are involved in mutual links. Additionally, 51.73% of indirect links between
families with same-gender children pass through a family with a same-gender
child and 66.45% of these indirect links are also direct. The significance of
same-gender children on indirect link formation indicates that not only does
the gender of children of families that are one-step away connections matter
for network formation but more widely of the community of families that are
connected to these direct links. It is also worth highlighting that the
indirect links we consider here can also be direct links, meaning that $i$ can
be connected to $j$ and $k$, and $j$ also connected to $k$, so from $i$’s
perspective $j$ and $k$ can be considered as both direct and indirect links
(same applies to $j$ and $k$). This makes the interpretation of the
coefficients we estimate complicated, which is why turning to counterfactual
simulations is insightful, as it allows us to trace out the full implications
of changes in one of the factors on the structure of networks.

In terms of other characteristics, several noteworthy observations stand out:
income gaps between families play a significant role for both direct and
mutual links. Age gaps between fathers and age gaps between mothers seem to
have significant and opposing effects (for mutual and indirect links). Having
similar education levels is important for indirect links and having similar
occupation has ambiguous effects.

Table 4: Summary of the parameter posterior distribution

| (1) | (2) | (3) | (4)
---|---|---|---|---
Direct links |  |  |  |  |  |  |  |
Intercept | $-1.003^{***}$ | (0.069) | $-1.005^{***}$ | (0.067) | $-2.257^{***}$ | (0.070) | $-2.261^{***}$ | (0.073)
$\text{Child Gender}_{ij}$ | $0.021$ | (0.015) |  |  | $0.010$ | (0.025) |  |
$\text{Child Male}_{ij}$ |  |  | $0.043^{**}$ | (0.018) |  |  | $0.020$ | (0.026)
$\text{Child Female}_{ij}$ |  |  | $0.012$ | (0.017) |  |  | $-0.026$ | (0.026)
Adiff(Father’s Age) | $0.002$ | (0.002) | $0.002$ | (0.002) | $\textless{}0.000$ | (0.003) | $\textless 0.000$ | (0.003)
$\text{Age Father}_{i}$ | $-0.002$ | (0.002) | $-0.002$ | (0.002) | $0.001$ | (0.002) | $-0.002$ | (0.002)
$\text{Age Father}_{j}$ | $-0.005^{**}$ | (0.002) | $-0.005^{**}$ | (0.002) | $-0.005^{**}$ | (0.002) | $-0.003^{*}$ | (0.002)
Adiff(Mother’s Age) | $-0.005^{**}$ | (0.002) | $-0.005^{**}$ | (0.002) | $-0.007^{**}$ | (0.003) | $-0.006^{*}$ | (0.003)
$\text{Age Mother}_{i}$ | $0.002$ | (0.002) | $0.002$ | (0.003) | $-0.001$ | (0.003) | $0.003$ | (0.002)
$\text{Age Mother}_{j}$ | $0.006^{**}$ | (0.003) | $0.006^{**}$ | (0.002) | $0.007^{***}$ | (0.003) | $0.004^{***}$ | (0.002)
Adiff(Income) | $-0.016^{***}$ | (0.002) | $-0.016^{***}$ | (0.002) | $-0.008^{***}$ | (0.002) | $-0.008^{***}$ | (0.002)
$\text{Income}_{i}$ | $0.012^{***}$ | (0.002) | $0.012^{***}$ | (0.002) | $0.008^{***}$ | (0.002) | $0.008^{***}$ | (0.002)
$\text{Income}_{j}$ | $0.013^{***}$ | (0.002) | $0.013^{***}$ | (0.002) | $0.010^{***}$ | (0.002) | $0.010^{***}$ | (0.002)
Adiff(Homestead land) | $-0.408^{***}$ | (0.121) | $-0.418^{***}$ | (0.125) | $0.060$ | (0.121) | $0.072$ | (0.113)
$\text{Homestead land}_{i}$ | $0.304^{***}$ | (0.116) | $0.305^{***}$ | (0.117) | $0.492^{***}$ | (0.109) | $0.483^{***}$ | (0.097)
$\text{Homestead land}_{j}$ | $0.187^{*}$ | (0.111) | $0.203^{*}$ | (0.116) | $0.307^{***}$ | (0.106) | $0.299^{***}$ | (0.111)
Adiff(Farming land) | $-0.185^{***}$ | (0.018) | $-0.187^{***}$ | (0.017) | $-0.112^{***}$ | (0.015) | $-0.110^{***}$ | (0.014)
