# Universal growth of social groups: empirical analysis and modeling

Ana Vranić  Institute of Physics Belgrade, University of Belgrade, Pregrevica
118, 11080 Belgrade, Serbia<EMAIL_ADDRESS>Jelena Smiljanić  Integrated
Science Lab, Department of Physics, Umeå University, SE-901 87 Umeå, Sweden
Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080
Belgrade, Serbia<EMAIL_ADDRESS>Marija Mitrović Dankulov
Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080
Belgrade, Serbia<EMAIL_ADDRESS>
###### Abstract

Social groups are fundamental elements of any social system. Their emergence
and evolution are closely related to the structure and dynamics of a social
system. Research on social groups was primarily focused on the growth and the
structure of the interaction networks of social system members and how
members’ group affiliation influences the evolution of these networks. The
distribution of groups’ size and how members join groups has not been
investigated in detail. Here we combine statistical physics and complex
network theory tools to analyze the distribution of group sizes in three data
sets, Meetup groups based in London and New York and Reddit. We show that all
three distributions exhibit log-normal behavior that indicates universal
growth patterns in these systems. We propose a theoretical model that combines
social and random diffusion of members between groups to simulate the roles of
social interactions and members’ interest in the growth of social groups. The
simulation results show that our model reproduces growth patterns observed in
empirical data. Moreover, our analysis shows that social interactions are more
critical for the diffusion of members in online groups, such as Reddit, than
in offline groups, such as Meetup. This work shows that social groups follow
universal growth mechanisms that need to be considered in modeling the
evolution of social systems.

##  1 Introduction

The need to develop methods and tools for their analysis and modeling comes
with massive data sets. Methods and paradigms from statistical physics have
proven to be very useful in studying the structure and dynamics of social
systems [1]. The main argument for using statistical physics to study social
systems is that they consist of many interacting elements. Due to this, they
exhibit different patterns in their structure and dynamics, commonly known as
collective behavior. While various properties can characterize a social
system’s building units, only a few enforce collective behavior in the
systems. The phenomenon is known as universality in physics and is commonly
observed in social systems such as in voting behavior [2], or scientific
citations [3]. It indicates the existence of the universal mechanisms that
govern the dynamics of the system [1].

Social groups, informal or formal, are mesoscopic building elements of every
socio-economic system that direct its emergence, evolution, and disappearance
[4]. The examples span from countries, economies, and science to society.
Settlements, villages, towns, and cities are formal and highly structured
social groups of countries. Their organization and growth determine the
functioning and sustainability of every society [5]. Companies are the
building blocks of an economic system, and their dynamics are essential
indicators of the level of its development [6]. Scientific conferences, as
scientific groups, enable fast dissemination of the latest results, exchange,
and evaluation of ideas as well as a knowledge extension, and thus are an
integral part of science [7]. The membership of individuals in various social
groups, online and offline, can be essential when it comes to the quality of
their life [8, 9, 10]. Therefore, it is not surprising that the social group
emergence and evolution are at the center of the attention of many researchers
[11, 12, 13, 14].

The availability of large-scale and long-term data on various online social
groups has enabled the detailed empirical study of their dynamics. The focus
was mainly on the individual groups and how structural features of social
interaction influence whether individuals will join the group [15] and remain
its active members [7, 16]. The study on LiveJournal [15] groups has shown
that decision of an individual to join a social group is greatly influenced by
the number of her friends in the group and the structure of their
interactions. The conference attendance of scientists is mainly influenced by
their connections with other scientists and their sense of belonging [7]. The
sense of belonging of an individual in social groups is achieved through two
main mechanisms [16]: expanding the social circle at the beginning of joining
the group and strengthening the existing connections in the later phase.
Analysis of the evolution of large-scale social networks has shown that edge
locality plays a critical role in the growth of social networks [17]. The
dynamics of social groups depend on their size [18]. Small groups are more
cohesive with continued long-term, while large groups change their active
members constantly [18]. These findings help us understand the growth of a
single group, the evolution of its social network, and the influence of the
network structure on group growth. However, how the growth mechanisms
influence the distribution of members of one social system among groups is yet
to be understood.

Furthermore, it is not clear whether the growth mechanisms of social groups
are universal or system-specific. The size distribution of social groups has
not been extensively studied. Rare empirical evidence of the size distribution
of social groups indicates that it follows power-law behavior [19]. However,
the distribution of company sizes follows log-normal behavior and remains
stable over decades [20, 21]. Analysis of the cities’ sizes shows that all
cities’ distribution also follows a log-normal distribution . In contrast, the
distribution of the largest cities resembles Zipf’s distribution [22].

A related question that should be addressed is whether we can create a unique
yet relatively simple microscopic model that reproduces the distribution of
members between groups and explains the differences observed between social
systems. French economist Gibrat proposed a simple growth model to produce
companies’ and cities’ observed log-normal size distribution. However, the
analysis of the growth rate of the companies [20] has shown that growth
mechanisms are different from those assumed by Gibrat. In addition, the
analysis of the growth of the online social networks showed that the
population size and spatial factors do not determine population growth, and it
deviates from Gibrat’s law [23]. Other mechanisms, for instance, growth
through diffusion, have been used to model and predict rapid group growth
[24]. However, the growth mechanisms of various social groups and the source
of the scaling observed in socio-economic systems remain hidden.

