# Information Access Equality on
Network Generative Models

Xindi Wang
Network Science Institute
Northeastern University
Boston, MA
<EMAIL_ADDRESS>
&Onur Varol
Faculty of Engineering and Natural Sciences
Sabancı University
Istanbul, Turkey
<EMAIL_ADDRESS>
&Tina Eliassi-Rad
Network Science Institute
Khoury College of Computer Sciences
Northeastern University
Boston, MA
<EMAIL_ADDRESS>

###### Abstract

It is well known that networks generated by common mechanisms such as
preferential attachment and homophily can disadvantage the minority group by
limiting their ability to establish links with the majority group. This has
the effect of limiting minority nodes’ access to information. We present the
results of an empirical study on the equality of information access in network
models with different growth mechanisms and spreading processes. For growth
mechanisms, we focus on the majority/minority dichotomy, homophily,
preferential attachment, and diversity. For spreading processes, we
investigate simple vs. complex contagions, different transmission rates within
and between groups, and various seeding conditions. We observe two phenomena.
First, information access equality is a complex interplay between network
structures and the spreading processes. Second, there is a trade-off between
equality and efficiency of information access under certain circumstances
(e.g., when inter-group edges are low and information transmits
asymmetrically). Our findings can be used to make recommendations for
mechanistic design of social networks with information access equality.

##  1 Introduction

The early hopes that an increasingly interconnected world will provide greater
equality of opportunity and remove information barriers are tempered by
observations of echo chambers and polarisation on social networks [1, 2, 3, 4,
5, 6]. Recent studies have also found that discrimination in social networks
can arise through simple mechanisms, for example, homophily and minority group
size can lead to a glass-ceiling effect [7, 8], a chasm effect [9], influence
the ranking of minority nodes [10, 11], and create perception bias in social
networks [12]. Furthermore, a user’s popularity and prestige on social
networking platforms can be altered through the use of automation and social
bots to gain unfair advantages. [13, 14, 15, 16, 17].

An important role of networks is that they disseminate information, from job
and business opportunities [18] to medical resources that can be accessed
[19], and such information is critical to people’s lives. However, inequality
in access to information in a network has not been fully explored. This is a
topic that has been discussed in the social and political sciences, with
examples such as economically disadvantaged people having less access to new
technologies such as the Internet Internet [20], and groups starting out with
poorer employment status being more likely to experience persistent
unemployment in the labour market [21].

A closely related research topic to information dissemination is information
maximization, which aims to find optimal starting (or seeding) nodes in a
network that maximize information spreading [22]. This problem has been
studied over the years, but the fairness aspect in this problem is relatively
new. For example, Stoica and Chaintreau [23] investigate two fairness criteria
for information maximization: fairness for early adopters (where seeds should
be proportional to the group population) and fairness in outreach (where final
outreach nodes should reflect the group population). They experiment on a
social network extracted from Instagram. In subsequent work, Stoica et al.
[24] proposed diversity-enhancing interventions in seeding and explored the
complicated relationship between diversity and efficiency. Other research
explored how fairness constraints can be incorporated into the information
maximization algorithm, including the maximin constraint (which requires that
the least wealthy group should be improved during the optimization process)
[25, 26], diversity constraint [27], welfare theory [28], and adding a time-
sensitive constraint on information [29]. These papers shed some initial light
on information equality in networks, but they mostly consider the problem as
an algorithmic problem (given a fixed graph, what are the optimal information
seeds), rather than a characteristic problem (what kind of network can better
promote information equality). Moreover, the spreading process they consider
is rather limited. Most of the work is concerned with studying the information
cascade model, which may not be representative of real-world processes.
Different from previous works, Jalali et al. [30] define a criterion for
information unfairness (based on whether information flows equally among all
groups in a network) and present an algorithm that adds edges to the network
to reduce the information unfairness criterion. They experiment on a social
network extracted from DBLP. This study does not provide much information
about the performance of various spreading processes on different networks. In
2021, Venkatasubramanian et al. [31] produced a much-needed tutorial on
fairness in networks. Their tutorial covers social capital, information
access, and interventions.

Here, we study the problem of information access equality in networks from a
characteristic point of view. Using several network models with two mutually
exclusive groups in the population (majority vs. minority), we generate
networks with different properties and constraints that are representative of
mechanisms in real networks. We define information access equality as follows:
_for a given process and seeds, the majority and minority nodes should receive
information at similar rates across various stages of the spreading
processes_. Our goal is to provide insight into which network characteristics
may affect information access equality and in what ways, and to make
recommendations for systematic mechanism design. We find that, in general,
homophily and preferential attachment can harm information access equality,
while introducing diversity can promote information access equality. However,
too much diversity can affect the efficiency of information spreading. We also
find that information access equality depends not only on the network, but
also on the characteristics of the spreading process.