$\text{Farming land}_{i}$ | $0.179^{***}$ | (0.017) | $0.181^{***}$ | (0.017) | $0.110^{***}$ | (0.015) | $0.107^{***}$ | (0.013)
$\text{Farming land}_{j}$ | $0.190^{***}$ | (0.017) | $0.191^{***}$ | (0.017) | $0.125^{***}$ | (0.015) | $0.122^{***}$ | (0.013)
$\text{Father\textquoteright s education}_{ij}$ | $0.055^{***}$ | (0.016) | $0.055^{***}$ | (0.015) | $0.010$ | (0.028) | $0.003$ | (0.023)
$\text{Mother\textquoteright s education}_{ij}$ | $0.031^{**}$ | (0.014) | $0.031^{**}$ | (0.015) | $-0.021$ | (0.024) | $-0.008$ | (0.027)
$\text{Father is daily laborer}_{ij}$ | $0.268^{***}$ | (0.021) | $0.267^{***}$ | (0.021) | $0.021$ | (0.038) | $0.016$ | (0.041)
$\text{Father is farmer}_{ij}$ | $0.151^{***}$ | (0.030) | $0.152^{***}$ | (0.030) | $0.026$ | (0.053) | $0.033$ | (0.043)
$\text{Father is self-employed}_{ij}$ | $-0.099^{***}$ | (0.024) | $-0.096^{***}$ | (0.024) | $0.079^{**}$ | (0.040) | $0.088^{**}$ | (0.039)
$\text{Mother stays home}_{ij}$ | $-0.073^{***}$ | (0.020) | $-0.072^{***}$ | (0.021) | $-0.080^{**}$ | (0.037) | $-0.067^{*}$ | (0.038)
Mutual links |  |  |  |  |  |  |  |
Intercept |  |  |  |  | $2.093^{***}$ | (0.074) | $2.124^{***}$ | (0.060)
$\text{Gender}_{ij}$ |  |  |  |  | $0.002$ | (0.050) |  |
$\text{Male}_{ij}$ |  |  |  |  |  |  | $0.008$ | (0.048)
$\text{Female}_{ij}$ |  |  |  |  |  |  | $-0.005$ | (0.047)
Adiff(Father’s Age) |  |  |  |  | $-0.006$ | (0.005) | $-0.007$ | (0.005)
Adiff(Mother’s Age) |  |  |  |  | $0.014^{**}$ | (0.007) | $0.013^{**}$ | (0.007)
Adiff(Income) |  |  |  |  | $-0.011^{***}$ | (0.004) | $-0.010^{**}$ | (0.004)
Adiff(Homestead land) |  |  |  |  | $-0.586^{***}$ | (0.164) | $-0.606^{***}$ | (0.179)
Adiff(Farming land) |  |  |  |  | $-0.015$ | (0.010) | $-0.014$ | (0.010)
$\text{Father\textquoteright s education}_{ij}$ |  |  |  |  | $0.034$ | (0.053) | $0.045$ | (0.046)
$\text{Mother\textquoteright s education}_{ij}$ |  |  |  |  | $0.066$ | (0.048) | $0.033$ | (0.047)
$\text{Father is daily laborer}_{ij}$ |  |  |  |  | $-0.108$ | (0.068) | $-0.104$ | (0.075)
$\text{Father is farmer}_{ij}$ |  |  |  |  | $0.094$ | (0.099) | $0.074$ | (0.078)
$\text{Father is self-employed}_{ij}$ |  |  |  |  | $0.146^{**}$ | (0.071) | $0.142^{**}$ | (0.065)
$\text{Mother stays home}_{ij}$ |  |  |  |  | $0.060$ | (0.058) | $0.042$ | (0.057)
Indirect links |  |  |  |  |  |  |  |
Intercept |  |  |  |  | $0.026^{***}$ | (0.004) | $0.025^{***}$ | (0.003)
$\text{Gender}_{ij}$ |  |  |  |  | $0.007^{**}$ | (0.003) |  |
$\text{Male}_{ik}$ |  |  |  |  |  |  | $0.005^{**}$ | (0.002)
$\text{Female}_{ik}$ |  |  |  |  |  |  | $0.011^{***}$ | (0.002)
Adiff(Father’s Age) |  |  |  |  | $0.001^{***}$ | (0.000) | $0.001^{***}$ | (0.000)
Adiff(Mother’s Age) |  |  |  |  | $-0.001^{***}$ | (0.000) | $-0.001^{***}$ | (0.000)
Adiff(Income) |  |  |  |  | $0.215$ | (0.190) | $0.130$ | (0.192)
Adiff(Homestead land) |  |  |  |  | $-0.049^{***}$ | (0.009) | $-0.047^{***}$ | (0.008)
Adiff(Farming land) |  |  |  |  | $-0.073$ | (0.700) | $0.001$ | (0.681)
$\text{Father\textquoteright s education}_{ik}$ |  |  |  |  | $0.006^{**}$ | (0.003) | $0.005^{**}$ | (0.002)
$\text{Mother\textquoteright s education}_{ik}$ |  |  |  |  | $0.008^{***}$ | (0.002) | $0.008^{***}$ | (0.002)
$\text{Father is daily laborer}_{ik}$ |  |  |  |  | $0.031^{***}$ | (0.003) | $0.031^{***}$ | (0.003)
$\text{Father is farmer}_{ik}$ |  |  |  |  | $0.002$ | (0.003) | $0.003$ | (0.003)
$\text{Father is self-employed}_{ik}$ |  |  |  |  | $-0.034^{***}$ | (0.004) | $-0.036^{***}$ | (0.004)
$\text{Mother stays home}_{ik}$ |  |  |  |  | $-0.001$ | (0.003) | $-0.001$ | (0.003)