Here we analyze the size distribution of formal social groups in three data
sets: Meetup groups based in London and New York and subreddits on Reddit. We
are interested in the scaling behavior of size distributions and the
distribution of growth rates. Empirical analysis of the dependence of growth
rates, shown in this work, indicates that growth cannot be explained through
Gibrat’s model. Here we contribute with a simple microscopic model that
incorporates some of the findings of previous research [15, 19]. We show that
the model can reproduce size and growth rate distributions for both studied
systems. Moreover, the model is flexible and can produce a broad set of log-
normal size distributions depending on the value of model parameters.

The paper is organized as follows: in Section 2 we describe the data, while in
Section 3 we present our empirical results. In Section 4 we introduce model
parameter and principles. In section 5 we demonstrate that model can reproduce
the growth of social groups in both systems and show the results for different
values of model parameters. Finally, in Section 6, we present concluding
remarks and discuss our results.

##  2 Data

We analyze the growth of social groups from two widely used online platforms:
Reddit and Meetup. Reddit 111https://www.reddit.com/ enables sharing of
diverse web content, and members of this platform interact exclusively online
through posts and comments. The Meetup 222www.meetup.com allows people to use
online tools to organize offline meetings. The building elements of the Meetup
system are topic-focused groups, such as food lovers or data science
professionals. Due to their specific activity patterns - events where members
meet face-to-face - Meetup groups are geographically localized, and
interactions between members are primarily offline.

We compiled the Reddit data from https://pushshift.io/. This site collects
data daily and, for each month, publishes merged comments and submissions in
the form of JSON files. Specifically, we focus on subreddits - social groups
of Reddit members interested in a specific topic. We selected subreddits
created between $2006$ and $2011$ that were active in $2017$ and followed
their growth from their beginning until $2011$. The considered dataset
contains $17073$ subreddits with $2195677$ active members, with the oldest
originating from $2006$ and the youngest being from $2011$. For each post
under a subreddit, we extracted the information about the member-id of the
post owner, subreddit-id, and timestamp. As we are interested in the
subreddits growth in the number of members, for each subreddit and member-id,
we selected the timestamp when a member made a post for the first time.
Finally, in the dataset, we include only subreddits active for at least two
months.

The Meetup data were downloaded in $2018$ using public API. The Meetup
platform was launched in 2003, and when we accessed the data, there were more
than 240 000 active groups. For each group, we extracted information about the
date it had been founded, its location, and the total number of members. We
focused on the groups founded in a period between $2003$ and $2017$ in big
cities, London and New York, where the Meetup platform achieved considerable
popularity. We considered groups active for at least two months. There were
4673 groups with 831685 members in London and 4752 groups with 1059632 members
in New York. In addition, we extracted the ids of group members, the
information about organized events, and which members attended these events.
Based on this, we obtained the date when a member joined a group, the first
time she participated in a group event.

For all systems, we extracted the timestamp when the member joined the group.
Based on this information, we can calculate the number of new members per
month $N_{i}(t)$, the group size $S_{i}(t)$ at each time step, and the growth
rate for each group. The time step for all three data sets is one month. The
size of the group $i$ at time step $t$ is the number of members that joined
that group ending with the month, i.e.,
$S_{i}(t)=\sum^{k=t}_{k=t_{i0}}N_{i}(t)$, where $t_{i0}$ is the time step in
which the group $i$ was created. Once the member joins the group, it has an
active status by default, which remains permanent. For these reasons, the size
of considered groups is a non-decreasing function. The growth rate $R_{i}(t)$
at step $i$ is obtained as logarithm of successive sizes
$R_{i}(t)=log(S_{i}(t)/S_{i}(t-1))$.

While the forms of communication between members and activities that members
engage in differ for considered systems, some common properties exist between
them. Members can form new groups and join the existing ones. Furthermore,
each member can belong to an unlimited number of groups. For these reasons, we
can use the same methods to study and compare the formation of groups on
Reddit and Meetup.

##  3 Empirical analysis of social group growth

Figure 1: The number of groups over time, normalized sizes distribution,
normalized log-rates distribution and dependence of log-rates and group sizes
for Meetup groups created in London and New York and subreddits. The number of
groups grows exponentially over time, while the group size distributions, and
log-rates distributions follow log-normal. Logrates depend on the size of the
group, implying that the growth cannot be explained by Gibrt law.