##  2 Network Models

Our study includes several generative models for complex networks. Each model
generates a network with two groups of nodes: majority and minority. The
proportion of minority nodes is $m$. The majority/minority dichotomy is based
on population size: minority nodes make up less than 50% of the network (i.e.
$m<0.5$). A real-world example of such a dichotomy is a computer science
collaboration network where male scientists are in the majority and female
scientists are in the minority.

The generative models produce undirected, unweighted networks. Starting from
an initialized network, a new node enters the network at each time step and
connects to some nodes according to a mechanism mandated by the model. The
initialization procedure is the same across the models: a node from the
majority group connects to a node from the minority group. Table 1 summarizes
the properties of each model and provides comparison between the proposed
models.

Random Network |  | Preferential
---
Attachment
Homophily | Diversity | Resultant Network
✓ | ✓ |  |  | BA
✓ |  | ✓ |  | Random Homophily
✓ | ✓ | ✓ |  | Homophily BA
✓ |  | ✓ | ✓ |  | Diversified
---
Homophily
✓ | ✓ | ✓ | ✓ |  | Diversified
---
Homophily BA
Table 1: Relationship between the Network Models. All network models are based
on Random Network, a growing network adaptation of the Erdős-Rényi random
network [32]. We focus on three mechanisms: preferential attachment,
homophily, and diversity (through adding inter-group edges). Starting from a
Random Network and adding preferential attachment results in the BA model.
Starting from a Random Network and adding homophily results in the Random
Homophily model. Starting from a Random Network and adding both preferential
attachment and homophily results in the Homophily BA model. Starting from
Random Homophily and adding diversity results in the Diversified Homophily
model. Starting from Homophily BA and adding diversity gives us Diversified
Homophily BA model. In our study, we treat BA and Random Homophily as the
variations of the Homophily BA model, and Diversified Homophily as a variation
of the Diversified Homophily BA.

### Random Network

In the Random Network model, at each time step a new node enters with
probability $m$ as the minority and $1-m$ as the majority. With uniform
probability, each node connects to $l$ existing nodes. This is a growing
version of the Erdős-Rényi Random Network [32].

### Homophily BA

In the Homophily BA model [10, 12],111BA is short for Barabási-Albert. there
are two ingredients in link formation: preferential attachment and homophily.
The parameters of Homophily BA include: (1) the proportion of minority nodes
$m$, (2) the number of edges $l$ for each new node, (3) the homophily matrix
$H$ with entries $h_{g_{i}g_{j}}$, and (4) the preferential attachment
strength $\alpha$. $g_{i}$ and $g_{j}$ denote the group memberships for nodes
$i$ and $j$, respectively. Each node $i$ can either belong to the majority
(maj) or the minority (min) group. Thus, we have the following homophily
matrix $H$:

$H=\begin{bmatrix}h_{maj,maj}&h_{maj,min}\\\
h_{min,maj}&h_{min,min}\end{bmatrix}$

For simplicity, we assume $H$ is a doubly stochastic matrix, with
$h_{maj,min}=h_{min,maj}$ and $h_{maj,maj}=h_{min,min}$. We define
$h=h_{maj,maj}=h_{min,min}$. When $h=1$, the network is perfectly homophilic,
i.e., the network has two distinct groups: majority and minority. The groups
are connected with a single inter-group edge (due to network initialization).
On the other hand, if $h=0$, the network is perfectly heterophilic. It has
been found that homophily can be different for different groups [33, 10]. We
leave the exploration of asymmetric homophily for future work.

The Homophily BA networks grow as follows.

  * •

At each time step, a new node $j$ enters with probability $m$ as the minority
and $1-m$ as the majority. Denote its group as $g_{j}$.

  * •

Node $j$ connects to $l$ nodes. Each connection to node $i$ in the network is
made with probability $\Pi_{i}$:

$\Pi_{i}=\frac{h_{g_{j}g_{i}}d_{i}^{\alpha}}{\sum_{i}{h_{g_{j}g_{i}}d_{i}^{\alpha}}},$
(1)

where $d_{i}$ is the degree of node $i$.

We are also interested in two special cases of Homophily BA: (1) without
preferential attachment (Random Homophily with $\alpha=0$) and (2) with random
mixing (BA [34] with $h=0.5$).

### Diversified Homophily BA

We propose the Diversified Homophily BA model to encourage inter-group
connections while maintaining some degree of homophily. The parameters of the
Diversified Homophily BA include: (1) the proportion of minority nodes $m$,
(2) the number of edges $l$ for each new node, (3) the homophily matrix $H$
(with the same assumption as in Homophily BA), (4) preferential attachment
strength $\alpha$, (5) the number of diversified edges for each node $l_{d}$,
and (6) the diversification probability $p_{d}$. The Diversified Homophily BA
networks grow as follows:

  * •

At each time step, a new node $j$ enters with probability $m$ as the minority
and $1-m$ as the majority. Denote its group as $g_{j}$.

  * •

Node $j$ forms $l-l_{d}$ links following Homophily BA mechanism
$\Pi_{i}=\frac{h_{g_{j}g_{i}}k_{i}^{\alpha}}{\sum_{i}{h_{g_{j}g_{i}}k_{i}^{\alpha}}}$.
Denote the nodes connected at this step as $S_{j}$.