  * Note: ***: pvalue $<1\%$, **: pvalue $<5\%$, *: pvalue $<10\%$
This table presents the mean of the posterior distribution of each parameter
followed by the standard deviation in brackets using the last 50,000
simulations of the MCMC. For any variable denoted “var", “varij” takes one if
var${}_{i}=$ varj and zero otherwise; “Adiff(var)” if the absolute difference
between vari and varj. The network is directed and the relationship we model
is the link from $j$ to $i$.

###  4.2 Counterfactual experiments

We next use the estimated model to perform counterfactual experiments that
enable us to gauge the impact of changes in characteristics of interest on
several key network structure measures: degree (number of links), asymmetry,
density, average distance, and clustering. Asymmetry measures variance in the
decision to form a link between two families chosen arbitrarily, that is, the
binary variable $a_{r,ij}$. Higher values indicate disparity in the decision
to form a relationship. We normalize the asymmetry for the highest value to be
equal to one. Network density represents the proportion of actual connections
within a network relative to all potential connections. Its values range
between 0 and 1, with higher values indicating denser networks. Network
clustering captures the tendency of families in a network to form clusters
(triangles), and ranges from 0 (no clustering) to 1 (maximum clustering). To
be precise, it measures the ratio of fully connected triples to the potential
triples that include at least two links.

Our counterfactual analysis is based on 1,000 independent draws from the set
constituted by the last 50,000 simulations on the MCMC process. For each draw,
we generate the network by varying the explanatory variable depending on the
targeted analysis. For example, in Figure 3, we vary the proportion of same-
gender children from 0% to 100%. We then plot various network characteristics,
including degree, asymmetry, density, average distance, and clustering.

#### Children’s Gender

Figure 3 illustrates our first counterfactual experiment in which we vary the
gender composition of the children in the sample. In panel (a), we vary the
gender composition of children from being split to being fully homogeneous
(either boys or girls), whereas in panel (b), we vary the proportion of male
children from 0 to 1 to assess whether there are gender differences in the
impact.

Starting with panel (a), we observe that as the gender composition becomes
more homogeneous, there is a significant impact on all the measures of network
structure that we consider. Notably, the average number of links (i.e.,
degree) increases by about 15% and, therefore, the network density and
asymmetry go up as well, along with a reduction in the average distance
between two nodes. We also see a significant increase in clustering by about
20% and a modest increase in asymmetry.

When we turn our attention to panel (b), we observe that girls have a stronger
effect on network structure. Comparing the scenario of all girls to either the
status quo or the scenario of all boys, we observe that the average degree is
higher by about 21% in the first case and 14% in the second case. Similarly,
girls have a stronger impact on the other dimensions, suggesting that they
give rise to denser and more clustered networks than the status quo or the
all-boys scenario.

(a) Impact of the proportion of same-gender children

(b) Impact of the proportion of male children

Notes:

Panel (a): Children’s gender is randomly generated as a dummy variable from a
Bernoulli distribution. The proportion of same-gender children varies from 50%
(meaning that they are 50% boys) to 100% (meaning that all the children are
either boys or girls).

Panel (b): Child being male or not is randomly generated as a dummy variable
from a Bernoulli distribution of parameter the proportion of male children
desired in the sample. Changing the variable “Male” also requires changing the
variable “Female,” which is defined as one minus the variable “Male.”

Figure 3: Counterfactual experiment on gender composition of children.

These results highlight a significant role of children’s gender for parents’
networks; however, they do not provide a precise sense of the effect’s
magnitude. To establish a benchmark against which we can gauge the
quantitative importance of children’s gender, we next present counterfactual
experiments in which we vary other important determinants of network
formation, such as households’ income, fathers’ occupation, fathers’ and
mothers’ education and age.