Figure 1 summarizes the properties of the groups in Meetup and Reddit systems.
The number of groups grows exponentially over time. Nevertheless, we notice
that Reddit has a substantially larger number of groups than Meetup. The
Reddit groups are prone to engage more members in a shorter period. The size
of the Meetup groups ranges from several members up to several tens of
thousands of members, while sizes of subreddits are between a few tens of
members up to several million. The distributions of normalized group sizes
follow the log-normal distribution (see Table S1 and Fig. S1 in SI)

$P(S)=\frac{1}{{\frac{S}{S_{0}}}\sigma\sqrt{2\pi}}exp\left(-\frac{(\ln(\frac{S}{S_{0}})-\mu)^{2}}{2\sigma^{2}}\right)\
,$ (1)

where $S$ is the group size, $S_{0}$ is the average group size in the system,
and $\mu$ and $\sigma$ are parameters of the distribution. We used power-law
package [25] to fit Eq. 1 to empirical data and found that distribution of
groups sizes for Meetup groups in London and New York follow similar
distributions with the values of parameters $\mu=-0.93$, $\sigma=1.38$ and
$\mu=-0.99$ and $\sigma=1.49$ for London and New York respectively. The
distribution of sizes of subreddits also has the log-normal shape with
parameters $\mu=-5.41$ and $\sigma=3.07$.

Multiplicative processes can generate the log-normal distributions [26]. If
there is a quantity with size $S_{i}(t)$ at time step $t$, it will grow so
after time period $\delta$ the size of the quantity is $S(t+\Delta t)=S(t)r$,
where $r$ represents a random number. The Gibrat law states that growth rates
$r$ are uncorrelated and do not depend on the current size. To describe the
growth of social groups, we calculate the logarithmic growth rates $R_{i}(t)$.
According to Gibrat law the distribution of logarithmic growth rates is
normal, or, as it is shown in many studies, it is better explained with
Laplacian (”tent-shaped ”) distribution [27], [28]. In Fig. 1 we show the
distributions of log-rates for all three data sets. Log-rates are very well
approximated with a log-normal distribution. Furthermore, the bottom panel of
Fig. 1 shows that log-rates are not independent of group size. Figure 1 shows
that these findings imply that the growth of Meetup and Reddit groups violates
the basic assumptions of the Gibrat’s law [29, 30] and that it can not be
explained as a simple multiplicative process.

We are considering a relatively significant period for online groups. The fast
expansion of information communications technologies (ICT) changed how members
access online systems. With the use of smartphones, online systems became more
available, which led to the exponential growth of ICTs systems and potential
change in the mechanisms that influence the social groups’ growth. For these
reasons, we aggregate groups according to the year they were founded for each
of the three data sets and look at the distributions of their sizes at the end
of 2017 for Meetup groups and 2011 for Reddit. For each year and each of the
three data sets, we calculate the average size of the groups created in a year
$y$ $\langle S^{y}\rangle$. We normalize the size of the groups originating in
year $y$ with the corresponding average size $s^{y}_{i}=S^{y}_{i}/\langle
S^{y}\rangle$ and calculate the distribution of the normalized sizes for each
year. The distribution of normalized sizes for all years and data sets is
shown in Fig. 2. All distributions exhibit log-normal behavior. Furthermore,
the distributions for the same data set and different years follow a universal
curve with the same value of parameters $\mu$ and $\sigma$. The universal
behavior is observed for the distribution of normalized log-rates as well, see
Fig. 2 (bottom panel). These results indicate that the growth of the social
groups did not change due to the increased growth of members in systems.
Furthermore, it implies that the growth is independent of the size of the
whole data set.

Figure 2: The figure shows the groups’ sizes distributions and log-rates
distributions. Figures in the top panel show the distribution of normalized
sizes of groups created in the same year. Distributions for the same system
and different years follow same log-normal distribution indicating existence
of universal growth patterns.

##  4 Model

The growth of social groups can not be explained by the simple rules of
Gibrat’s law. Previous research on group growth and longevity has shown that
social connections with members of a group influence individual’s choice to
join that group [24, 19]. Individuals’ interests and the need to discover new
content or activity also influence the diffusion of individuals between
groups. Furthermore, social systems constantly grow since new members join
every minute. The properties of the growth signal that describes the arrival
of new members influence both dynamics of the system [31, 32] and the
structure of social interactions [33]. The number of social groups in the
social systems is not constant. They are constantly created and destroyed.

In Ref. [19] authors propose the co-evolution model of the growth of social
networks. In this model, the authors assume that the social system evolves
through the co-evolution of two networks: a network of social contacts between
members and a network of members’ affiliations with groups. This model
addresses the problem of the growth of social networks that includes both
linking between members and social group formation. In this model, a member of
a social system selects to join a group either through random selection or
according to her social contacts. In the case of random selection, there is a
selection preference for larger groups. If a member chooses to select a group
according to her social contacts, the group is selected randomly from the list
of groups with which her friends are already affiliated.

In Ref.[19] authors demonstrate that mechanisms postulated in the model could
reproduce the power-law distribution of group sizes observed for some social
networks. However, as illustrated in Sec Ref sec:emp, the distribution of
group sizes in real systems is not necessarily power-law. Our rigorous
empirical analysis shows that the distribution of social group sizes exhibits
log-normal behavior. To fill the gap in understanding how social groups in the
social system grow, we propose a model of group growth that combines random
and social diffusion between groups but follows different rules than the co-
evolution model [19].