  * •

Node $j$ forms $l_{d}$ diversified links. Given diversification probability
$p_{d}$, we have the connecting probability of two nodes $j$ and $k$ as

$p_{jk}=\begin{cases}p_{d},&{g_{j}\neq g_{k}}\\\
1-p_{d},&{g_{j}=g_{k}}.\end{cases}$ (2)

For node $i\in S_{j}$, we obtain their neighbors, denoted as $N_{S_{j}}$.
Generate $l_{d}$ links by connecting to $k\in N_{S_{j}}$ with probability
$\Pi_{jk}\propto p_{jk}\times\frac{1}{|d_{k}-d_{i}|}$, where $d_{i}$ is the
degree of node $i\in S_{j}$. The idea behind this step is to connect to nodes
which are of opposite group, but with similar degree to existing neighbors.

We are interested in a special case under Diversified Homophily BA with
$\alpha=0$, which removes preferential attachment. We call this case
Diversified Homophily.

##  3 Information Access Equality

There is currently no consensus on how to measure information access equality
in a network. In this work, we choose to measure it experimentally, i.e., by
simulating spreading processes on the network and observing the differences in
spreading among majority and minority nodes. This allows us to explore more
possibilities in the process configurations, such as the type of contagion,
the transmission rate, and the information seeds.

### Varying Processes that Spread Information

We consider several variations of dynamic processes. The first variation we
consider is the distinction between simple vs. complex contagion. Studies have
found that while simple contagion is appropriate for most diseases and the
spread of information, complex contagion is more common for collective
behaviors, such as the spread of new technologies and innovations, the growth
of social movements, spread of misinformation, etc. [35, 36, 37, 38, 39, 40,
41, 42, 43, 44]. Same network properties may respond differently to these two
types of contagion. For example, in simple contagion, weak ties are considered
important because they can disseminate information to an isolated part of the
network. However, in complex contagion, weak ties might not be as useful
because one link is not sufficient to disseminate information [45, 35]. For
modeling, we rely on a Susceptible-Infectious (SI) model [46] for both simple
and complex contagion. For complex contagion, we add the parameter of
activation threshold $a$, where each node can only be infected if $a$ portion
of its neighborhood is infected.

The second variation we consider is the group difference in information
transmission rate. Although the transmission rate can be different at the node
level [47], we study the simple case where the transmission rate differs by
group. More specifically, we are interested in the case where the transmission
rate between all nodes is the same (symmetric) and the case where the
transmission rate between different groups is lower than within the same group
(asymmetric). There are four sets of transmission rates: $r_{maj\rightarrow
maj}$, $r_{min\rightarrow min}$, $r_{maj\rightarrow min}$ and
$r_{min\rightarrow maj}$. Here we assume that $r_{maj\rightarrow
maj}=r_{min\rightarrow min}=r_{within}$ and $r_{maj\rightarrow
min}=r_{min\rightarrow maj}=r_{between}$. We simulate the SI process with
symmetric transmission rate as $r_{within}=r_{between}$, and asymmetric
transmission rate as $r_{within}>r_{between}$.

The last variation we consider is the location where the information is
seeded. Different seeding locations can advantage or disadvantage a group. For
example, imagine a network with two groups (majority vs. minority) connected
by a single edge. Under this circumstance, if all seeds belong to the majority
community, we would expect the minority group to be disadvantaged. To test the
influence of seeding, we assume random seeding (as opposed to targeted
seeding, such as selecting higher degree nodes). We assume different portion
of minority seeds: low (minority seeds below 30%), mid (minority seeds between
30% and 70%), and high (minority seeds above 70%). For brevity, we show only
the low and high minority seeding portions. The results for mid and high
minority seeding portions are similar.

### Measuring Information Access Equality

Since spreading processes can take different times in different networks, we
first normalize the fraction of nodes in state I (Infected) at each time step
$I(t)$ by the length of the spreading process T to obtain $I(t/T)$. Then, we
separate majority ($I_{maj}(t/T)$) and minority ($I_{min}(t/T)$) groups and
calculate the relative difference between them as

$\Delta I(t/T)=\frac{I_{maj}(t/T)-I_{min}(t/T)}{I_{maj}(t/T)+I_{min}(t/T)}.$

We choose the denominator to be $I_{min}(t/T)+I_{maj}(t/T)$, which corresponds
to twice the mean of $I_{maj}(t/T)$ and $I_{min}(t/T)$, making this metric
symmetric and bounded between -1 and 1. When there is information access
equality, $\Delta I(t/T)=0$; and the closer $\Delta I(t/T)$ is to 0, the
greater the equality between the two groups. $\Delta I(t/T)>0$ means that the
majority group has a greater advantage and $\Delta I(t/T)<0$ means that the
minority group has a greater advantage. This measure is intuitive and it
allows us to inspect information access equality at different stages of the
spreading process.