#### Parents’ Income

Figure 4 presents our second counterfactual experiment in which we vary income
inequality across households. Specifically, the horizontal access measures
income inequality, with a 0 corresponding to the scenario of perfect equality,
1 to the observed income, and 2 to a scenario of high inequality (twice that
of the observed one). All measures vary monotonically with inequality in the
direction of families having fewer connections, and networks becoming less
dense and clustered. It is of interest to consider what are the quantitative
implications of income inequality and how do they compare to those of
children’s gender. By moving from the status quo (1) to perfect equality (0),
we would increase the average degree by about 7.3, which is within the
range—in fact, slightly less—than the all-girls scenario depicted in Figure 3.
Thus, our first take-away is that children’s gender is as important for the
structure of parental networks as income distribution.

Figure 4: Impact of income inequality

Note: For each simulation, we vary the income inequality by generating a new
income variable given by
$\hat{\text{Inc}}=\text{Inc}_{0}+\bar{\delta}\times(\text{Inc}-\text{Inc}_{0})$,
where Inc in the observed income in the original sample, $\text{Inc}_{0}$ is
the average income observed in the original sample, and $\bar{\delta}$ is the
inequality measure. If $\bar{\delta}=0$, then the generated income is the
observed average income, i.e. the generated income is the same for all
families. If $\bar{\delta}=1$, then the generated income coincides with the
observed income. If $\bar{\delta}=2$, then the standard deviation of the
generated income is twice that of the observed income.

#### Fathers’ Occupation

Our third counterfactual experiment focuses on varying the heterogeneity of
fathers’ occupations across families. We select two occupations that are most
prevalent in the sample—that is, daily laborers and self-employed—and we vary
the proportion of fathers who have these occupations. Panel (a) in Figure 5
displays the case of daily laborers. It is evident that more homogeneity has a
strong effect on network degree, since moving from the status quo (marked by
the vertical dashed line) to full homogeneity—that is, all fathers are daily
laborers—leads to a doubling of the average degree, the network density, and
the overall clustering. In contrast, asymmetry exhibits different behavior due
to the significant increase in degree.101010Asymmetry measures the variance of
the binary variable $a_{r,ij}$, which represents the decision to form a
relationship. The variance is highest when the probability of $a_{r,ij}$ being
one reaches 0.5. When this probability exceeds 0.5, the variance decreases.
This is in line with the asymmetry behavior. When the degree is below 10,
which is half of the average village size, asymmetry increases with the
degree. Conversely, when the degree exceeds half of the average village size,
asymmetry decreases.

On the other hand, the picture is quite different in the case of self-
employment. As we increase the proportion of self-employed fathers, we observe
in panel (b) of Figure 5 a significant and monotonic reduction in degree,
density, and clustering. This may be because self-employed individuals in this
village context may be small shop-keepers or engaged in handicraft production.
These occupations might place them in competition with each other in the
village market, suggesting that they might be less likely to form strong
connections with each other for risk-sharing and mutual support purposes.

In Figure A1 in the Appendix, we also present a similar experiment on varying
proportion of farmers, which has the same qualitative features as that of
daily laborers, though the magnitudes are lower.

(a) Impact of the proportion of daily laborer fathers

(b) Impact of the proportion of self-employed fathers

Notes: Father being a daily laborer (self-employed) is generated as a dummy
variable from a Bernoulli distribution. Changing the variable "daily laborer"
("self-employed") also requires changing the proportion of other occupations.
Once the variable "daily laborer fathers" ("self-employed") is generated, we
proportionally adjust the proportion of the other occupations for the total
proportion to equal one.

Figure 5: Counterfactual experiment on occupation of fathers.

#### Parents’ Education

The fourth policy experiment varies the proportion of fathers or mothers who
share the same level of education. In Figure 6, we vary the proportion of
fathers with the same education level (panel (a)) and the proportion of
mothers with the same education level (panel (b)). This allows us to observe
how changes in education homogeneity impact network characteristics.

When we increase the proportion of parents with similar education levels this
leads to networks with higher degree, density, and clustering. Interestingly,
the magnitude of the impact as we move from the status quo to having a
perfectly homogeneous composition of parents in terms of education is similar
to that found for same-gender children. This finding constitutes another
important benchmark, allowing us to distill the importance of children’s
gender for network structure.

(a) Impact of the proportion of fathers with the same education level

(b) Impact of the proportion of mothers with the same education level

Notes: We first generate the education level as a multinomial variable with
four categories (primary school, middle school, high school, and higher
education), and then we compute the proportion of fathers (mothers) with the
same education level. The lowest proportion possible is 25% because we
consider four categories.

Figure 6: Counterfactual experiment on education of parents.

#### Parents’ Age

In our last experiment, displayed in Figure 7 we vary the age of parents to
assess its influence on network structure. Panel (a) examines the impact of
variations in the age gap among fathers, while panel (b) focuses on the age
gap among mothers.