Figure 3: The top panel shows bipartite (member-group) and social (member-
member) network. Filled nodes are active members, while thick lines are new
links in this time step. In the social network dashed lines show that members
are friends but still do not share same groups. The lower panel shows model
schema. Example: member $u_{6}$ is a new member. First it will make random
link with node $u_{4}$, and then with probability $p_{g}$ makes new group
$g_{5}$. With probability $p_{a}$ member $u_{3}$ is active, while others stay
inactive for this time step. Member $u_{3}$ will with probability $1-p_{g}$
choose to join one of old groups and with probability $p_{aff}$ linking is
chosen to be social. As its friend $u_{2}$ is member of group $g_{1}$, member
$u_{3}$ will also join group $g_{1}$. Joining group $g_{1}$, member $u_{3}$
will make more social connections, in this case it is member $u_{1}$.

Figure 3 shows a schematic representation of our model. Similar to the co-
evolution model [19], we represent a social system with two evolving networks,
see Fig. 3. One network is a bipartite network that describes the affiliation
of individuals to social groups $\mathcal{B}(V_{U},V_{G},E_{UG})$. This
network consists of two partitions, members $V_{U}$ and groups $V_{G}$, and a
set of links $E_{UG}$, where a link $e(u,g)$ between a member $u$ and a group
$g$ represents the member’s affiliation with that group. Bipartite network
grows through three activities: the arrival of new members, the creation of
new groups, and members joining groups. In bipartite networks, links only
exist between nodes belonging to different partitions. However, as we
explained above, social connections affect whether a member will join a
certain group or not. In the simplest case, we could assume that all members
belonging to a group are connected. However, previous research on this subject
[16, 15, 19] has shown that the existing social connections of members in a
social group are only a subset of all possible connections. For these reasons,
we introduce another network $\mathcal{G}(V_{U},E_{UU})$ that describes social
connections between members. The social network grows by adding new members to
the set $V_{U}$ and creating new links between them. The member partition in
bipartite network $\mathcal{B}(V_{U},V_{G},E_{UG})$ and set of nodes in
members’ network $\mathcal{G}(V_{U},E_{UU})$ are identical.

For convenience, we represent the bipartite and social network of members with
adjacency matrices $B$ and $A$. The element of the matrix $B_{ug}$ equals one
if member $u$ is affiliated with group $g$, and zero otherwise. In matrix $A$,
the element $A_{u_{1}u_{2}}$ equals one if members $u_{1}$ and $u_{2}$ are
connected and zero otherwise. The neighborhood $\mathcal{N}_{u}$ of member $u$
is a set of groups with which the member is affiliated. On the other hand, the
neighborhood $\mathcal{N}_{g}$ of a group $g$ is a set of members affiliated
with that group. The size $S_{g}$ of set $\mathcal{N}_{g}$ equals to the size
of the group $g$.

In our model, the time is discrete, and networks evolve through several simple
rules. In each time step, we add $N_{U}(t)$ new members and increase the size
of the set $V_{U}$. For each newly added member, we create the link to a
randomly chosen old member in the social network $G$. This condition allows
each member to perform social diffusion [24], i.e., to select a group
according to her social contacts. Not all members from setting $V_{U}$ are
active in each time step. Only a subset of existing members is active in each
time step. The activity of old members is a stochastic process determined by
parameter $p_{a}$; every old member is activated with probability $p_{a}$. Old
members are activated in this way, and new members make a set of active
members $\mathcal{A}_{U}$ at time t.

The group partition $V_{G}$ grows through creating new groups. Each active
member $u\in\mathcal{A}_{U}$ can decide with probability $p_{g}$ to create a
new group or to join an already existing one with probability $1-p_{g}$.

If the active member $u$ decides that she will join an existing group, she
first needs to choose a group. A member $u$ with probability $p_{aff}$ decides
to select a group based on her social connections. For each active member, we
look at how many social contacts she has in each group. The number of social
contacts $s_{ug}$ that member $u$ has in the group $g$ equals the overlap of
members affiliated with a group $g$ and social contacts of member $u$, and is
calculated according to

$s_{ug}=\sum_{u_{1}\in\mathcal{N}_{g}}A_{uu_{1}}\ .$ (2)

Member $u$ selects an old group $g$ to join according to probability $P_{ug}$
that is proportional to $s_{ug}$. Member-only considers groups with which it
has no affiliation. However, if an active member decides to neglect her social
contacts in the choice of the social group, she will select a random group
from the set $V_{G}$ with which she is not yet affiliated.

After selecting the group $g$, a member joins that group, and we create a link
in the bipartite network between a member $u$ and a group $g$. At the same
time, the member selects $X$ members of a group $g$ which do not belong to her
social circle and creates social connections with them. As a consequence of
this action, we make $X$ new links in-network $\mathcal{G}$ between member $u$
and $X$ members from a group $g$.