## Network Measures

To understand the structure and properties of the network, and to gain insight
into possible roots of information access inequality, we examine several
network measures. The first set of measures is dyadicity and heterophilicity
[48]. The second set of measures are regarding the distance in the network,
which is directly related to information spreading. And the last set of
measures are regarding degree inequalities between the two groups, as a
measure of difference in social capital [49, 50].

### Homophily: Dyadicity and Heterophilicity

The homophily parameter $h$ in Homophily BA and Diversified Homophily BA may
not reflect the actual homophily level of the final network. To measure the
homophily effect more accurately, we calculate the dyadicity and
heterophilicity score of the network [48]. For a network with random mixing of
group membership, dyadicity and heterophilicity are expected to be close to 1.

### Distance: Average Shortest Path Length and Diameter

We measure the average shortest path length and the diameter of the network.
These two measures are directly related to the spreading efficiency. The
higher the shortest path length or diameter, the longer it takes for
information to spread to the entire network.

### Social Capital: Degree Equality

We are interested in whether there is a relationship between equality of
social capital and equality of information access. We use several measures of
degree equality.

##### Earth Mover Distance (EMD).

We calculate the Earth Mover Distance (EMD) between two distributions. The
smaller the distance, the more equal the two groups are.

##### Power Inequality (PI).

Power inequality [7] measures the “power" (average degree) ratio of the
minority and majority groups. Let $\bar{d}_{min}$ and $\bar{d}_{maj}$ denote
the average degrees of the minority and majority nodes, respectively. The
power inequality is defined as $PI=\frac{\bar{d}_{min}}{\bar{d}_{maj}}$. A
network with power equality should have $PI=1$; and the lower the $PI$, the
more disadvantaged are minority nodes.

##### Moment Glass Ceiling (g).

Moment glass ceiling measures the glass-ceiling effect in a network [51, 7].
The intuition behind it is that a larger second moment (and assuming a similar
average degree, i.e., no power inequality) leads to a larger variance in the
distribution and thus a significantly larger number of nodes with high degree
(i.e., hubs). Let $E(d_{min}^{2})$ and $E(d_{maj}^{2})$ denote the second
moment of the degree distribution for the minority and majority nodes,
respectively. The moment glass ceiling is defined as
$g=\frac{E(d_{min}^{2})}{E(d_{maj}^{2})}$. A network with no glass-ceiling
effect should have $g=1$; and the lower the $g$, the more disadvantaged are
minority nodes.

##  4 Experiments

For simplicity, we control the parameter space for the information spreading
simulations. For all synthetic models, we set the number of seed nodes to be
$s=10$. In symmetric transmission, $r_{within}=r_{between}=0.7$. In asymmetric
transmission, $r_{within}=0.7$ and $r_{between}=0.3$. The activation threshold
for complex contagion is $a=0.1$.

### Experiment 1: Information Access Equality across Different Network Models

We create networks with $N=5000$ nodes, $m=0.2$ (i.e., 20% of the nodes are
minority nodes) and $l=2$. We set $h=0.8$ and $\alpha=1$ in Homophily BA and
Diversified Homophily BA, including their variations (namely, Random Homophily
and Diversified BA). For Diversified Homophily BA, we set $l_{d}=1$ and
$p_{d}=0.6$.

#### Network Measures

In Fig. 1a and Fig. 1b, we compare the networks generated by the various
models based on their degree distributions and statistics on degrees, edges,
dyadicity, heterophilicity, and average shortest path and diameter. For the
degree distribution, we show the result of one sample network under each
model. For all other statistics, we show the average of 20 realizations.

As expected, we observe that Random Network has dyadicity and heterophilicity
of 1, and BA is similar. For the other models, we note that the dyadicity for
both groups is above 1 and the heterophilicity is below 1, suggesting that
there are fewer edges between the groups than in random mixing. We also note
that, except for the Random Network, the dyadicity in the minority group is
larger than the dyadicity in the majority group. Homophily BA and its
variation have higher dyadicity and lower heterophilicity compared to
Diversified Homophily BA and its variation. This observation makes sense since
Diversified Homophily BA intentionally introduces inter-group edges.

We observe that Homophily BA and BA have lower average shortest path lengths
and diameters due to the presence of hubs. Diversified Homophily BA has a
slightly higher average shortest path length and diameter than Random Network
and Random Homophily, while Diversified Homophily has even higher values,
possibly due to the elimination of preferential attachment.

Figure 1c shows the social captial (degree) equality measures across different
models. For Earth Mover Distance, Power Inequality, and Moment Glass Ceiling,
the rankings of the models from more equality to less equality is Random
Network or BA, followed by Diversified Homophily or Diversified Homophily BA,
and lastly Random Homophily and Homophily BA.