We find some interesting opposing patterns for the two parents. While for
fathers, a more dispersed age distribution increases connections, and enhances
the density and clustering of networks, the opposite is true for mothers’ age.
This suggests that women tend to form connections with other women of the same
age while the opposite is true for men. In terms of magnitude, increasing age
inequality for fathers or decreasing that of mothers has a similar impact on
network structure as the child’s gender.

(a) Impact of father’s age gap

(b) Impact of mother’s age gap

Notes: For each simulation, we vary the father’s (mother’s) age gap by
regenerating the variable father’s (mother’s) age as
$\hat{\text{Age}}=\text{Age}_{0}+\bar{\delta}\times(\text{Age}-\text{Age}_{0})$,
where Age in the observed father’s (mother’s) age in the original sample,
$\text{Inc}_{0}$ is the average age observed in the original sample, and
$\bar{\delta}$ is the inequality measure. If $\bar{\delta}=0$, then the
generated age is the observed average age, i.e. the generated age is the same
for all families. If $\bar{\delta}=1$, then the generated age coincides with
the observed age. If $\bar{\delta}=2$, then the standard deviation of the
generated age is twice that of the observed age.

Figure 7: Counterfactual experiment on age of parents.

##  5 Conclusion

This paper investigates the role of children’s gender on shaping parental
network structure. We draw on a rich network dataset of households with
children in early childhood comprising 4,500 households across 222 villages in
Bangladesh. We estimate a structural model of network formation and simulate
it to gain insight as to how the gender composition of children in a village
affects the structure of social ties among parents. Our findings reveal that
children’s gender plays a pivotal role in shaping the number of social
connections among parents, with particularly pronounced effects when families
have girls, suggesting a nuanced interplay between children’s gender and
parents’ social network formation. Importantly, the impact of children’s
gender on parental network formation is comparable in magnitude to other key
socioeconomic factors, including income distribution, parental occupation,
education, and age.

Our findings may have policy implications that favor single-sex schools.
However, this needs to be balanced by considering the impact on children’s
socialization into gender roles, which begins at an early age. Indeed, in the
psychology literature, it is well-documented that parents begin to socialize
their children as boys or girls without even knowing it (Lindsey, 2011). Our
study shows that parents are more likely to interact with their peers—here,
other parents with children of the same gender—which may reinforce gender-
typed expectations and stereotypes, so that parents will have a differential
treatment of daughters and sons. Thus, promoting single-sex schools may
increase the social interactions of parents and eventually that of children of
the same gender but it may also inadvertently promote gender stereotypes.

While our study offers significant insights into parental network formation in
a rural Bangladeshi context, future research could extend the analysis to
other settings and cultural contexts to enrich our understanding of these
dynamics. Additionally, expanding the study to cover families with children
from a broader range of age groups could shed light on how these influences on
parents’ networks evolve throughout children’s development. Furthermore, given
that parents’ social interactions are influenced by their children’s gender, a
natural next step is to explore the extent to which this channel can explain
the formation of social links among children. These are promising avenues that
we intent to pursue in future research.

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APPENDIX

##  Appendix A Estimation method

The MCMC approach proposed by Mele (2017) avoids the need of computing the
denominator in the distribution of $A_{r}$ given by Equation (4). This
approach requires simulating at each iteration an artificial network
$\tilde{A}_{r}$ with the same likelihood than that of the observed network
$A_{r}$. Let $\boldsymbol{\beta}\in\mathbb{R}^{K}$ be the vector of parameters
to be estimated, for some integer $K>0$. With the specification

$\displaystyle u(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})$
$\displaystyle=\beta^{u}_{1}Gender_{r,ij}+\boldsymbol{x}^{u\prime}_{ij}\boldsymbol{\beta}^{u}_{2},$
$\displaystyle m(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,j})$
$\displaystyle=\beta^{m}_{1}Gender_{r,ij}+\boldsymbol{x}^{m\prime}_{ij}\boldsymbol{\beta}^{m}_{2},$
$\displaystyle v(\boldsymbol{x}_{r,i},\boldsymbol{x}_{r,k})$
$\displaystyle=\beta^{v}_{1}Gender_{r,ik}+\boldsymbol{x}^{v\prime}_{ik}\boldsymbol{\beta}^{v}_{2},$

we use the notation $Q_{r}(A_{r},\mathcal{X}_{r},\boldsymbol{\beta})$ to
denote the potential function in Equation (3) instead of
$Q_{r}(A_{r},\mathcal{X}_{r})$. Given the high number of observations in our
sample, we set non-informative flat prior for $\boldsymbol{\beta}$, i.e., the
prior density function of $\boldsymbol{\beta}$ is
$p(\boldsymbol{\beta})\propto 1$. We propose an adaptive version of the MCMC
process.