The evolution of bipartite and social networks, and consequently growth of
social groups, is determined by parameters $p_{a}$, $p_{g}$ and $p_{aff}$.
Parameter $p_{a}$ determines the activity level of members and takes values
between $0$ and $1$. Higher values of $p_{a}$ result in a higher number of
active members and thus faster growth of the number of links in both networks
and the size and number of groups. Parameter $p_{g}$ in combination with
parameter $p_{a}$ determines the growth of the set $V_{G}$. $p_{g}=1$ means
that members only create new groups, and the existing network consists of
star-like subgraphs with members being central nodes and groups as leaves. On
the other hand, $p_{g}=0$ means that there is no creation of new groups, and
the bipartite network only grows through adding new members and creating new
links between members and groups.

Parameter $p_{aff}$ determines the importance of social diffusion. $p_{aff}=0$
means that social connections are irrelevant, and the group choice is random.
On the other hand, $p_{aff}=1$ means that only social contacts become
important for group selection.

Several differences exist between the model presented in this work and the co-
evolution model Ref. [19]. In our model, $p_{aff}$ is constant and the same
for all members. In the co-evolution model, this probability depends on
members’ degrees. The members are activated in our model with probability
$p_{a}$. In contrast, in the co-evolution model, members are constantly active
from the moment they are added to a set $V_{U}$ until they become inactive
after time $t_{a}$. Time $t_{a}$ differs for every member and is drawn from an
exponential distribution. In the co-evolution model, the number of social
contacts members have within the group is irrelevant to its selection. On the
other hand, in our model, members tend to choose groups more often in which
there is a greater number of social contacts. While in our model, in the case
of a random selection of a group, a member selects with equal probability a
group that she is not affiliated with, in the co-evolution model, the choice
of group is preferential.

##  5 Results

The distribution of group sizes produced by our and co-evolution models
significantly differ. The distribution of group sizes in the co-evolution
model is a power-law. Our model enables us to create groups with log-normal
size distribution and expand classes of social systems that can be modeled.

###  5.1 Model properties

First, we explore the properties of size distribution depending on parameters
$p_{g}$ and $p_{aff}$, for the fixed value of activity parameter $p_{a}$ and
constant number of members added in each step $N(t)=30$. We set the value of
parameter $X$ to $25$ for all simulations presented in this work. Our detailed
analysis of the results for different parameter values $X$ shows that these
results are independent of their value.

Figure 4: The distribution of sizes for different values of $p_{g}$ and
$p_{aff}$ and constant $p_{a}$ and growth of the system. The combination of
the values of parameters of $p_{g}$ and $p_{aff}$ determine the shape and the
width of the distribution of group sizes.

Figure 4 shows some of the selected results and their comparison with power-
law and log-normal fits. We see that values of both $p_{g}$ and $p_{aff}$
parameters, influence the type and properties of size distribution. For low
values of parameter $p_{g}$, left column in Fig. 4, the obtained distribution
is log-normal. The width of the distribution depends on $p_{aff}$. Higher
values of $p_{aff}$ lead to a broader distribution.
As we increase $p_{g}$, right column Fig. 4, the size distribution begins to
deviate from log-normal distribution. The higher the value of parameter
$p_{g}$, the total number of groups grows faster. For $p_{g}=0.5$, half of the
active members in each time step create a group, and the number of groups
increases fast. How members are distributed in these groups depends on the
parameter $p_{aff}$ value. When $p_{aff}=0$, social connections are irrelevant
to the group’s choice, and members select groups randomly. The obtained
distribution slightly deviates from log-normal, especially for large group
sizes. In this case, large group sizes become more probable than in the case
of the log-normal distribution. The non-zero value of parameter $p_{aff}$
means that the choice of a group becomes dependent on social connections. When
a member chooses a group according to her social connections, larger groups
have a higher probability of being affiliated with the social connections of
active members, and thus this choice resembles preferential attachment. For
these reasons, the obtained size distribution has more broad tail than log-
normal distribution and begins to resemble power-law distribution.

The top panel of Fig. S3 in SI shows how the shape of distribution is changing
with the value of parameter $p_{aff}$ and fixed values of $p_{a}=0.1$ and
$p_{g}=0.1$. Preferential selection groups according to their size instead of
one where a member selects a group with equal probability leads to a drastic
change in the shape of the distribution, bottom panel Fig. S3 in SI. As is to
be expected, the distribution of group sizes with preferential attachment
follows power-law behavior.

###  5.2 Modeling real systems

The social systems do not grow at a constant rate. In Ref. [33] authors have
shown that features of growth signal influence the structure of social
networks. For these reasons, we use the real growth signal from Meetup groups
located in London and New York and Reddit to simulate the growth of the social
groups in these systems. Figure 5 top panel shows the time series of the
number of new members that join each of the considered systems each month. All
three data sets have relatively low growth at the beginning, and then the
growth accelerates as the system becomes more popular.

Figure 5: The time series of the number of new members (top panel). The time
series of the ratio between several old active members and total members in
the system (middle panel); its median value approximates the parameter
$p_{a}$, the probability that the user is active. The bottom panel shows the
time series of the ratio between new groups and active members; its median
value approximates the probability that active users create a new group,
parameter $p_{g}$.