Figure 1: Statistics of Network Models. All networks have $N=5000$ nodes,
where each node joins the network with $l=2$ edges and 20% of the nodes are
minority. We set $h=0.8$, $e_{o}=1$, $\alpha=1$ and $p_{d}=0.6$. Degree
distributions (a) and Basic Statistics (b). Compared to Random Network, we see
that Homophily BA has a longer tail in the degree distribution, a larger gap
in average and maximum degree between the two groups, and a much smaller
proportion of majority-minority edges, leading to an increase in dyadicity and
a decrease in heterophilicity. Random Homophily decreases the portion of
majority-minority edges more than Homophily BA, but it has a smaller gap in
degree for the two groups, resulting from the absence of Preferential
Attachment, leading to heavy-tail degree distribution. BA model has a larger
gap in maximum degree differences between the two groups compared to Random
Network, but smaller than Homophily BA. The edge composition is similar to
Random Network. Compared to Homophily BA, Diversified Homophily BA has a
smaller gap in the degrees and also a higher portion of majority-minority
edges, resulting in a higher heterophilicity score. A similar phenomenon is
observed in Diversified Homophily but with a smaller maximum degree for both
groups due to the absence of Preferential Attachment. Looking at the average
shortest path length and diameter, we find that Diversified Homophily BA and
its variation have significantly higher average shortest path length and
diameter, while Homophily BA and BA have lower average shortest path length
and diameter. (c) Social Captial Equality Measures. For each equality measure,
we sort the generative models from most equal (left) to least equal (right).
The earth mover distance is the distance between the degree distributions of
the minority and majority groups. The power inequality is the ratio of the
average degree in the minority group to the average degree in the majority
group. The moment glass ceiling is the ratio of the second moment of degree
distribution in the minority group to the second moment of degree distribution
in the majority group. Across all three equality measures, from more equal to
less equal, the rankings are Random Network or BA, followed by Diversified
Homophily or Diversified Homophily BA, and lastly Random Homophily or
Homophily BA.

#### Information Access Equality

In Fig. 2, we plot information access equality as a heatmap, where the x-axis
represents the different stages of the process $t/T$, and the y-axis is the
different models. The color of each cell shows $\Delta I(t/T)$. $\Delta
I(t/T)=0$ is colored white (denotes equality); red color means $\Delta
I(t/T)>0$ and the majority group is at an advantage, while blue color means
$\Delta I(t/T)<0$ and the minority group is at an advantage. We omit the
performances under mid minority seeding because we found they are very close
to performance under high seeding portion.

Under low minority seeding portion, the majority group initially has an
advantage ($\Delta I(t/T)>0$) for all models, and subsequently converges to
equality ($\Delta I(t/T)=0$). Comparing the models, we observe that
Diversified Homophily BA and its variation take shorter time to reach
equality, while Homophily BA and Random Homophily take longer time. When the
minority seeding portion is high, the minority group initially has an
advantage ($\Delta I(t/T)<0$) and then converges to equality ($\Delta
I(t/T)=0$). Under asymmetric transmission, majority group has an advantage
($\Delta I(t/T)>0$) in the middle of the process for some models. We again
observe that Diversified Homophily BA and its variation take shorter time to
reach equality, while Homophily BA and Random Homophily take longer time.

In summary, Homophily BA and Random Homophily achieve the lowest information
access equality, followed by Random Network, BA and Diversified Homophily BA
and their variation. The ranking between Random Network, BA, and Diversified
Homophily BA depends on the process. Although Random Network and BA are most
equal in degree equality measures and have the highest heterophilicity score,
they are not always the most equal in information access equality (e.g., under
complex contagion and asymmetric transmission rate).

Figure 2: Information Spreading Equality of Different Network Models. We plot
the spreading equality with low and high minority seeding portions. Each plot
is a heatmap, where the x-axis represents the relative time $t/T$, the y-axis
represents the different models, and the color represents $\Delta I(t/T)$.
Recall that $\Delta I(t/T)=0$ represents equality. When the minority seeding
portion is low, $\Delta I(t/T)$ is initially positive (i.e., the majority
group has a greater advantage); it then decreases to 0 for all models. When
the minority seeding portion is high, $\Delta I(t/T)$ is initially negative
(i.e., the minority group has a greater advantage). Under symmetric
transmission, $\Delta I(t/T)$ increases to 0. However under asymmetric
transmission, $\Delta I(t/T)$ increases to 0, then becomes positive, and
eventually becomes 0. We observe that regardless of the contagion type and the
seeding condition, Homophily BA and Random Homophily take longer to reach
$\Delta I(t/T)=0$, indicating less equality. On the contrary, Diversified
Homophily BA and Diversified Homophily reach $\Delta I(t/T)=0$ faster,
sometimes even faster than Random Network and BA, indicating more equality.
The differences between the models are more pronounced under complex
contagion.

### Experiment 2: The Effects of Different Network Parameters on Information
Access Equality

We investigate the information access equality of network models with
different parameters. Specifically, we investigate Homophily BA under
different homophily $h$, preferential attachment strength $\alpha$, and
minority portion $m$. We also investigate Diversified Homophily BA under
different diversity $p_{d}$.