Algorithm 1 Simulation from the parameter posterior distribution

Initialize $\boldsymbol{\beta}^{(0)}\in\mathbb{R}^{K}$ and choose
$\tau^{(0)}>0$ and $\boldsymbol{\Sigma}$ a positive definite matrix dimension
$K\times K$.
For t from 1 to $T$:

  1. 1.

Draw a proposal $\tilde{\boldsymbol{\beta}}$ from
$N\left(\boldsymbol{\beta}^{(t-1)},\tau^{(t-1)}\boldsymbol{\boldsymbol{\Sigma}}\right)$.

  2. 2.

For $r$ from 1 to $\bar{r}=222$, simulate the artificial network
$\tilde{A}_{r}$ using Algorithm 2, where $\boldsymbol{\beta}$ is set to
$\tilde{\boldsymbol{\beta}}$ and $\tilde{A}_{r}^{(0)}$ is the observed network
$A_{r}$. We perform this step in parallel for many villages simultaneously
because the simulations of $\tilde{A}_{r}$ are independent across $r$,
conditional on $\boldsymbol{\beta}$.

  3. 3.

Accept the proposal $\tilde{\boldsymbol{\beta}}$ with the probability

$\min\left\\{1,\dfrac{\exp\left\\{\sum_{r=1}^{\bar{r}}\left(Q_{r}(\tilde{A}_{r},\mathcal{X}_{r},\boldsymbol{\beta}^{(t-1)})+Q_{r}(A_{r},\mathcal{X}_{r},\tilde{\boldsymbol{\beta}})\right)\right\\}}{\exp\left\\{\sum_{r=1}^{\bar{r}}\left(Q_{r}(\tilde{A}_{r},\mathcal{X}_{r},\tilde{\boldsymbol{\beta}})+Q_{r}(A_{r},\mathcal{X}_{r},\boldsymbol{\beta}^{(t-1)})\right)\right\\}}\right\\}.$

Update $\boldsymbol{\beta}^{(t)}=\tilde{\boldsymbol{\beta}}$ if
$\tilde{\boldsymbol{\beta}}$ is accepted, and
$\boldsymbol{\beta}^{(t)}=\boldsymbol{\beta}^{(t-1)}$ otherwise.

  4. 4.

Update $\tau^{(t)}$ following Atchadé and Rosenthal (2005) for the acceptance
rate equals 23.4%.

We first run Algorithm 1, defining $\boldsymbol{\Sigma}$ as an identity
matrix, and $\tau^{(0)}=1$. This means that the proposal
$\tilde{\boldsymbol{\beta}}$ does not take into account the correlation
between the components of $\boldsymbol{\beta}$. We perform $T=100,000$
simulations and define a new $\boldsymbol{\Sigma}$ as a covariance matrix of
the last 50,000 simulations. We run the algorithm a second time with the new
$\boldsymbol{\Sigma}$ and perform $T=100,000$ simulations again. The last
50,000 simulations are considered draws from the posterior distribution of
$\boldsymbol{\beta}$. As we rely on an adaptive MCMC in which $\tau^{(t)}$ is
updated, we do not care about the choice of $\tau^{(0)}$. We set
$\tau^{(0)}=1$ every time. We do this exercise using several starting values
$\boldsymbol{\beta}^{(0)}$ and converge to the same posterior distribution.
This suggests that the model does not suffer from the identification problem
pointed out by M17.

To simulate the artificial network $\tilde{A}_{r}$ in step 2 of Algorithm 1,
we rely on a Gibbs sampler, where only one entry of the network (randomly
chosen) is updated at each iteration. We also try to update two and three
entries simultaneously at each iteration. We notice that the results are
robust to the number of entries we select. The convergence is faster when we
select many entries, but the computations become more intensive.

Algorithm 2 Simulation of artificial networks in the village $r$

Inputs: Assign a value to $\boldsymbol{\beta}$ and initialize
$\tilde{A}_{r}^{(0)}$ is the observed network $A_{r}$.
Perform the following steps for t from 1 to $R=7,000$.

  1. 1.

Randomly chosen the entry $(i,j)$ to update in $\tilde{A}_{r}^{(t)}$. We have
two proposals for $\tilde{A}_{r}^{(t)}$. The $(i,j)$-th entry is zero for the
first proposal denoted $A_{r}^{0}$ and is one for the second proposal denoted
$A_{r}^{1}$. The other entries in $A_{r}^{0}$ and $A_{r}^{1}$ keep the same
values as in $\tilde{A}_{r}^{(t-1)}$.

  2. 2.