We also use empirical data to estimate $p_{a}$, $p_{g}$ and $p_{aff}$. The
data can approximate the probability that old members are active $p_{a}$ and
that new groups are created $p_{g}$. Activity parameter $p_{a}$ is the ratio
between the number of old members active in month $t$ and the total number of
members in the system at time $t$. Figure 5 middle row shows the variation of
parameter $p_{a}$ during the considered time interval for each system. The
value of this parameter fluctuates between $0$ and $0.2$ for London and New
York based Meetup groups, while its value is between $0$ and $0.15$ for
Reddit. To simplify our simulations, we assume that $p_{a}$ is constant in
time and estimate its value as its median value during the $170$ months for
Meetup and $80$ months for Reddit systems. For Meetup groups based in London
and New York $p_{a}=0.05$, while Reddit members are more active on average and
$p_{a}=0.11$ for this system.
Figure 5 bottom row shows the evolution of parameter $p_{g}$ for the
considered systems. The $p_{g}$ in month $t$ is estimated as the ratio between
the groups created in month $t$ $Ng_{new}(t)$ and the total number of groups
in that month $Ng_{new}(t)+Ng_{old}(t)$, i.e.,
$p_{g}(t)=\frac{Ng_{new}(t)}{N_{new}(t)+N_{old}(t)}$. We see from Fig. 5 that
$p_{g}(t)$ has relatively high values at the beginning of the system’s
existence. This is not surprising. Initially, these systems have a relatively
small number of groups and often cannot meet the needs of the content of all
their members. As the time passes, the number of groups and content scope
within the system grows, and members no longer have a high need to create new
groups. Figure Ref fig:fig5 shows that $p_{g}$ fluctuates less after the first
few months, and thus we again assume that $p_{g}$ is constant in time and set
its value to the median value during 170 months for Meetup and 80 months for
Reddit. For all three systems $p_{g}$ has the value of $0.003$.
The affiliation parameter $p_{aff}$ cannot estimate directly from the
empirical data. For these reasons, we simulate the growth of social groups for
each data set with the time series of new members obtained from the real data
and estimated values of parameters $p_{a}$ and $p_{g}$, while we vary the
value of $p_{aff}$. We compare the distribution of group sizes obtained from
simulations for different values of $p_{aff}$ with ones obtained from
empirical analysis using Jensen Shannon (JS) divergence. The JS divergence
[34] between two distributions $P$ and $Q$ is defined as

$JS(P,Q)=H\left(\frac{P+Q}{2}\right)-\frac{1}{2}\left(H(P)+H(Q)\right)$ (3)

where $H(p)$ is Shannon entropy $H(p)=\sum_{x}p(x)log(p(x)$. The JS divergence
is symmetric and if $P$ is identical to $Q$, $JS=0$. The smaller the value of
JS divergence, the better is the match between empirical and simulated group
size distributions. Table 1 shows the value of JS divergence for all three
data sets. We see that for London based Meetup groups the affiliation
parameter is $p_{aff}=0.5$, for New York groups $p_{aff}=0.4$, while the
affiliation parameter for Reddit $p_{aff}=0.8$. Our results show that social
diffusion is important in all three data sets. However, Meetup members are
more likely to join groups at random, while for the Reddit members their
social connections are more important when it comes to choice of the
subreddit.

$p_{aff}$ | JS cityLondon | JS cityNY | JS reddit2012
---|---|---|---
0.1 | 0.0161 | 0.0097 | 0.00241
0.2 | 0.0101 | 0.0053 | 0.00205
0.3 | 0.0055 | 0.0026 | 0.00159
0.4 | 0.0027 | 0.0013 | 0.00104
0.5 | 0.0016 | 0.0015 | 0.00074
0.6 | 0.0031 | 0.0035 | 0.00048
0.7 | 0.0085 | 0.0081 | 0.00039
0.8 | 0.0214 | 0.0167 | 0.00034
0.9 | 0.0499 | 0.0331 | 0.00047
Table 1: Jensen Shannon divergence between group sizes distributions from
model and data. In model we vary affiliation parameter $p_{aff}$ and find its
optimal value (bold text).

Figure Ref fig:fig6 compares the empirical and simulation distribution of
group sizes for considered systems. We see that empirical distributions for
Meetup groups based in London and New York are well reproduced by the model
and chosen values of parameters. In the case of Reddit, the distribution is
broad, and the model reproduces the tail of the distribution well. Figure S2
and Table S2 in SI confirm that the distribution of group sizes follow a log-
normal distribution.
The bottom row of Fig. 6 shows the distribution of logarithmic values of
growth rates of groups obtained from empirical and simulated data. We see that
the tails of empirical distributions for all three data sets are well emulated
by the ones obtained from the model. The deviations we observe are the most
likely consequence of using median values of parameters $p_{a}$, $p_{g}$, and
$p_{aff}$.

Figure 6: The comparison between empirical and simulation distribution for
group sizes (top panel) and log-rates (bottom panel).