##### What is the influence of homophily on information access equality?

We first investigate the impact of homophily on information access equality
(see Fig. 3). We vary homophily $h$ in the range between 0.5 (random mixing)
and 1 (perfectly homophilic) under Homophily BA, setting $m=0.2$, $l=2$,
$\alpha=1$. At low minority seeding portion, we observe that for all processes
lower $h$ values have more information access equality. At high minority
seeding portion, under symmetric transmission rate (Fig. 3a, c), lower $h$
values also achieve more information access equality. However, under
asymmetric transmission rate and high minority seeding portion, (Fig. 3b, d),
we observe that the majority nodes gain an advantage around $t/T=40$, keep
that advantage until $t/T=70$ before equality is established.

Figure 3: Information Spreading Equality for Homophily BA with Different
Levels of Homophily $h$. We plot the spreading equality for Homophly BA
networks with low and high minority seeding portions. Each plot is a heatmap,
where the x-axis represents the relative time $t/T$, the y-axis represents the
graph’s homophily $h$, and the color represents $\Delta I(t/T)$. Recall that
$\Delta I(t/T)=0$ represents equality. At low minority seeding, the lower $h$
values, the more information access equality. This holds across all processes.
However at high minority seeding portion, the behavior depends on the symmetry
of the transmission. Under symmetric transmission rates (shown in (a) and (c))
with high minority seeding portion, lower $h$ values achieve more information
access equality. Under asymmetric transmission rates (shown in (b) and (d))
with high minority seeding portion, lower $h$ values increase to zero, then
become positive, before finally decreasing to zero. This makes it hard to
judge which $h$ value achieves more equality when one has an asymmetric
transmission and high minority seeding.

##### What is the influence of preferential attachment strength on information
access equality?

We experiment with $\alpha$ between 0 and 1.4, from no preferential attachment
($\alpha$ = 0) to preferential attachment ($\alpha>1$). Fig. 4 shows the
impact of $\alpha$ on information access equality . When $\alpha<1$, we find
limited differences in information equality, implying that for sublinear
preferential attachment, other factors such as homophily are more important
for information equality. However, for $\alpha\geq 1$, under simple contagion
(Fig. 4a, b) we observe a slight decrease and then an increase in equality.
$\alpha=1.4$ achieves the most information access equality. However, under
complex contagion (Fig. 4c, d), we find that $\alpha=1.4$ achieves the least
information access equality. This could be related to the different functions
hubs play under simple and complex contagion. Under simple contagion, hubs are
pathways, while under complex contagion, hubs are bottlenecks. The observation
suggests that the occurrence of hubs on information access equality seems to
be significant only under super-linear preferential attachment.

Figure 4: Information Spreading Equality for Homophily BA with Different
Preferential Attachment Strengths $\alpha$. We plot the spreading equality for
Homophly BA networks with low and high minority seeding portions. Each plot is
a heatmap, where the x-axis represents the relative time $t/T$, the y-axis
represents the graph’s preferential attachment strength $\alpha$, and the
color represents $\Delta I(t/T)$. Recall that $\Delta I(t/T)=0$ represents
equality. When $\alpha<1$, we observe few differences in information equality,
implying that for sublinear preferential attachment, other factors such as
homophily are more important for information access equality. However, for
$\alpha\geq 1$ (which is often the case in real-world networks), the type of
contagion is important. For example when $\alpha=1.4$, simple contagion
achieves information access equality quickly (shown in (a) and (b)). This is
not true under complex contagion (shown in (c) and (d)). This highlights the
importance of the type of contagion on the network.

##### What is the influence of minority portion on information access
equality?

For the minority portion $m$, we experiment in the range between 0.05 (almost
no minority nodes) and 0.5 (no population difference between majority and
minority nodes). We find that the influence of $m$ on information access
equality is strongly dependent on minority seeding portion, independent of
contagion type and transmission rate (see Fig. 5). At low minority seeding
portion, more equality is achieved with the lowest $m$ (0.05 in our
experiments). At high seeding proportion, the higher $m$ values achieve more
equality (0.5 under our experiment). This can be explained by the fact that
minority seeding portion (as an absolute value), has a different relative
effect among different minority populations. For example, minority seeding
portion of 0.3 is considered lower when $m=0.5$ compared to $m=0.1$.

Figure 5: Information Spreading Equality for Homophily BA with Different
Minority Portions $m$. We plot the spreading equality for Homophly BA networks
with low and high minority seeding portions. Each plot is a heatmap, where the
x-axis represents the relative time $t/T$, the y-axis represents the portion
of the nodes that are in the minority group $m$, and the color represents
$\Delta I(t/T)$. Recall that $\Delta I(t/T)=0$ represents equality. The
influence of $m$ on information access equality is strongly dependent on the
minority seeding portion and is independent of contagion type and transmission
rate. At low minority seeding portion, more equality is achieved with the
lowest $m$ (0.05 in our experiments).