Accept $A_{r}^{0}$ with the probability
$\dfrac{\exp\left\\{Q_{r}(A_{r}^{0},\mathcal{X}_{r},\boldsymbol{\beta})\right\\}}{\exp\left\\{Q_{r}(A_{r}^{0},\mathcal{X}_{r},\boldsymbol{\beta})\right\\}+\exp\left\\{Q_{r}(A_{r}^{1},\mathcal{X}_{r},\boldsymbol{\beta})\right\\}}$.
Set $\tilde{A}_{r}^{(t)}=A_{r}^{0}$ if $A_{r}^{0}$ is accepted and
$\tilde{A}_{r}^{(t)}=A_{r}^{1}$ otherwise.

The last simulation $\tilde{A}_{r}^{(R)}$ is the generated artificial network
$\tilde{A}_{r}$.

##  Appendix B Counterfactual analysis

Our counterfactual analysis consists of varying family observable
characteristics and examining how these variations influence the network. For
example, we can regenerate the variable $Gender$ from a Bernoulli distribution
to change the proportion of same-gender families from 50% to 100%. Another
example is to vary the proportion of boys in the villages from 0% to 100%.
Each of these examples yields a set of family observable characteristics
denoted as $\hat{\mathcal{X}}_{r}$, which serves as the counterpart of
$\mathcal{X}_{r}$ in our counterfactual experiment. We use
$\hat{\mathcal{X}}_{r}$ to simulate the village networks.

We replicate the following steps 1,000 times.

  1. 1.

We randomly draw $\boldsymbol{\beta}$ from the last 50,000 simulations of
Algorithm 1.

  2. 2.

We simulate $\tilde{A}_{r}$ using Algorithm 2, where $\boldsymbol{\beta}$ is
set to the simulation of step 1 and $\mathcal{X}_{r}$ is replaced with
$\hat{\mathcal{X}}_{r}$.

  3. 3.

We compute some statistics of $\tilde{A}_{r}$, such as degree, clustering,
diameter, etc.

For each statistic, we have 1,000 replications, which can be used to compute
the confidence interval.

##  Appendix C Supplementary results

Table A1: Summary of the parameter posterior distribution (continued)