##  6 Discussion and conclusions

The results of empirical analysis show that there are universal growth rules
that govern the growth of social systems. We analysed the growth of social
groups for three data sets, Meetup groups located in London and New York and
Reddit. We showed that the distribution of group sizes has log-normal
behaviour. The empirical distributions of normalised sizes of groups created
in different years in a single system fall on top of each other, following the
same log-normal distributions. Due to a limited data availability, we only
study three data sets which may affect the generality of our results. However,
the substantial differences between Reddit and Meetup social systems when it
comes to their popularity, size and purpose, demonstrate that observed growth
patterns are universal.

Even though the log-normal distribution of group sizes can originate from the
proportional growth model, Gibrat law, we show that it does not apply to the
growth of online social groups. The monthly growth rates are log-normally
distributed and dependent on the size of a group. Gibrat law was proposed to
describe the growth of various socio-economical systems, including the cities
and firms. Recent studies showed that the growth of cities and firms [35, 21,
36] goes beyond Gibrat law. Still, our findings confirm the existence of
universal growth patterns, indicating the presence of the general law in the
social system’s growth.

The model proposed in Ref. [19] is able to produce only power-law
distributions of group sizes. However, our empirical analysis shows that these
distributions can also have a log-normal behavior. Thus, we propose a new
model that emulate log-normal distributions. The analysed groups grow through
two mechanisms [19]: members join a group that is chosen according to their
interests or by social relations with the group’s members. The number of
members in the system is growing as well as the number of groups. While the
processes that govern the growth of social groups are the same, their
importance varies among the systems. The distributions for Meetup groups
located in the London and New York have similar log-normal distribution
parameter values, while for Reddit, the distribution is broader. Numerical
simulations further confirm these findings. Different modalities of
interactions between their members can explain the observed differences.

Meetup members need to invest more time and resources to interact with their
peers. The events are localised in time and space, and thus the influence of
peers in selecting another social group may be limited. On the other hand,
Reddit members do not have these limitations. The interactions are online,
asynchronous, and thus not limited in time. The influence of peers in choosing
new subreddits and topics thus becomes more important. The values of $p_{aff}$
parameters for Meetup and Reddit imply that social connections in diffusion
between groups are more critical in Reddit than in Meetup.

The results presented in this paper contribute to our knowledge of the growth
of socio-economical systems. The previous study analysed the social systems in
which size distributions follow the power-law, which is the consequence of a
preferential choice of groups during the random diffusion of members. Our
findings show that preferential selection of groups during social diffusion
and uniform selection during random diffusion result in log-normal
distribution of groups sizes. Furthermore, we show that broadness of the
distribution depends on the involvement of social diffusion in the growth
process. Our model increases the number of systems that can be modelled and
help us better understand the growth and segmentation of social systems and
predict their evolution.