##### What is the influence of diversity on information access equality?

We investigate the behavior of Diversified Homophily BA with different $p_{d}$
values: 0.01, 0.05, 0.1, 0.2, 0.4, 0.6 and 0.8. We set $m=0.2$, $h=0.8$,
$l_{d}=1$, and $\alpha=1$. We see that, in general, higher $p_{d}$ values
achieves higher information access equality, especially for low minority
seeding portion (Fig. 6), which matches our observation that higher $p_{d}$
has more degree equality and less homophily.

Figure 6: Information Spreading Equality for Diversified BA with Different
Diversification Probabilities $p_{d}$. We plot the spreading equality for
Diversified BA networks with low and high minority seeding portions. Each plot
is a heatmap, where the x-axis represents the relative time $t/T$, the y-axis
represents the diversification probability $p_{d}$, and the color represents
$\Delta I(t/T)$. Recall that $\Delta I(t/T)=0$ represents equality. In
general, we observe that the higher the $p_{d}$, the higher the information
access equality, especially for low minority seeding portion.

### Experiment 3: Information Access Equality in Real-world Networks

We experiment with three real-world networks: (1) the Github follower network,
(2) the DBLP collaboration network, and (3) the APS citation network [12]. The
Github and DBLP networks have gender as a grouping attribute. APS has research
field as the grouping attribute. We filter the networks by including only
labeled nodes and selecting the largest connected component. The basic
statistics of the three networks are in Table 7b. For the information access
simulation, we keep the parameter the same as in Experiments 1 and 2, but
change the seed number $s$ to 0.2% of the number of nodes in each network. For
APS, 0.2% of the nodes is only 2 nodes, which makes it difficult to enforce a
different portion of minority seeding, so we set $s=5$.

Table 7b lists the dyadicity and heterophilicity of the real networks. We note
that all networks have dyadicity greater than 1 and heterophilicity less than
1, indicating homophily in the network. Comparing across the networks, we
observe that APS has the highest dyadicity and the lowest heterophilicity,
followed by Github and DBLP.

Figure 7(a) shows the degree distributions of of the real networks. Basic
statistics for these networks are in (Fig. 7b), and social capital (degree)
equality measures are in (Fig. 7c) . We observe that all networks have heavy-
tailed degree distributions, and in terms of degree equality, Github, DBLP,
and APS are each from more to less equal.

Figure 7: Statistics of Real Networks. (a) Degree Distributions. We see that
all networks have heavy-tail degree distributions, but we note that the
minority group in APS has a much shorter tail. (b) Basic Statistics. The table
lists basic statistics, including node composition, average and maximum
degree, edge composition, dyadicity and heterophilicity scores for the three
networks. Github has the lowest proportion of minority nodes, while APS has
the highest proportion of minority nodes. DBLP has the highest average and
maximum degree. All the networks have dyadicity greater than 1 and
heterophilicity less than 1, indicating homophilic behavior. Among them, APS
has the highest dyadicity value and the lowest heterophilicity value.  (c)
Social Captial Equality Measures. The Earth Mover Distance of the three
networks are all positive, indicating that the degree distribution is
different between the groups. APS has the highest Earth Mover Distance. All
networks have power inequality less than 1, indicating that minority groups
have a lower average degree. For the moment glass ceiling, GitHub is slightly
higher than 1, but the other two networks both have a moment glass ceiling
lower than 1, indicating that there are fewer minority nodes as hubs.

Figure 8 shows information access equality on real networks. For Github, we
find that initially $\Delta I(t/T)<0$, indicating that nodes that are in the
minority have a greater advantage. This is because Github has the smallest
minority portion, and this is consistent with the observation for influence on
equality with different $m$ (see Fig. 5). For all different process settings,
we find that APS takes the longest to reach equality, which is consistent with
our previous observation that APS is less equal in degree and has higher
homophily level. DBLP is most equal in information access, which is consistent
with lower homophily. We find that the information access equality landscape
depends on different process settings. For example, under asymmetric
transmission rate, we notice that $\Delta I(t/T)$ becomes positive for Github
and DBLP under high seeding portion, which is not the case under symmetric
transmission rate. We also find that achieving equality is much harder under
complex contagion and asymmetric transmission rate. These variations are also
consistent with our earlier observation that information access is not only
related to the network but also to the process setting (see Fig. 8).

Figure 8: Information Spreading Equality of Real Networks. We plot the
spreading equality with low and high minority seeding portions. Each plot is a
heatmap, where the x-axis represents the relative time $t/T$, the y-axis
represents the different real-world networks, and the color represents $\Delta
I(t/T)$. Recall that $\Delta I(t/T)=0$ represents equality. For Github, we
find that initially $\Delta I(t/T)<0$, indicating that minority nodes have a
greater advantage. This is because Github has the smallest minority portion,
and this is consistent with the observation for influence on equality across
different $m$ values (see Fig. 5). Across all process settings, we find that
APS takes the longest to reach equality, which is consistent with our previous
observation that APS is less equal in degree and has a higher homophily level.
DBLP is the most equal in information access, which is consistent with lower
homophily. We find that the information access equality landscape depends on
different process settings. For example, under asymmetric transmission rate,
we notice that $\Delta I(t/T)$ becomes positive for Github and DBLP under high
seeding portion, which is not the case under symmetric transmission rate. We
also find that achieving equality is much harder under complex contagion and
asymmetric transmission rate.