| Model 7 | Model 8 | Model 9 | Model 10
---|---|---|---|---
Direct links |  |  |  |  |  |  |  |
Intercept |  |  |  |  | $-1.749^{***}$ | (0.069) | $-1.734^{***}$ | (0.073)
$\text{Gender}_{ij}$ |  |  |  |  | $0.010$ | (0.023) |  |
$\text{Male}_{ij}$ |  |  |  |  |  |  | $0.007$ | (0.026)
$\text{Female}_{ij}$ |  |  |  |  |  |  | $-0.012$ | (0.028)
Adiff(Father’s Age) |  |  |  |  | $0.003$ | (0.003) | $0.004$ | (0.003)
$\text{Age Father}_{i}$ |  |  |  |  | $0.001$ | (0.002) | $0.001$ | (0.002)
$\text{Age Father}_{j}$ |  |  |  |  | $-0.005^{**}$ | (0.002) | $-0.005^{**}$ | (0.003)
Adiff(Mother’s Age) |  |  |  |  | $-0.009^{***}$ | (0.004) | $-0.010^{**}$ | (0.004)
$\text{Age Mother}_{i}$ |  |  |  |  | $-0.001$ | (0.003) | $-0.001$ | (0.003)
$\text{Age Mother}_{j}$ |  |  |  |  | $0.007^{**}$ | (0.003) | $0.007^{***}$ | (0.003)
Adiff(Income) |  |  |  |  | $-0.007^{***}$ | (0.003) | $-0.007^{***}$ | (0.002)
$\text{income}_{i}$ |  |  |  |  | $0.008^{***}$ | (0.002) | $0.007^{***}$ | (0.002)
$\text{Income}_{j}$ |  |  |  |  | $0.009^{***}$ | (0.002) | $0.010^{***}$ | (0.002)
Adiff(Homestead land) |  |  |  |  | $0.004$ | (0.113) | $0.007$ | (0.121)
$\text{Homestead land}_{i}$ |  |  |  |  | $0.242^{**}$ | (0.097) | $0.246^{**}$ | (0.097)
$\text{Homestead land}_{j}$ |  |  |  |  | $0.079$ | (0.099) | $0.076$ | (0.101)
Adiff(Farming land) |  |  |  |  | $-0.121^{***}$ | (0.015) | $-0.123^{***}$ | (0.013)
$\text{Farming land}_{i}$ |  |  |  |  | $0.120^{***}$ | (0.014) | $0.122^{***}$ | (0.013)
$\text{Farming land}_{j}$ |  |  |  |  | $0.133^{***}$ | (0.014) | $0.134^{***}$ | (0.013)
$\text{Father\textquoteright s education}_{ij}$ |  |  |  |  | $0.018$ | (0.022) | $0.015$ | (0.026)
$\text{Mother\textquoteright s education}_{ij}$ |  |  |  |  | $-0.003$ | (0.022) | $-0.002$ | (0.024)
$\text{Father is daily laborer}_{ij}$ |  |  |  |  | $0.199^{***}$ | (0.033) | $0.202^{***}$ | (0.037)
$\text{Father is farmer}_{ij}$ |  |  |  |  | $0.053$ | (0.047) | $0.049$ | (0.045)
$\text{Father is sel-femployed}_{ij}$ |  |  |  |  | $-0.115^{***}$ | (0.037) | $-0.112^{***}$ | (0.035)
$\text{Mother stays home}_{ij}$ |  |  |  |  | $-0.080^{**}$ | (0.034) | $-0.081^{***}$ | (0.034)
Mutual links |  |  |  |  |  |  |  |
Intercept | $-0.258^{***}$ | (0.042) | $-0.266^{***}$ | (0.044) | $2.139^{***}$ | (0.082) | $2.109^{***}$ | (0.083)
$\text{Gender}_{ij}$ | $0.026$ | (0.023) |  |  | $0.010$ | (0.045) |  |
$\text{Male}_{ij}$ |  |  | $0.059^{**}$ | (0.027) |  |  | $0.050$ | (0.052)
$\text{Female}_{ij}$ |  |  | $0.024$ | (0.027) |  |  | $0.047$ | (0.057)
Adiff(Father’s Age) | $-0.001$ | (0.003) | $-0.001$ | (0.003) | $-0.005$ | (0.005) | $-0.005$ | (0.005)
Adiff(Mother’s Age) | $\textless{}0.000$ | (0.003) | $\textless 0.000$ | (0.003) | $0.013^{*}$ | (0.007) | $0.014^{**}$ | (0.007)
Adiff(Income) | $-0.007^{***}$ | (0.002) | $-0.007^{***}$ | (0.002) | $-0.010^{**}$ | (0.005) | $-0.009^{**}$ | (0.004)
Adiff(Homestead land) | $-0.268^{***}$ | (0.093) | $-0.261^{***}$ | (0.094) | $-0.668^{***}$ | (0.167) | $-0.676^{***}$ | (0.169)
Adiff(Farming land) | $-0.003$ | (0.005) | $-0.003$ | (0.005) | $-0.016^{*}$ | (0.010) | $-0.015$ | (0.010)
$\text{Father\textquoteright s education}_{ij}$ | $0.091^{***}$ | (0.024) | $0.089^{***}$ | (0.025) | $0.042$ | (0.045) | $0.051$ | (0.053)
$\text{Mother\textquoteright s education}_{ij}$ | $0.061^{***}$ | (0.024) | $0.063^{**}$ | (0.024) | $0.055$ | (0.044) | $0.056$ | (0.046)
$\text{Father is daily laborer}_{ij}$ | $0.268^{***}$ | (0.034) | $0.266^{***}$ | (0.033) | $-0.038$ | (0.066) | $-0.046$ | (0.070)
$\text{Father is farmer}_{ij}$ | $0.211^{***}$ | (0.046) | $0.206^{***}$ | (0.046) | $0.103$ | (0.097) | $0.112$ | (0.089)
$\text{Father is sel-femployed}_{ij}$ | $-0.057$ | (0.038) | $-0.061$ | (0.039) | $0.109$ | (0.074) | $0.099$ | (0.070)
$\text{Mother stays home}_{ij}$ | $-0.073^{**}$ | (0.033) | $-0.073^{**}$ | (0.034) | $0.071$ | (0.066) | $0.070$ | (0.069)

  * Note: ***: pvalue $<1\%$, **: pvalue $<5\%$, *: pvalue $<10\%$
This table presents the mean of the posterior distribution of each parameter
followed by the standard deviation in brackets using the last 50,000
simulations of the MCMC. For any variable denoted "var", "varij" takes one if
var${}_{i}=$ varj and zero otherwise; "Adiff(var)" if the absolute difference
between vari and varj. The network is directed and the relationship we model
is the link from $j$ to $i$.

Figure A1: Impact of the proportion of farming fathers

Note: Father being a farmer is generated as a dummy variable from a Bernoulli
distribution. Changing the variable "farming fathers" also requires changing
the proportion of other occupations. Once the variable "farming fathers" is
generated, we proportionally adjust the proportion of the other occupations
for the total proportion to equal one.

Figure A2: Impact of the proportion of mothers who stay at home

Note: We vary the proportion of mothers who stay at home by simulating the
variable "Mother stays home" from a Bernoulli.