## References

## References

  * [1] Claudio Castellano, Santo Fortunato, and Vittorio Loreto.  Statistical physics of social dynamics.  Reviews of modern physics, 81(2):591, 2009.
  * [2] Arnab Chatterjee, Marija Mitrović, and Santo Fortunato.  Universality in voting behavior: an empirical analysis.  Scientific reports, 3(1):1–9, 2013.
  * [3] Filippo Radicchi, Santo Fortunato, and Claudio Castellano.  Universality of citation distributions: Toward an objective measure of scientific impact.  Proceedings of the National Academy of Sciences, 105(45):17268–17272, 2008.
  * [4] Raymond Firth.  Elements of social organisation.  Routledge, 2013.
  * [5] Marc Barthelemy.  The structure and dynamics of cities.  Cambridge University Press, 2016.
  * [6] César A Hidalgo and Ricardo Hausmann.  The building blocks of economic complexity.  Proceedings of the national academy of sciences, 106(26):10570–10575, 2009.
  * [7] Jelena Smiljanić, Arnab Chatterjee, Tomi Kauppinen, and Marija Mitrović Dankulov.  A theoretical model for the associative nature of conference participation.  PloS one, 11(2):e0148528, 2016.
  * [8] Ali Montazeri, Soghra Jarvandi, Shahpar Haghighat, Mariam Vahdani, Akram Sajadian, Mandana Ebrahimi, and Mehregan Haji-Mahmoodi.  Anxiety and depression in breast cancer patients before and after participation in a cancer support group.  Patient education and counseling, 45(3):195–198, 2001.
  * [9] Kathryn P Davison, James W Pennebaker, and Sally S Dickerson.  Who talks? the social psychology of illness support groups.  American Psychologist, 55(2):205, 2000.
  * [10] Wendy K Tam Cho, James G Gimpel, Daron R Shaw, et al.  The tea party movement and the geography of collective action.  Quarterly Journal of Political Science, 7(2):105–133, 2012.
  * [11] Sinan Aral and Dylan Walker.  Identifying influential and susceptible members of social networks.  Science, 337(6092):337–341, 2012.
  * [12] Sandra González-Bailón, Javier Borge-Holthoefer, and Yamir Moreno.  Broadcasters and hidden influentials in online protest diffusion.  American behavioral scientist, 57(7):943–965, 2013.
  * [13] János Török, Gerardo Iñiguez, Taha Yasseri, Maxi San Miguel, Kimmo Kaski, and János Kertész.  Opinions, conflicts, and consensus: modeling social dynamics in a collaborative environment.  Physical review letters, 110(8):088701, 2013.
  * [14] Taha Yasseri, Robert Sumi, András Rung, András Kornai, and János Kertész.  Dynamics of conflicts in wikipedia.  PloS one, 7(6):e38869, 2012.
  * [15] Lars Backstrom, Dan Huttenlocher, Jon Kleinberg, and Xiangyang Lan.  Group formation in large social networks: membership, growth, and evolution.  In Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 44–54, 2006.
  * [16] Jelena Smiljanić and Marija Mitrović Dankulov.  Associative nature of event participation dynamics: A network theory approach.  PloS one, 12(2):e0171565, 2017.
  * [17] Jure Leskovec, Lars Backstrom, Ravi Kumar, and Andrew Tomkins.  Microscopic evolution of social networks.  In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 462–470, 2008.
  * [18] Gergely Palla, Albert-László Barabási, and Tamás Vicsek.  Quantifying social group evolution.  Nature, 446(7136):664–667, 2007.
  * [19] Elena Zheleva, Hossam Sharara, and Lise Getoor.  Co-evolution of social and affiliation networks.  In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 1007–1016, 2009.
  * [20] Luís A Nunes Amaral, Sergey V Buldyrev, Shlomo Havlin, Heiko Leschhorn, Philipp Maass, Michael A Salinger, H Eugene Stanley, and Michael HR Stanley.  Scaling behavior in economics: I. empirical results for company growth.  Journal de Physique I, 7(4):621–633, 1997.
  * [21] Michael HR Stanley, Luis AN Amaral, Sergey V Buldyrev, Shlomo Havlin, Heiko Leschhorn, Philipp Maass, Michael A Salinger, and H Eugene Stanley.  Scaling behaviour in the growth of companies.  Nature, 379(6568):804–806, 1996.
  * [22] Giorgio Fazio and Marco Modica.  Pareto or log-normal? best fit and truncation in the distribution of all cities.  Journal of Regional Science, 55(5):736–756, 2015.
  * [23] Konglin Zhu, Wenzhong Li, Xiaoming Fu, and Jan Nagler.  How do online social networks grow?  Plos one, 9(6):e100023, 2014.
  * [24] Sanjay Ram Kairam, Dan J Wang, and Jure Leskovec.  The life and death of online groups: Predicting group growth and longevity.  In Proceedings of the fifth ACM international conference on Web search and data mining, pages 673–682, 2012.
  * [25] Jeff Alstott, Ed Bullmore, and Dietmar Plenz.  powerlaw: A python package for analysis of heavy-tailed distributions.  PLOS ONE, 9(1):1–11, 01 2014.
  * [26] Michael Mitzenmacher.  A brief history of generative models for power law and lognormal distributions.  Internet mathematics, 1(2):226–251, 2004.
  * [27] Hernan Mondani, Petter Holme, and Fredrik Liljeros.  Fat-tailed fluctuations in the size of organizations: the role of social influence.  PLoS One, 9(7):e100527, 2014.
  * [28] Dongfeng Fu, Fabio Pammolli, Sergey V Buldyrev, Massimo Riccaboni, Kaushik Matia, Kazuko Yamasaki, and H Eugene Stanley.  The growth of business firms: Theoretical framework and empirical evidence.  Proceedings of the National Academy of Sciences, 102(52):18801–18806, 2005.
  * [29] Gerald F Frasco, Jie Sun, Hernán D Rozenfeld, and Daniel Ben-Avraham.  Spatially distributed social complex networks.  Physical Review X, 4(1):011008, 2014.
  * [30] Jiang-Hai Qian, Qu Chen, Ding-Ding Han, Yu-Gang Ma, and Wen-Qing Shen.  Origin of gibrat law in internet: Asymmetric distribution of the correlation.  Physical Review E, 89(6):062808, 2014.
  * [31] Marija Mitrović, Georgios Paltoglou, and Bosiljka Tadić.  Quantitative analysis of bloggers’ collective behavior powered by emotions.  Journal of Statistical Mechanics: Theory and Experiment, 2011(02):P02005, 2011.
  * [32] Marija Mitrović Dankulov, Roderick Melnik, and Bosiljka Tadić.  The dynamics of meaningful social interactions and the emergence of collective knowledge.  Scientific reports, 5(1):1–10, 2015.
  * [33] Ana Vranić and Marija Mitrović Dankulov.  Growth signals determine the topology of evolving networks.  Journal of Statistical Mechanics: Theory and Experiment, 2021(1):013405, 2021.
  * [34] Jop Briët and Peter Harremoës.  Properties of classical and quantum jensen-shannon divergence.  Phys. Rev. A, 79:052311, May 2009.
  * [35] Edwin Mansfield.  Entry, gibrat’s law, innovation, and the growth of firms.  The American economic review, 52(5):1023–1051, 1962.
  * [36] Marc Barthelemy.  The statistical physics of cities.  Nature Reviews Physics, 1(6):406–415, 2019.