### Experiment 4: Relationship Between Information Equality and Spreading
Efficiency

Research on fairness in AI has shown a phenomenon called the “price of
fairness” [52, 53], where researchers have found that fairness comes at a cost
to other performance measures of interest. For information spreading, the
relationship between equality and efficiency is a rather important one,
especially when the information is time-sensitive. Past work has found the
“price of fairness” under information maximization [27].

To measure spreading efficiency, we inspect the value of $I(t)$ in the
spreading process. Since comparing efficiency among various spreading
processes is affected by network statistics such as the size of the network,
the average degree, etc., we examine spreading efficiency for the networks
created for Experiment 1.

Figure 9 shows the information spreading efficiency of different network
models. Similar to equality, we plot the spreading process as a heatmap, where
the x-axis represents the actual time $t$, the y-axis represents the different
models, and the color represents the fraction of already infected nodes
$I(t)$. We notice that the driving force for spreading efficiency is the type
of contagion. In simple contagion (Fig. 9a, b), Homophily BA and BA are much
more efficient at spreading, while Diversified Homophily is the least
efficient. In complex contagion (Fig. 9c, d), we observe that Random Network
and Random Homophily are the most efficient, followed by BA and Homophily BA,
possibly due to the fact that hubs are not beneficial in complex contagion.
Diversified Homophily BA and its variant are still the least efficient in
spreading. Recall that Diversified Homophily BA and its variation achieve more
information access equality, indicating that inter-group edges may lead to a
trade-off between equality and efficiency in information spreading.

Figure 9: Information Spreading Efficiency of Different Generative Models of
Complex Networks. Similar to the information access equality, we also inspect
information access efficiency as a heatmap. Here the x-axis is the actual time
$t$, the y-axis is the different models, and the color represents the fraction
of nodes are in state I (i.e., infected by the information) at a given time.
We notice that the driving force of the spreading efficiency is the type of
contagion. For the case of simple contagion (shown in (a) and (b)), we observe
that Homophily BA and BA spread much faster, while Diversified Homophily is
the slowest. In the case of complex contagion (shown in (c) and (d)), we
observe that Random Network and Random Homophily are the most efficient,
followed by BA and Homophily BA. We conjecture that this is due to the fact
that hubs become bottlenecks in complex contagion. Diversified Homophily and
Diversified Homophily BA are much slower in spreading, indicating the trade-
off between equality and efficiency of information access.

By inspecting different network parameters, we observe that except for the
minority population $m$, all other parameters can affect the information
spreading efficiency. For homophily $h$, $h=1$ always has the lowest
efficiency, possibly due to the extreme case where only one edge connects the
two groups. We observe that for asymmetric transmission rate, a lower $h$ has
a slightly slower spreading speed due to more inter-group edges, again showing
the equality and efficiency trade-off of inter-group edges.

We observe a clear trend of the spreading efficiency and $\alpha$. Under
simple contagion, the higher the $\alpha$, the faster the spreading. In
contrast, under complex contagion, the higher the $\alpha$, the slower the
spreading speed. This is to be expected because the emergence of hubs promotes
simple contagion but hinders complex contagion. Recall that we see that higher
$\alpha$ has higher equality in simple contagion and lower equality in complex
contagion, this shows that hubs do not create a trade-off between efficiency
and equality.

We observe a slight decrease in spreading efficiency as $p_{d}$ varies under
simple contagion. Under complex contagion, higher $p_{d}$ significantly
decreases the spreading speed, again showing the trade-off between efficiency
and equality with increasing inter-group edges.

In short, we find that there is a trade-off between information access
equality and information spreading efficiency. This is mainly due to the
inter-group edges. However, similar to the observation that equality is
related to process setting, this trade-off is also dependent on process
setting.

##  5 Conclusion and Discussion

In this paper, we focused on information access equality in complex networks
when the population is divided into two mutually exclusive groups: majority
vs. minority. We measured the information access equality of various processes
on different network models with different parameters and on three real
networks. We find that information access equality depends not only on the
network structure, but also on the spreading process, different contagion
types (i.e., simple vs. complex), different transmission rates (symmetric vs.
asymmetric), and different minority proportions. Although it is very difficult
to draw a single conclusion about what type of networks can promote
information access equality, we find that, in general, more inter-group edges
can help achieve equality. However, we note one drawback to more inter-group
edges: they may reduce the efficiency of information spreading. Designing a
network with information access equality requires more knowledge about the
spreading process itself. Our findings can be used to guide recommendations
for mechanistic design of social networks that foster more equal information
access. For example, social networking platforms, such as Facebook and
LinkedIn, can recommend new connections the lead to more equal information
access for various spreading processes on their platforms. Deployment of our
findings in a real-world system is part of our future work.

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